GCSE Maths Fractions (Foundation): worked examples and questions with answers
By the end of this chapter you will be able to
- Read, write and draw fractions, and tell proper, improper and mixed apart
- Find equivalent fractions and write fractions in their simplest form
- Order and compare fractions
- Add, subtract, multiply and divide fractions and mixed numbers
- Find a fraction of an amount, and find the whole from a fraction
- Change between fractions, decimals and percentages
You should already know
- Times tables up to 12 × 12
- Factors, multiples, HCF and LCM (Chapter 2)
- Division with remainders
Where this leads
- Percentages, ratio and proportion, probability, algebraic fractions, and every area of the course
Edexcel GCSE (9–1) Mathematics 1MA1, Foundation tier. Content references N1, N2, N8, N10, N12, R3. (Check references against the latest published specification before publishing.)
- 3.1 Understanding fractions: Worksheet (PDF) · Answers (PDF) · lesson page
- 3.2 Equivalent fractions and simplifying: Worksheet (PDF) · Answers (PDF) · lesson page
- 3.3 Ordering and comparing fractions: Worksheet (PDF) · Answers (PDF) · lesson page
- 3.4 Mixed numbers and improper fractions: Worksheet (PDF) · Answers (PDF) · lesson page
- 3.5 Adding and subtracting fractions: Worksheet (PDF) · Answers (PDF) · lesson page
- 3.6 Multiplying fractions: Worksheet (PDF) · Answers (PDF) · lesson page
- 3.7 Dividing fractions: Worksheet (PDF) · Answers (PDF) · lesson page
- 3.8 Fractions of amounts: Worksheet (PDF) · Answers (PDF) · lesson page
- 3.9 Fractions, decimals and percentages: Worksheet (PDF) · Answers (PDF) · lesson page
- Check-up and exam-style questions: Worksheet (PDF) · Answers (PDF) · lesson page
Exercise P Do you remember?
Non-calculatorMethod. A factor of a number divides into it exactly, with no remainder. Factors come in pairs that multiply to make the number.
- Always start with 1 and the number itself. Every number can be divided by 1 and by itself, so these two are always factors.1 × 12 = 12
- Test 2. Check whether 12 is in the 2 times table. It is, so 2 is a factor and it comes with a partner.2 × 6 = 12
- Test 3. Check the 3 times table. 12 is in it, so 3 is a factor with a partner.3 × 4 = 12
- Test 4, 5, then stop. 4 is already in our list (as a partner of 3). 5 does not divide 12. Once the numbers we test meet the partners we have already found, there are no more factors to find.4 × 3 was found already; 12 ÷ 5 = 2 remainder 2, so 5 is not a factor
- Write the factors in order. List every number from the pairs, smallest first.1, 2, 3, 4, 6, 12
Check. Multiply each pair: 1 × 12, 2 × 6 and 3 × 4 all equal 12. Six numbers are listed because there are three pairs. ✓
Watch out. Do not forget 1 and the number itself, and do not list a factor twice.
Method. The highest common factor (HCF) is the biggest number that is a factor of both numbers.
- List the factors of 12. We need all the numbers that divide into 12 exactly.Factors of 12: 1, 2, 3, 4, 6, 12
- List the factors of 18. Do the same for the other number.Factors of 18: 1, 2, 3, 6, 9, 18
- Find the numbers that are in both lists. These are the common factors.Common factors: 1, 2, 3, 6
- Choose the biggest common factor. Highest means biggest.HCF = 6
Check. 12 ÷ 6 = 2 and 18 ÷ 6 = 3. The numbers 2 and 3 have no common factor except 1, so no bigger number could work. ✓
Watch out. HCF means the highest factor in both lists. Do not pick the first common factor (1).
Method. The lowest common multiple (LCM) is the smallest number that is a multiple of both numbers.
- List the multiples of 4. Multiples are the numbers in the times table: keep adding 4.Multiples of 4: 4, 8, 12, 16, 20, 24
- List the multiples of 6. Keep adding 6.Multiples of 6: 6, 12, 18, 24
- Find the numbers that are in both lists. These are common multiples.12 and 24 are in both lists
- Choose the smallest. Lowest means smallest.LCM = 12
Check. 12 ÷ 4 = 3 and 12 ÷ 6 = 2, so 12 is a multiple of both. No smaller number is in both lists. ✓
Watch out. LCM is the lowest number in both lists. Do not mix it up with HCF.
Method. Division is the inverse (opposite) of multiplication, so a division fact comes from a times-table fact.
- Work out 7 × 8 by doubling. Doubling is easy. 7 × 2 = 14, then double, then double again.7 × 2 = 14
7 × 4 = 28
7 × 8 = 56 - Check 7 × 8 using the 8 times table. Count in 8s seven times to see if we get the same answer.8, 16, 24, 32, 40, 48, 56
8 × 7 = 56, so 7 × 8 = 56 - Work out 54 ÷ 6. Ask: what number times 6 makes 54? 6 × 10 = 60, which is 6 too much, so go one less than 10.6 × 9 = 54
- Write the division fact. The division fact comes from the multiplication fact.54 ÷ 6 = 9
Check. Inverse check: 56 ÷ 8 = 7 (we get back to 7) and 9 × 6 = 54 (we get back to 54). ✓
Watch out. Learn the times tables up to 12 × 12. Every fraction topic uses them.
Method. Division with a remainder: find the biggest multiple of the divisor that fits into the number, then see what is left over.
- List multiples of 4 near 29. We need the biggest multiple of 4 that is not more than 29.4, 8, 12, 16, 20, 24, 28, 32
- Choose the biggest multiple that fits. 28 is less than 29. 32 is more than 29, so it is too big.4 × 7 = 28
- Work out the remainder. The remainder is what is left after taking away that multiple.29 − 28 = 1
- Write the answer. Say how many whole 4s, then the remainder.29 ÷ 4 = 7 remainder 1
Check. Multiply back and add the remainder: 7 × 4 + 1 = 28 + 1 = 29. ✓
Watch out. The remainder must always be smaller than the number you divide by.
Understanding fractions
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
A fraction describes part of a whole. The whole could be a shape, a quantity, or a group of objects. To be a fraction, the whole must be split into equal parts.
Key ideas
- In 38, the numerator (top) says how many parts we have. The denominator (bottom) says how many equal parts the whole is split into.
- A unit fraction has a numerator of 1, for example 15.
- A proper fraction is less than 1 (numerator smaller than denominator). An improper fraction is 1 or more (numerator bigger than or equal to the denominator), for example 94.
- A fraction is also a division: 35 means 3 ÷ 5.
Method. A fraction of a shape = (number of shaded parts) over (number of equal parts altogether).
- Check that the parts are equal. A fraction only makes sense when the whole is split into equal parts. Every small square here is the same size, so we can go on.All 12 squares are the same size.
- Count all the equal parts. This total goes on the bottom of the fraction. It is the denominator. The grid has 3 rows of 4 squares, so we can multiply instead of counting one by one.3 × 4 = 12 parts altogether
- Count the shaded parts. This goes on the top of the fraction. It is the numerator. Count row by row so that none is missed.Row 1: 2 shaded
Row 2: 1 shaded
Row 3: 2 shaded
2 + 1 + 2 = 5 shaded - Write the shaded parts over the total parts. Top = how many parts we have, bottom = how many equal parts the whole is split into.512
- Check whether it can be simplified. A fraction is in its simplest form if no number except 1 divides into both numbers.5 is a prime number and it does not divide into 12, so 512 is already in its simplest form.
Check. Count the unshaded squares: there are 7. Then 5 shaded + 7 unshaded = 12, the total, so every square has been counted exactly once. ✓
Watch out. Count all the parts for the denominator, not just the unshaded ones.
Method. To shade a fraction of a bar: the denominator tells you how many equal groups to make, the numerator tells you how many groups to shade.
- Read the fraction. The denominator is 3, so the bar must be split into 3 equal groups. The numerator is 2, so we shade 2 of those groups.23: make 3 equal groups, shade 2 of them
- Find how many parts are in each group. The bar has 6 equal parts and we need 3 equal groups, so share the 6 parts out equally.6 ÷ 3 = 2 parts in each group
- Work out how many parts to shade. We shade 2 groups, and each group has 2 parts.2 × 2 = 4 parts
- Shade the parts. It does not matter which parts we pick, as long as 4 of the 6 are shaded.Shade 4 of the 6 parts.
Check. 4 parts out of 6 is 46. Divide the top and the bottom by 2: 46 = 23. This is the fraction we were asked for. ✓
Watch out. Do not shade just 2 parts. The 2 on top counts groups of parts, and each group here is 2 parts big.
Method. The fraction of a group is (the number we want) over (the total number in the group).
- Find the total number of counters. The total goes on the bottom. It is every counter in the bag, of all colours.5 + 3 + 2 = 10 counters
- Find the number of blue counters. This is the part we want, so it goes on the top.3 blue counters
- Write the part over the total. Numerator = blue counters, denominator = all counters.310
- Check whether it can be simplified. See if any number except 1 divides into both 3 and 10.The factors of 3 are 1 and 3, and 3 does not divide 10, so 310 is already in its simplest form.
Check. Find the other fractions too: red 510, green 210. 510 + 310 + 210 = 1010 = 1 whole bag. ✓
Watch out. The denominator is the total of all the counters (10), not the number of the other colours (7).
Watch out
The parts must be equal. A shape cut into 4 unequal pieces does not show quarters, even if one piece is shaded.Exercise 3A Understanding fractions
Non-calculatorMethod. A fraction of a shape = (shaded parts) over (total equal parts).
- Check the parts are equal. A fraction only works when the whole is split into equal parts. The diagram is drawn with equal parts, so we can go on.The parts are all the same size.
- Count all the equal sections in the bar. The total number of equal parts goes on the bottom of the fraction. This is the denominator.Total parts = 8
- Count the shaded parts. The shaded parts go on the top of the fraction. This is the numerator.Shaded parts = 3
- Write the numerator over the denominator. Top = how many parts we are counting, bottom = how many equal parts there are altogether.38
Check. Count the unshaded parts: 5. Then 3 + 5 = 8, the total number of parts, so nothing has been missed. ✓
Watch out. Count all the parts for the denominator, not just the unshaded ones.
Method. A fraction of a shape = (shaded parts) over (total equal parts).
- Check the parts are equal. A fraction only works when the whole is split into equal parts. The diagram is drawn with equal parts, so we can go on.The parts are all the same size.
- Count all the equal sections in the bar. The total number of equal parts goes on the bottom of the fraction. This is the denominator.Total parts = 6
- Count the shaded parts. The shaded parts go on the top of the fraction. This is the numerator.Shaded parts = 5
- Write the numerator over the denominator. Top = how many parts we are counting, bottom = how many equal parts there are altogether.56
Check. Count the unshaded parts: 1. Then 5 + 1 = 6, the total number of parts, so nothing has been missed. ✓
Watch out. Count all the parts for the denominator, not just the unshaded ones.
Method. A fraction of a shape = (shaded parts) over (total equal parts).
- Check the parts are equal. A fraction only works when the whole is split into equal parts. The diagram is drawn with equal parts, so we can go on.The parts are all the same size.
- Count all the equal slices in the circle. The total number of equal parts goes on the bottom of the fraction. This is the denominator.Total parts = 4
- Count the shaded parts. The shaded parts go on the top of the fraction. This is the numerator.Shaded parts = 1
- Write the numerator over the denominator. Top = how many parts we are counting, bottom = how many equal parts there are altogether.14
Check. Count the unshaded parts: 3. Then 1 + 3 = 4, the total number of parts, so nothing has been missed. ✓
Watch out. Count all the parts for the denominator, not just the unshaded ones.
Method. A fraction of a shape = (shaded parts) over (total equal parts).
- Check the parts are equal. A fraction only works when the whole is split into equal parts. The diagram is drawn with equal parts, so we can go on.The parts are all the same size.
- Count all the equal slices in the circle. The total number of equal parts goes on the bottom of the fraction. This is the denominator.Total parts = 10
- Count the shaded parts. The shaded parts go on the top of the fraction. This is the numerator.Shaded parts = 7
- Write the numerator over the denominator. Top = how many parts we are counting, bottom = how many equal parts there are altogether.710
Check. Count the unshaded parts: 3. Then 7 + 3 = 10, the total number of parts, so nothing has been missed. ✓
Watch out. Count all the parts for the denominator, not just the unshaded ones.
Method. A fraction of a shape = (shaded parts) over (total equal parts).
- Check the parts are equal. A fraction only works when the whole is split into equal parts. The diagram is drawn with equal parts, so we can go on.The parts are all the same size.
- Count all the equal squares in the grid. The grid has 2 rows of 5 squares, so we can multiply. The total goes on the bottom of the fraction. This is the denominator.2 × 5 = 10 parts
- Count the shaded parts. The shaded parts go on the top of the fraction. This is the numerator.Shaded parts = 4
- Write the numerator over the denominator. Top = how many parts we are counting, bottom = how many equal parts there are altogether.410
Check. Count the unshaded parts: 6. Then 4 + 6 = 10, the total number of parts, so nothing has been missed. It can also be written as 25 in its simplest form. ✓
Watch out. Count all the parts for the denominator, not just the unshaded ones.
Method. A fraction of a shape = (shaded parts) over (total equal parts).
- Check the parts are equal. A fraction only works when the whole is split into equal parts. The diagram is drawn with equal parts, so we can go on.The parts are all the same size.
- Count all the equal squares in the grid. The grid has 3 rows of 3 squares, so we can multiply. The total goes on the bottom of the fraction. This is the denominator.3 × 3 = 9 parts
- Count the shaded parts. The shaded parts go on the top of the fraction. This is the numerator.Shaded parts = 5
- Write the numerator over the denominator. Top = how many parts we are counting, bottom = how many equal parts there are altogether.59
Check. Count the unshaded parts: 4. Then 5 + 4 = 9, the total number of parts, so nothing has been missed. ✓
Watch out. Count all the parts for the denominator, not just the unshaded ones.
Method. Shading 34 means splitting the bar into 4 equal groups and shading 3 of the groups.
- Read the fraction. The denominator 4 tells us how many equal groups to make. The numerator 3 tells us how many of those groups to shade.Make 4 equal groups, shade 3
- Find how many parts are in each group. The bar has 4 parts and we need 4 equal groups, so share the parts out equally.4 ÷ 4 = 1 part in each group
- Work out how many parts to shade. We shade 3 groups, and each group has 1 part.3 × 1 = 3 parts
- Shade the parts. Start from the left and shade that many parts.Shade 3 of the 4 parts
Check. 3 out of 4 parts is 34. ✓
Watch out. The number of parts shaded must match the numerator when the bar already has the right number of parts.
Method. Shading 23 means splitting the bar into 3 equal groups and shading 2 of the groups.
- Read the fraction. The denominator 3 tells us how many equal groups to make. The numerator 2 tells us how many of those groups to shade.Make 3 equal groups, shade 2
- Find how many parts are in each group. The bar has 6 parts and we need 3 equal groups, so share the parts out equally.6 ÷ 3 = 2 parts in each group
- Work out how many parts to shade. We shade 2 groups, and each group has 2 parts.2 × 2 = 4 parts
- Shade the parts. Start from the left and shade that many parts.Shade 4 of the 6 parts
Check. 4 out of 6 parts is 46, and 46 = 23 when we divide the top and bottom by 2. ✓
Watch out. The numerator 2 counts groups, not parts, when the bar is split into more parts than the denominator.
Method. Shading 58 means splitting the bar into 8 equal groups and shading 5 of the groups.
- Read the fraction. The denominator 8 tells us how many equal groups to make. The numerator 5 tells us how many of those groups to shade.Make 8 equal groups, shade 5
- Find how many parts are in each group. The bar has 8 parts and we need 8 equal groups, so share the parts out equally.8 ÷ 8 = 1 part in each group
- Work out how many parts to shade. We shade 5 groups, and each group has 1 part.5 × 1 = 5 parts
- Shade the parts. Start from the left and shade that many parts.Shade 5 of the 8 parts
Check. 5 out of 8 parts is 58. ✓
Watch out. The number of parts shaded must match the numerator when the bar already has the right number of parts.
Method. Shading 35 means splitting the bar into 5 equal groups and shading 3 of the groups.
- Read the fraction. The denominator 5 tells us how many equal groups to make. The numerator 3 tells us how many of those groups to shade.Make 5 equal groups, shade 3
- Find how many parts are in each group. The bar has 10 parts and we need 5 equal groups, so share the parts out equally.10 ÷ 5 = 2 parts in each group
- Work out how many parts to shade. We shade 3 groups, and each group has 2 parts.3 × 2 = 6 parts
- Shade the parts. Start from the left and shade that many parts.Shade 6 of the 10 parts
Check. 6 out of 10 parts is 610, and 610 = 35 when we divide the top and bottom by 2. ✓
Watch out. The numerator 3 counts groups, not parts, when the bar is split into more parts than the denominator.
Method. In a fraction, the number on top is the numerator and the number on the bottom is the denominator.
- Look at the fraction. A fraction has two numbers separated by a line.59
- Read the top number. The numerator counts how many parts we have.Numerator = 5
- Read the bottom number. The denominator tells us how many equal parts the whole is split into.Denominator = 9
- Say it in words. This shows we understand what the two numbers mean: the whole is cut into equal parts of one size, and we have some of them.5 out of 9 equal parts, which is 5 ninths
Check. Swap them to check: the numerator 5 can be bigger than the denominator 9 only for an improper fraction, and 59 is proper. ✓
Watch out. Remember: Denominator is Down. Do not swap the two names.
Method. In a fraction, the number on top is the numerator and the number on the bottom is the denominator.
- Look at the fraction. A fraction has two numbers separated by a line.712
- Read the top number. The numerator counts how many parts we have.Numerator = 7
- Read the bottom number. The denominator tells us how many equal parts the whole is split into.Denominator = 12
- Say it in words. This shows we understand what the two numbers mean: the whole is cut into equal parts of one size, and we have some of them.7 out of 12 equal parts, which is 7 twelfths
Check. Swap them to check: the numerator 7 can be bigger than the denominator 12 only for an improper fraction, and 712 is proper. ✓
Watch out. Remember: Denominator is Down. Do not swap the two names.
Method. In a fraction, the number on top is the numerator and the number on the bottom is the denominator.
- Look at the fraction. A fraction has two numbers separated by a line.1120
- Read the top number. The numerator counts how many parts we have.Numerator = 11
- Read the bottom number. The denominator tells us how many equal parts the whole is split into.Denominator = 20
- Say it in words. This shows we understand what the two numbers mean: the whole is cut into equal parts of one size, and we have some of them.11 out of 20 equal parts, which is 11 twentieths
Check. Swap them to check: the numerator 11 can be bigger than the denominator 20 only for an improper fraction, and 1120 is proper. ✓
Watch out. Remember: Denominator is Down. Do not swap the two names.
Method. In a fraction, the number on top is the numerator and the number on the bottom is the denominator.
- Look at the fraction. A fraction has two numbers separated by a line.13
- Read the top number. The numerator counts how many parts we have.Numerator = 1
- Read the bottom number. The denominator tells us how many equal parts the whole is split into.Denominator = 3
- Say it in words. This shows we understand what the two numbers mean: the whole is cut into equal parts of one size, and we have some of them.1 out of 3 equal parts, which is 1 third
Check. Swap them to check: the numerator 1 can be bigger than the denominator 3 only for an improper fraction, and 13 is proper. ✓
Watch out. Remember: Denominator is Down. Do not swap the two names.
Method. A unit fraction has a numerator of 1. It is one single piece of the whole.
- Recall the definition. Unit means one. A unit fraction has 1 on the top.Unit fraction: numerator = 1
- Check the numerator of each fraction. Only look at the top number.15: top is 1 ✓
25: top is 2 ✗
112: top is 1 ✓
34: top is 3 ✗
1100: top is 1 ✓ - Explain the ones that fail. 25 is two fifths and 34 is three quarters. They are made of more than one piece.Not unit fractions
- Write the unit fractions. List the ones with 1 on top.15, 112 and 1100
Check. A unit fraction is one piece, so 15 is one fifth, 112 is one twelfth and 1100 is one hundredth. The others are several pieces. ✓
Watch out. The bottom number does not matter for a unit fraction. Only the top number must be 1.
Method. Work out the size of one step first, then count the steps from 0.
- Count the equal steps from 0 to 1. The line from 0 to 1 is split into 5 equal steps. So each step is one fifth (15).Each step = 15
- Count the steps from 0 to A. The number of steps is the numerator of the fraction.1 step
- Write the fraction. 1 steps of size 15.15
Check. A is 4 steps before 1, and 15 + 45 = 55 = 1. ✓
Watch out. The numerator counts the steps from 0, not from the nearest whole number.
Method. Work out the size of one step first, then count the steps from 0.
- Count the equal steps from 0 to 1. The line from 0 to 1 is split into 5 equal steps. So each step is one fifth (15).Each step = 15
- Count the steps from 0 to B. The number of steps is the numerator of the fraction.3 steps
- Write the fraction. 3 steps of size 15.35
Check. B is 2 steps before 1, and 35 + 25 = 55 = 1. ✓
Watch out. The numerator counts the steps from 0, not from the nearest whole number.
Method. Work out the size of one step first, then count the steps from 0.
- Count the equal steps from 0 to 1. The line from 0 to 1 is split into 5 equal steps. So each step is one fifth (15).Each step = 15
- Count the steps from 0 to C. The number of steps is the numerator of the fraction.4 steps
- Write the fraction. 4 steps of size 15.45
Check. C is 1 step before 1, and 45 + 15 = 55 = 1. ✓
Watch out. The numerator counts the steps from 0, not from the nearest whole number.
Method. Work out the size of one step first, then count the steps from 0.
- Count the equal steps from 0 to 1. The line from 0 to 1 is split into 4 equal steps, so each step is one quarter.Each step = 14
- Count the steps from 0 to P. Keep counting along the line, past 1. Each step is still one quarter.3 steps
- Write the fraction. 3 steps of size 14.34
Check. There are 4 quarters in each whole, so 3 ÷ 4 = 0.75. P should be at 0.75 on the number line. ✓
Watch out. The numerator counts the steps from 0, not from the nearest whole number.
Method. Work out the size of one step first, then count the steps from 0.
- Count the equal steps from 0 to 1. The line from 0 to 1 is split into 4 equal steps, so each step is one quarter.Each step = 14
- Count the steps from 0 to Q. Keep counting along the line, past 1. Each step is still one quarter.5 steps
- Write the fraction. 5 steps of size 14.54 = 114
Check. There are 4 quarters in each whole, so 5 ÷ 4 = 1.25. Q should be at 1.25 on the number line. ✓
Watch out. The numerator counts the steps from 0, not from the nearest whole number.
Method. Work out the size of one step first, then count the steps from 0.
- Count the equal steps from 0 to 1. The line from 0 to 1 is split into 4 equal steps, so each step is one quarter.Each step = 14
- Count the steps from 0 to R. Keep counting along the line, past 1. Each step is still one quarter.7 steps
- Write the fraction. 7 steps of size 14.74 = 134
Check. There are 4 quarters in each whole, so 7 ÷ 4 = 1.75. R should be at 1.75 on the number line. ✓
Watch out. The numerator counts the steps from 0, not from the nearest whole number.
Method. A fraction of a group = (the number we want) over (the total number in the group).
- Find the part. The students who wear glasses are the part we are counting: 3 of them.Part = 3
- Find the whole (the total). All the students are the whole group: 7.Total = 7
- Write the part over the whole. Numerator = part, denominator = total.37
Check. The rest (4 students without glasses) is 47, and 37 + 47 = 77 = 1 whole. ✓
Watch out. The denominator is the whole group (7), not the number of the other kind (4).
Method. A fraction of a group = (the number we want) over (the total number in the group).
- Find the part. The red flowers are the part: 4 of them.Part = 4
- Find the whole (the total). All the flowers are the whole: 9.Total = 9
- Write the part over the whole. Numerator = part, denominator = total.49
Check. The rest (5 flowers that are not red) is 59, and 49 + 59 = 99 = 1 whole. ✓
Watch out. The denominator is all the flowers (9).
Method. The total is every sweet in the bag. Add the colours together first.
- Find the total number of sweets. The denominator counts all the sweets, so add both colours.2 yellow + 7 white = 9 sweets
- Find the part. We want the fraction that are yellow, so the part is the yellow sweets.Part = 2
- Write yellow over the total. Numerator = yellow, denominator = all sweets.29
Check. White sweets are 79, and 29 + 79 = 99 = 1 whole bag. ✓
Watch out. The denominator is the total (9). Do not use 7, the number of white sweets.
Method. A week is a fixed number of days. Compare the days in the question with the days in a week.
- Find the part. The question says 5 days.Part = 5 days
- Find the whole. A week has 7 days.Whole = 7 days
- Write the fraction. Numerator = days we count, denominator = days in the whole week.57
Check. The remaining 2 days make 27, and 57 + 27 = 77 = 1 whole week. ✓
Watch out. The whole is always the thing the part is taken from: here, a week.
Method. A fraction bar means divide: the first number is divided by the second, so it goes on top.
- Identify the number being divided and the number we divide by. In 3 ÷ 5 the first number, 3, is shared out. The second number, 5, is the number of shares.First number 3, second number 5
- Put the first number on top and the second on the bottom. The fraction bar means divide, so top ÷ bottom.3 ÷ 5 = 35
- Say what it means. If 3 pizzas are shared equally between 5 people, each person gets three fifths of a pizza.3 ÷ 5 = 35
Check. Divide: 3 ÷ 5 = 0.6 and the fraction 3 over 5 is also 0.6. ✓
Watch out. The first number goes on top. 3 ÷ 5 is not 5 over 3.
Method. A fraction bar means divide: the first number is divided by the second, so it goes on top.
- Identify the two numbers. In 7 ÷ 8 the first number is 7 and the second is 8.First number 7, second number 8
- Put the first number on top and the second on the bottom. The fraction bar means divide.7 ÷ 8 = 78
- Say what it means. 7 is less than 8, so the answer is a proper fraction (less than 1).78 is less than 1
Check. 7 ÷ 8 = 0.875 and 7 over 8 is 0.875 too. ✓
Watch out. Check which number is being divided. It goes on top.
Method. A fraction bar means divide: the top is divided by the bottom.
- Identify the numerator and the denominator. The top number is the one being divided.Top = 4, bottom = 7
- Write the top divided by the bottom. The fraction bar is a division sign.47 = 4 ÷ 7
- Say what it means. 4 is less than 7, so this is less than 1.4 ÷ 7 is a little over a half
Check. 4 ÷ 7 = 0.571… and the fraction 4 over 7 is the same number. ✓
Watch out. Read the line as “divided by”. The top number comes first.
Method. A fraction bar means divide: the top is divided by the bottom.
- Identify the numerator and the denominator. The top number is the one being divided.Top = 11, bottom = 3
- Write the top divided by the bottom. The fraction bar is a division sign.113 = 11 ÷ 3
- Say what it means. 11 is bigger than 3, so the fraction is improper: more than one whole.11 ÷ 3 = 3 remainder 2
Check. 11 ÷ 3 = 3 remainder 2, so 113 = 323. A little more than 3, which fits. ✓
Watch out. An improper fraction is just a division where the top is larger than the bottom.
Method. Proper fraction: top smaller than bottom. Improper fraction: top bigger than (or equal to) the bottom.
- Read the top and the bottom numbers. We compare the numerator with the denominator.Top = 5, bottom = 3
- Compare them. If the top is smaller than the bottom, the fraction is less than 1 (proper). If the top is the same as or bigger than the bottom, it is 1 or more (improper).5 is bigger than 3
- Decide. Use the rule.improper
Check. Divide: 5 ÷ 3 = 1.6667. This is more than 1, which matches. ✓
Watch out. A fraction with the top equal to the bottom, like 9/9, equals 1 and counts as improper.
Method. Proper fraction: top smaller than bottom. Improper fraction: top bigger than (or equal to) the bottom.
- Read the top and the bottom numbers. We compare the numerator with the denominator.Top = 3, bottom = 8
- Compare them. If the top is smaller than the bottom, the fraction is less than 1 (proper). If the top is the same as or bigger than the bottom, it is 1 or more (improper).3 is smaller than 8
- Decide. Use the rule.proper
Check. Divide: 3 ÷ 8 = 0.375. This is less than 1, which matches. ✓
Watch out. A fraction with the top equal to the bottom, like 9/9, equals 1 and counts as improper.
Method. Proper fraction: top smaller than bottom. Improper fraction: top bigger than (or equal to) the bottom.
- Read the top and the bottom numbers. We compare the numerator with the denominator.Top = 9, bottom = 9
- Compare them. If the top is smaller than the bottom, the fraction is less than 1 (proper). If the top is the same as or bigger than the bottom, it is 1 or more (improper).9 is equal to 9
- Decide. Use the rule.improper (it equals 1 whole)
Check. Divide: 9 ÷ 9 = 1. This is equal to 1, which matches. ✓
Watch out. A fraction with the top equal to the bottom, like 9/9, equals 1 and counts as improper.
Method. Proper fraction: top smaller than bottom. Improper fraction: top bigger than (or equal to) the bottom.
- Read the top and the bottom numbers. We compare the numerator with the denominator.Top = 12, bottom = 7
- Compare them. If the top is smaller than the bottom, the fraction is less than 1 (proper). If the top is the same as or bigger than the bottom, it is 1 or more (improper).12 is bigger than 7
- Decide. Use the rule.improper
Check. Divide: 12 ÷ 7 = 1.7143. This is more than 1, which matches. ✓
Watch out. A fraction with the top equal to the bottom, like 9/9, equals 1 and counts as improper.
Method. Add the slices eaten to find the fraction eaten. The slices left are the total slices take away the slices eaten.
- Add the slices eaten. Ali and Ben each ate some slices. Together they ate both amounts.3 + 2 = 5 slices
- Write the eaten fraction. The pizza has 8 equal slices altogether, so the denominator is 8.Eaten = 58
- Find the slices left. Take the eaten slices away from the total.8 − 5 = 3 slices
- Write the fraction left. Left slices over total slices.Left = 38
Check. 58 + 38 = 88 = 1 whole pizza. The two fractions add to 1. ✓
Watch out. Both fractions have the same denominator, 8, because they are both parts of the same pizza.
Method. Write the number we want over the total.
- Find the total number of counters. The denominator is all the counters, of every colour.5 + 3 + 2 = 10
- Find the red counters. These are the part we want.5 red
- Write red over the total. Numerator = red, denominator = all counters.510 (which simplifies to 12)
Check. Red 510 + blue 310 + green 210 = 1010 = 1. ✓
Watch out. The denominator is 10, the total, not 5 or 3.
Method. “Not green” means red and blue together.
- Find the counters that are not green. Red and blue counters are the ones we want.5 red + 3 blue = 8
- Find the total number of counters. The denominator is all the counters.5 + 3 + 2 = 10
- Write the fraction. Numerator = counters that are not green, denominator = all counters.810 (which simplifies to 45)
Check. A different route: green is 210, so not green is 1 − 210 = 1010 − 210 = 810. ✓
Watch out. Read the question carefully: “not green” is everything except the green ones.
Method. A fraction is only correct when the whole is split into equal parts. One quarter means one of four equal parts.
- Recall what a quarter is. A quarter is one of four parts that are all exactly the same size.14 = 1 of 4 equal parts
- Look at what the diagram shows. The rectangle is cut into four pieces, but the pieces are different sizes.4 pieces, different sizes
- Test the claim. If the shaded piece really were a quarter, it would be the same size as each of the other three. It is not.The shaded piece is not the same size as the others.
- Conclude and say how to fix it. The shaded part is not a quarter. To get quarters the rectangle must be cut into four pieces of equal size.Sam is wrong.
Check. Fold or measure: four equal quarters would all cover each other exactly. These pieces do not, so they are not quarters. ✓
Watch out. Always check the parts are equal before you name a fraction.
Method. Use the information about the numerator and the denominator to build the fraction one number at a time.
- Write the numerator. The question tells us this directly.Numerator = 3
- Work out the denominator. It is 4 more than the numerator, so add 4.3 + 4 = 7
- Write the fraction. Numerator on top, denominator on the bottom.37
Check. The denominator 7 is 4 more than the numerator 3 (7 − 3 = 4). ✓
Watch out. “More than” means add. Do not subtract.
Method. The fraction must have denominator 6, be proper (top smaller than bottom) and not be a unit fraction (top not 1).
- Use the denominator. The question says the denominator is 6.?6
- Use “proper”. A proper fraction has a top smaller than the bottom, so the numerator must be less than 6.Numerator can be 1, 2, 3, 4 or 5
- Use “not a unit fraction”. A unit fraction has numerator 1, so we must not use 1.Numerator can be 2, 3, 4 or 5
- Choose one numerator. Any of these works. We pick 2.26
Check. 26 has denominator 6, 2 is less than 6, and 2 is not 1. All three conditions are met. ✓ (There are four correct answers: 26, 36, 46, 56.)
Watch out. There is more than one correct answer, so any fraction that fits all three rules is right.
Method. The number of equal parts gives the denominator. The relationship between the numerator and the denominator gives the numerator.
- Find the denominator. The whole is drawn with 11 equal parts, so the denominator is 11.Denominator = 11
- Use the clue about the numerator. The denominator is 3 more than the numerator, so the numerator is 3 less than the denominator.11 − 3 = 8
- Check the fraction is proper. The question says it is proper, so the numerator must be smaller than the denominator.8 < 11
- Write the fraction. Numerator 8 over denominator 11.811
Check. The denominator 11 is 3 more than the numerator 8, and 811 is a proper fraction. ✓
Watch out. There is only one possible numerator because the denominator is fixed at 11.
Equivalent fractions and simplifying
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Equivalent fractions are different-looking fractions that have the same value. Look at the bars: they have the same amount shaded.
Key ideas
- To make an equivalent fraction, multiply or divide the numerator and the denominator by the same number.
- To simplify a fraction, divide the top and bottom by a common factor. A fraction is in its simplest form when no number (other than 1) divides into both.
- The quickest way: divide by the highest common factor (HCF) of the numerator and denominator.
Method. Equivalent fractions: whatever you multiply the bottom by, multiply the top by the same number.
- Read the question. We have a fraction with a missing top number. The new bottom number is given, so we must find out how the bottom changed.35 = ?20
- Find the multiplier for the bottom: 5 → 20. Ask: what do we multiply the old bottom by to get the new bottom? Divide the new bottom by the old bottom.20 ÷ 5 = 4, so the bottom was multiplied by 4.
- Multiply the top by 4 as well. To keep the fraction the same size we must multiply the top and the bottom by the same number.3 × 4 = 12
- Write the full equation. Now put the new top into the gap.35 = 1220
Check. Divide the top and bottom of 1220 by 4 and we get 35 back. Also 3 × 20 = 60 and 12 × 5 = 60, which are equal. ✓
Watch out. Never add the same number to the top and the bottom. You must multiply (or divide).
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.1830
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 3, 6
HCF = 6 - Divide the top by 6. Whatever we do to the top we must also do to the bottom.18 ÷ 6 = 3
- Divide the bottom by 6. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.30 ÷ 6 = 5
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.1830 = 35
Check. A different route: divide in small steps (÷ 2, then ÷ 3). 1830 → 915 → 35. We reach the same fraction. As decimals, 1830 = 0.6 and 35 = 0.6. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Two fractions are equivalent if they have the same value. Simplify both fully: if they reach the same fraction, they are equivalent.
- Say what we need to show. Equivalent fractions look different but have the same value. The clearest test is to reduce each one to its simplest form and compare.46 and 1015
- Simplify 46. The biggest number that divides into both 4 and 6 is 2 (factors of 4: 1, 2, 4. Factors of 6: 1, 2, 3, 6). Divide the top and the bottom by it.4 ÷ 2 = 2 and 6 ÷ 2 = 3
46 = 23 - Simplify 1015. The biggest number that divides into both 10 and 15 is 5 (factors of 10: 1, 2, 5, 10. Factors of 15: 1, 3, 5, 15). Divide the top and the bottom by it.10 ÷ 5 = 2 and 15 ÷ 5 = 3
1015 = 23 - Compare the simplest forms. Both fractions have reduced to the same fraction, so they have the same value.23 = 23
- Write the conclusion. Answer the question that was asked.Yes, 46 and 1015 are equivalent.
Check. A different test: cross-multiply. 4 × 15 = 60 and 6 × 10 = 60. The two answers are equal, so the fractions are equivalent. Also both equal 0.667 as a decimal. ✓
Watch out. Fractions with different numbers can still be the same fraction. Always simplify before you decide.
Method. If the highest common factor is hard to spot, divide the top and the bottom by any common factor you can see, then repeat until no number except 1 divides into both.
- Look for an easy common factor. Both 84 and 126 are even, so both are in the 2 times table. We can always divide two even numbers by 2.84 ÷ 2 = 42 and 126 ÷ 2 = 63
84126 = 4263 - Look at the new fraction and divide again. 42 and 63 are both in the 3 times table (the digits of 42 add to 6 and the digits of 63 add to 9, and both totals are in the 3 times table).42 ÷ 3 = 14 and 63 ÷ 3 = 21
4263 = 1421 - Divide again. 14 and 21 are both in the 7 times table.14 ÷ 7 = 2 and 21 ÷ 7 = 3
1421 = 23 - Stop when nothing is left to divide out. 2 and 3 have no common factor except 1, so the fraction cannot be simplified any more.84126 = 23
Check. The three divisions were by 2, 3 and 7, and 2 × 3 × 7 = 42. So 42 is the highest common factor. One division by 42 gives 84 ÷ 42 = 2 and 126 ÷ 42 = 3, the same answer 23. ✓
Watch out. Do not stop too soon. 4263 is simpler than 84126 but it is not yet in its simplest form.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. £2.40 is 240 pence. A fraction compares two amounts, so they must be in the same units. Change the pounds to pence.Part = 36p, whole = 240p
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).36240
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
HCF = 12 - Divide the top and the bottom by 12. Dividing both by the same number keeps the value the same but gives smaller numbers.36 ÷ 12 = 3 and 240 ÷ 12 = 20
36240 = 320
Check. 36 ÷ 240 = 0.15 and 320 = 0.15. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Watch out
Adding the same number to the top and bottom does not make an equivalent fraction. 12 → 34 = 34 changes the value. You must multiply or divide.Exercise 3B Equivalent fractions and simplifying
Non-calculatorMethod. Equivalent fractions: whatever you multiply the bottom by, multiply the top by the same number.
- Read the question. We have a fraction with a missing top number. The new bottom number is given, so we must find out how the bottom changed.12 = ?6
- Find the multiplier for the bottom: 2 → 6. Ask: what do we multiply the old bottom by to get the new bottom? Divide the new bottom by the old bottom.6 ÷ 2 = 3, so the bottom was multiplied by 3.
- Multiply the top by 3 as well. To keep the fraction the same size we must multiply the top and the bottom by the same number.1 × 3 = 3
- Write the full equation. Now put the new top into the gap.12 = 36
Check. Divide the top and bottom of 36 by 3 and we get 12 back. Also 1 × 6 = 6 and 3 × 2 = 6, which are equal. ✓
Watch out. Never add the same number to the top and the bottom. You must multiply (or divide).
Method. Equivalent fractions: whatever you multiply the bottom by, multiply the top by the same number.
- Read the question. We have a fraction with a missing top number. The new bottom number is given, so we must find out how the bottom changed.13 = ?12
- Find the multiplier for the bottom: 3 → 12. Ask: what do we multiply the old bottom by to get the new bottom? Divide the new bottom by the old bottom.12 ÷ 3 = 4, so the bottom was multiplied by 4.
- Multiply the top by 4 as well. To keep the fraction the same size we must multiply the top and the bottom by the same number.1 × 4 = 4
- Write the full equation. Now put the new top into the gap.13 = 412
Check. Divide the top and bottom of 412 by 4 and we get 13 back. Also 1 × 12 = 12 and 4 × 3 = 12, which are equal. ✓
Watch out. Never add the same number to the top and the bottom. You must multiply (or divide).
Method. Equivalent fractions: whatever you multiply the bottom by, multiply the top by the same number.
- Read the question. We have a fraction with a missing top number. The new bottom number is given, so we must find out how the bottom changed.25 = ?15
- Find the multiplier for the bottom: 5 → 15. Ask: what do we multiply the old bottom by to get the new bottom? Divide the new bottom by the old bottom.15 ÷ 5 = 3, so the bottom was multiplied by 3.
- Multiply the top by 3 as well. To keep the fraction the same size we must multiply the top and the bottom by the same number.2 × 3 = 6
- Write the full equation. Now put the new top into the gap.25 = 615
Check. Divide the top and bottom of 615 by 3 and we get 25 back. Also 2 × 15 = 30 and 6 × 5 = 30, which are equal. ✓
Watch out. Never add the same number to the top and the bottom. You must multiply (or divide).
Method. Equivalent fractions: whatever you multiply the bottom by, multiply the top by the same number.
- Read the question. We have a fraction with a missing top number. The new bottom number is given, so we must find out how the bottom changed.34 = ?20
- Find the multiplier for the bottom: 4 → 20. Ask: what do we multiply the old bottom by to get the new bottom? Divide the new bottom by the old bottom.20 ÷ 4 = 5, so the bottom was multiplied by 5.
- Multiply the top by 5 as well. To keep the fraction the same size we must multiply the top and the bottom by the same number.3 × 5 = 15
- Write the full equation. Now put the new top into the gap.34 = 1520
Check. Divide the top and bottom of 1520 by 5 and we get 34 back. Also 3 × 20 = 60 and 15 × 4 = 60, which are equal. ✓
Watch out. Never add the same number to the top and the bottom. You must multiply (or divide).
Method. Equivalent fractions: whatever you multiply the top by, multiply the bottom by the same number.
- Read the question. We have a fraction with a missing bottom number. The new top number is given, so we must find how the top changed.78 = 21?
- Find the multiplier for the top: 7 → 21. Divide the new top by the old top to see what the top was multiplied by.21 ÷ 7 = 3, so the top was multiplied by 3.
- Multiply the bottom by 3 as well. Top and bottom must be scaled by the same number or the fraction changes.8 × 3 = 24
- Write the full equation. Put the new bottom into the gap.78 = 2124
Check. As decimals, both are 0.875. Also 7 × 24 = 168 and 21 × 8 = 168, which are equal. ✓
Watch out. The same multiplier must be used on the top and the bottom.
Method. Equivalent fractions: whatever you multiply the top by, multiply the bottom by the same number.
- Read the question. We have a fraction with a missing bottom number. The new top number is given, so we must find how the top changed.49 = 20?
- Find the multiplier for the top: 4 → 20. Divide the new top by the old top to see what the top was multiplied by.20 ÷ 4 = 5, so the top was multiplied by 5.
- Multiply the bottom by 5 as well. Top and bottom must be scaled by the same number or the fraction changes.9 × 5 = 45
- Write the full equation. Put the new bottom into the gap.49 = 2045
Check. As decimals, both are 0.4444. Also 4 × 45 = 180 and 20 × 9 = 180, which are equal. ✓
Watch out. The same multiplier must be used on the top and the bottom.
Method. Equivalent fractions: whatever you multiply the top by, multiply the bottom by the same number.
- Read the question. We have a fraction with a missing bottom number. The new top number is given, so we must find how the top changed.56 = 20?
- Find the multiplier for the top: 5 → 20. Divide the new top by the old top to see what the top was multiplied by.20 ÷ 5 = 4, so the top was multiplied by 4.
- Multiply the bottom by 4 as well. Top and bottom must be scaled by the same number or the fraction changes.6 × 4 = 24
- Write the full equation. Put the new bottom into the gap.56 = 2024
Check. As decimals, both are 0.8333. Also 5 × 24 = 120 and 20 × 6 = 120, which are equal. ✓
Watch out. The same multiplier must be used on the top and the bottom.
Method. Equivalent fractions: whatever you multiply the top by, multiply the bottom by the same number.
- Read the question. We have a fraction with a missing bottom number. The new top number is given, so we must find how the top changed.27 = 12?
- Find the multiplier for the top: 2 → 12. Divide the new top by the old top to see what the top was multiplied by.12 ÷ 2 = 6, so the top was multiplied by 6.
- Multiply the bottom by 6 as well. Top and bottom must be scaled by the same number or the fraction changes.7 × 6 = 42
- Write the full equation. Put the new bottom into the gap.27 = 1242
Check. As decimals, both are 0.2857. Also 2 × 42 = 84 and 12 × 7 = 84, which are equal. ✓
Watch out. The same multiplier must be used on the top and the bottom.
Method. To make equivalent fractions, multiply the top and the bottom by the same number.
- Choose multipliers. Any whole number bigger than 1 works. We use 2, 3 and 4.Multipliers: 2, 3, 4
- Multiply the top and the bottom by 2. We must multiply both numbers by the same multiplier.14 = 28 (1 × 2 = 2, 4 × 2 = 8)
- Multiply the top and the bottom by 3. Same idea with a different multiplier.14 = 312 (1 × 3 = 3, 4 × 3 = 12)
- Multiply the top and the bottom by 4. And once more.14 = 416 (1 × 4 = 4, 4 × 4 = 16)
Check. Each fraction is 0.25 as a decimal. Also, simplifying 28, 312 and 416 (dividing by 2, 3 and 4) gives 14 each time. ✓
Watch out. Multiply, do not add. 1/4 is not equal to 3/6 (adding 2 to each number).
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.48
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 4: 1, 2, 4
Factors of 8: 1, 2, 4, 8 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 4
HCF = 4 - Divide the top by 4. Whatever we do to the top we must also do to the bottom.4 ÷ 4 = 1
- Divide the bottom by 4. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.8 ÷ 4 = 2
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.48 = 12
Check. A different route: divide in small steps (÷ 2, then ÷ 2). 48 → 24 → 12. We reach the same fraction. As decimals, 48 = 0.5 and 12 = 0.5. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.69
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 6: 1, 2, 3, 6
Factors of 9: 1, 3, 9 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 3
HCF = 3 - Divide the top by 3. Whatever we do to the top we must also do to the bottom.6 ÷ 3 = 2
- Divide the bottom by 3. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.9 ÷ 3 = 3
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.69 = 23
Check. A different route: divide in small steps (÷ 3). 69 → 23. We reach the same fraction. As decimals, 69 = 0.6667 and 23 = 0.6667. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.1015
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 10: 1, 2, 5, 10
Factors of 15: 1, 3, 5, 15 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 5
HCF = 5 - Divide the top by 5. Whatever we do to the top we must also do to the bottom.10 ÷ 5 = 2
- Divide the bottom by 5. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.15 ÷ 5 = 3
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.1015 = 23
Check. A different route: divide in small steps (÷ 5). 1015 → 23. We reach the same fraction. As decimals, 1015 = 0.6667 and 23 = 0.6667. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.1216
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 4
HCF = 4 - Divide the top by 4. Whatever we do to the top we must also do to the bottom.12 ÷ 4 = 3
- Divide the bottom by 4. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.16 ÷ 4 = 4
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.1216 = 34
Check. A different route: divide in small steps (÷ 2, then ÷ 2). 1216 → 68 → 34. We reach the same fraction. As decimals, 1216 = 0.75 and 34 = 0.75. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.1824
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 3, 6
HCF = 6 - Divide the top by 6. Whatever we do to the top we must also do to the bottom.18 ÷ 6 = 3
- Divide the bottom by 6. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.24 ÷ 6 = 4
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.1824 = 34
Check. A different route: divide in small steps (÷ 2, then ÷ 3). 1824 → 912 → 34. We reach the same fraction. As decimals, 1824 = 0.75 and 34 = 0.75. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.2050
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 50: 1, 2, 5, 10, 25, 50 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 5, 10
HCF = 10 - Divide the top by 10. Whatever we do to the top we must also do to the bottom.20 ÷ 10 = 2
- Divide the bottom by 10. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.50 ÷ 10 = 5
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.2050 = 25
Check. A different route: divide in small steps (÷ 2, then ÷ 5). 2050 → 1025 → 25. We reach the same fraction. As decimals, 2050 = 0.4 and 25 = 0.4. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.2842
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 28: 1, 2, 4, 7, 14, 28
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 7, 14
HCF = 14 - Divide the top by 14. Whatever we do to the top we must also do to the bottom.28 ÷ 14 = 2
- Divide the bottom by 14. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.42 ÷ 14 = 3
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.2842 = 23
Check. A different route: divide in small steps (÷ 2, then ÷ 7). 2842 → 1421 → 23. We reach the same fraction. As decimals, 2842 = 0.6667 and 23 = 0.6667. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.3660
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 3, 4, 6, 12
HCF = 12 - Divide the top by 12. Whatever we do to the top we must also do to the bottom.36 ÷ 12 = 3
- Divide the bottom by 12. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.60 ÷ 12 = 5
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.3660 = 35
Check. A different route: divide in small steps (÷ 2, then ÷ 2, then ÷ 3). 3660 → 1830 → 915 → 35. We reach the same fraction. As decimals, 3660 = 0.6 and 35 = 0.6. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.4575
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 75: 1, 3, 5, 15, 25, 75 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 3, 5, 15
HCF = 15 - Divide the top by 15. Whatever we do to the top we must also do to the bottom.45 ÷ 15 = 3
- Divide the bottom by 15. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.75 ÷ 15 = 5
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.4575 = 35
Check. A different route: divide in small steps (÷ 3, then ÷ 5). 4575 → 1525 → 35. We reach the same fraction. As decimals, 4575 = 0.6 and 35 = 0.6. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.6381
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 63: 1, 3, 7, 9, 21, 63
Factors of 81: 1, 3, 9, 27, 81 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 3, 9
HCF = 9 - Divide the top by 9. Whatever we do to the top we must also do to the bottom.63 ÷ 9 = 7
- Divide the bottom by 9. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.81 ÷ 9 = 9
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.6381 = 79
Check. A different route: divide in small steps (÷ 3, then ÷ 3). 6381 → 2127 → 79. We reach the same fraction. As decimals, 6381 = 0.7778 and 79 = 0.7778. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify each fraction. If it simplifies to the same fraction as the target, it is equivalent to it.
- Decide the test. To check whether a fraction is equal to 23, simplify it fully and compare with 23.Target: 23
- Simplify each fraction fully. Divide the top and bottom by their highest common factor.46 ÷ 2 = 23
610 ÷ 2 = 35
1015 ÷ 5 = 23
812 ÷ 4 = 23
1220 ÷ 4 = 35 - Compare each with the target. Keep those that simplify to 23.46: = 23 ✓
610: is 35, not 23 ✗
1015: = 23 ✓
812: = 23 ✓
1220: is 35, not 23 ✗ - Write the answer. List the fractions that are equal to the target.46, 1015, 812
Check. A different test, as decimals: 46 = 0.6667; 610 = 0.6; 1015 = 0.6667; 812 = 0.6667; 1220 = 0.6. Those equal to 0.6667 are the ones listed. ✓
Watch out. 610 is a close one: it simplifies to 35, not 23.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. There are 60 minutes in 1 hour, so compare 15 minutes with 60 minutes. Both are already in minutes.Part = 15, whole = 60
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1560
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 15: 1, 3, 5, 15
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
HCF = 15 - Divide the top and the bottom by 15. Dividing both by the same number keeps the value the same but gives smaller numbers.15 ÷ 15 = 1 and 60 ÷ 15 = 4
1560 = 14
Check. 15 ÷ 60 = 0.25 and 14 = 0.25. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. 1 m = 100 cm. Both amounts must be in the same units, so change the metre to centimetres.Part = 40, whole = 100
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).40100
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 20 - Divide the top and the bottom by 20. Dividing both by the same number keeps the value the same but gives smaller numbers.40 ÷ 20 = 2 and 100 ÷ 20 = 5
40100 = 25
Check. 40 ÷ 100 = 0.4 and 25 = 0.4. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. 1 kg = 1000 g. Both amounts must be in the same units, so change the kilogram to grams.Part = 250, whole = 1000
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).2501000
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 250: 1, 2, 5, 10, 25, 50, 125, 250
Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
HCF = 250 - Divide the top and the bottom by 250. Dividing both by the same number keeps the value the same but gives smaller numbers.250 ÷ 250 = 1 and 1000 ÷ 250 = 4
2501000 = 14
Check. 250 ÷ 1000 = 0.25 and 14 = 0.25. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. 1 minute = 60 seconds. Both must be in seconds.Part = 45, whole = 60
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).4560
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
HCF = 15 - Divide the top and the bottom by 15. Dividing both by the same number keeps the value the same but gives smaller numbers.45 ÷ 15 = 3 and 60 ÷ 15 = 4
4560 = 34
Check. 45 ÷ 60 = 0.75 and 34 = 0.75. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. 1 year = 12 months. Both must be in months.Part = 8, whole = 12
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).812
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
HCF = 4 - Divide the top and the bottom by 4. Dividing both by the same number keeps the value the same but gives smaller numbers.8 ÷ 4 = 2 and 12 ÷ 4 = 3
812 = 23
Check. 8 ÷ 12 = 0.6667 and 23 = 0.6667. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. £2.40 = 240p. Both must be in pence.Part = 36, whole = 240
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).36240
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
HCF = 12 - Divide the top and the bottom by 12. Dividing both by the same number keeps the value the same but gives smaller numbers.36 ÷ 12 = 3 and 240 ÷ 12 = 20
36240 = 320
Check. 36 ÷ 240 = 0.15 and 320 = 0.15. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. Both numbers are counts of pupils, so the units already match.Part = 18, whole = 30
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1830
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
HCF = 6 - Divide the top and the bottom by 6. Dividing both by the same number keeps the value the same but gives smaller numbers.18 ÷ 6 = 3 and 30 ÷ 6 = 5
1830 = 35
Check. 18 ÷ 30 = 0.6 and 35 = 0.6. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. Equivalent fractions come from multiplying or dividing the top and bottom by the same number. Adding the same number does not keep the value.
- Test what Tom did. He added 3 to the top and 3 to the bottom.37 → 3 + 37 + 3 = 610
- Compare the values as decimals. If two fractions are equivalent they have the same value.37 = 0.43 (to 2 d.p.) but 610 = 0.6
- Compare by simplifying. 610 simplifies to 35. Is that the same as 37? The denominators 5 and 7 are different and the tops are the same, so no.610 = 35, which is not 37
- Show a correct equivalent fraction. Multiply the top and bottom by 2.37 × 2 = 614, so 37 = 614
Check. Cross-multiply Tom's claim: 3 × 10 = 30 but 7 × 6 = 42. These are not equal, so 37 ≠ 610. ✓
Watch out. Adding to the top and the bottom changes the fraction. Multiply or divide.
Method. Equivalent to 2/3 means the top is 2k and the bottom is 3k for some multiplier k.
- Write the fraction using a multiplier k. To make an equivalent fraction we multiply the top 2 and the bottom 3 by the same number k.2k3k
- Use the sum 35. The numerator and denominator add up to 35.2k + 3k = 35
5k = 35 - Solve for k. Divide both sides by 5.k = 35 ÷ 5 = 7
- Find the numerator and the denominator. Multiply 2 and 3 by the multiplier 7.2 × 7 = 14 and 3 × 7 = 21
1421
Check. 14 + 21 = 35 and 1421 simplifies to 23 (divide both by 7). ✓
Watch out. Use the same multiplier on the top and the bottom. The sum of 35 tells us the total number of parts, 2k + 3k.
Method. A fraction is in simplest form when its top and bottom have no common factor except 1.
- Look for a common factor. 17 is a prime number, so its only factors are 1 and 17. The only way 17 and 51 share a factor other than 1 is if 17 divides into 51.Test: 51 ÷ 17
- Do the division. If it divides exactly, 17 is a common factor.17 × 3 = 51, so 51 ÷ 17 = 3 exactly
- Divide the top and the bottom by 17. Dividing both by the same number does not change the value.17 ÷ 17 = 1 and 51 ÷ 17 = 3
1751 = 13 - Write the conclusion. It simplifies, so it was not in simplest form.No, 1751 is not in simplest form. Its simplest form is 13.
Check. As decimals, 1751 = 0.333… and 13 = 0.333…. They are equal. ✓
Watch out. Big numbers can still simplify. Test whether the smaller number divides into the larger one.
Method. Divide the top and the bottom by their highest common factor (HCF).
- List what we know about the numbers. 84 = 2 × 42 and 126 = 3 × 42, so 42 divides into both.Both are multiples of 42
- Find the HCF of 84 and 126. Check that no bigger number works. The multiples of 42 near these are 84 (42 × 2) and 126 (42 × 3), and 2 and 3 have no common factor except 1, so 42 is the highest.HCF = 42
- Divide the top and the bottom by 42. The same number is divided into both.84 ÷ 42 = 2 and 126 ÷ 42 = 3
- Write the answer. Show that you found the HCF first, as the question asks.84126 = 23
Check. A different route: divide in steps. 84126 → (÷ 2) 4263 → (÷ 3) 1421 → (÷ 7) 23. Same answer. ✓
Watch out. If you divide by a common factor that is not the HCF, you must keep simplifying until no common factor is left.
Ordering and comparing fractions
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
A fraction wall helps us see which fractions are bigger. Each row is the same length, split into different equal parts.
Key ideas
- If the denominators are the same, the bigger numerator means the bigger fraction.
- If the numerators are the same (for unit fractions especially), the bigger denominator means the smaller fraction: 18 < 15.
- Otherwise, write the fractions with a common denominator. Use the LCM of the denominators.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 3 and 5. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 3: 3, 6, 9, 12, 15
Multiples of 5: 5, 10, 15
LCM = 15 - Rewrite the first fraction with denominator 15. Multiply the top and the bottom by the same number: 15 ÷ 3 = 5.23 = 1015 (× 5)
- Rewrite the second fraction with denominator 15. 15 ÷ 5 = 3, so multiply the top and the bottom by 3.35 = 915 (× 3)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.10 > 9
- Write the answer using the original fractions. The first fraction is bigger.23 > 35
Check. A different route, as decimals: 23 = 0.6667 and 35 = 0.6, so 0.6667 > 0.6. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. To put fractions in order, give them all the same denominator, then order the numerators.
- Find the LCM of the denominators 2, 5, 10, 4. This is the common denominator.LCM = 20
- Rewrite every fraction with denominator 20. Multiply the top and bottom of each by the same number.12 = 1020
25 = 820
710 = 1420
34 = 1520 - Put the numerators in order. The order is smallest first.8, 10, 14, 15
- Write the original fractions in that order. The question asks for the original fractions, not the converted ones.25 , 12 , 710 , 34
Check. As decimals: 25 = 0.4, 12 = 0.5, 710 = 0.7, 34 = 0.75.
Watch out. Always give your final answer using the original fractions, not the ones with the common denominator. And never order fractions by their denominators alone.
Method. When two fractions have the same numerator, compare the size of the pieces. A bigger denominator means smaller pieces.
- Look at the numerators and denominators. Both fractions have the same top number, 3. So we are comparing 3 pieces of one size with 3 pieces of another size.37 and 35: both have numerator 3
- Compare the size of one piece of each. If a whole is cut into 7 equal pieces, each piece is smaller than if the whole is cut into only 5 equal pieces.17 < 15 (sevenths are smaller than fifths)
- Take 3 pieces of each. Three small pieces make less than three bigger pieces.3 sevenths < 3 fifths
- Write the comparison. Put the correct sign between the original fractions.37 < 35, so 35 is bigger.
Check. Use a common denominator of 35: 37 = 1535 and 35 = 2135. 15 < 21, so 35 is bigger. ✓
Watch out. A bigger denominator does not mean a bigger fraction. When the numerators match, the fraction with the bigger denominator is smaller.
Method. Rewrite both fractions with a common denominator that is big enough to leave a gap, then choose a numerator between the two numerators.
- Say what “between” means. We need a fraction that is bigger than one of them and smaller than the other.12 < ? < 34
- Try the lowest common denominator, 4. 12 = 24, and 34 stays the same. But there is no whole number between 2 and 3, so there is no room for a fourth.24 and 34: no whole number between 2 and 3
- Use a bigger common denominator, 8. Doubling the top and bottom of both fractions makes more room (8 is a multiple of 4).12 = 48 and 34 = 68
- Pick a numerator between 4 and 6. The only whole number between 4 and 6 is 5.48 < 58 < 68
- Write the answer. Any correct fraction is allowed, for example this one.58
Check. As decimals: 12 = 0.5, 58 = 0.625, 34 = 0.75, and 0.5 < 0.625 < 0.75. ✓
Watch out. The lowest common denominator might leave no room. If so, use a bigger one such as double.
Exam tip
Always give your final answer using the original fractions, not the ones with the common denominator, unless the question says otherwise.Exercise 3C Ordering and comparing fractions
Non-calculatorMethod. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 3 and 5. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 3: 3, 6, 9, 12, 15
Multiples of 5: 5, 10, 15
LCM = 15 - Rewrite the first fraction with denominator 15. Multiply the top and the bottom by the same number: 15 ÷ 3 = 5.13 = 515 (× 5)
- Rewrite the second fraction with denominator 15. 15 ÷ 5 = 3, so multiply the top and the bottom by 3.15 = 315 (× 3)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.5 > 3
- Write the answer using the original fractions. The first fraction is bigger.13 > 15
Check. A different route, as decimals: 13 = 0.3333 and 15 = 0.2, so 0.3333 > 0.2. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. When the denominators are the same, the pieces are the same size, so the bigger numerator is the bigger fraction.
- Check the denominators. If the denominators match, the pieces are the same size and we can compare them straight away.Both denominators are 8.
- Compare the numerators. More pieces of the same size means a bigger fraction.3 and 5: 5 is bigger than 3
- Write the comparison. Put the correct sign between the two fractions.38 < 58
Check. As decimals: 0.375 < 0.625. ✓
Watch out. This shortcut only works when the denominators are the same.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 3 and 4. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Rewrite the first fraction with denominator 12. Multiply the top and the bottom by the same number: 12 ÷ 3 = 4.23 = 812 (× 4)
- Rewrite the second fraction with denominator 12. 12 ÷ 4 = 3, so multiply the top and the bottom by 3.34 = 912 (× 3)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.8 < 9
- Write the answer using the original fractions. The second fraction is bigger.23 < 34
Check. A different route, as decimals: 23 = 0.6667 and 34 = 0.75, so 0.6667 < 0.75. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 6 and 9. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 6: 6, 12, 18
Multiples of 9: 9, 18
LCM = 18 - Rewrite the first fraction with denominator 18. Multiply the top and the bottom by the same number: 18 ÷ 6 = 3.56 = 1518 (× 3)
- Rewrite the second fraction with denominator 18. 18 ÷ 9 = 2, so multiply the top and the bottom by 2.79 = 1418 (× 2)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.15 > 14
- Write the answer using the original fractions. The first fraction is bigger.56 > 79
Check. A different route, as decimals: 56 = 0.8333 and 79 = 0.7778, so 0.8333 > 0.7778. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 5 and 10. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 5: 5, 10
Multiples of 10: 10
LCM = 10 - Rewrite the first fraction with denominator 10. Multiply the top and the bottom by the same number: 10 ÷ 5 = 2.45 = 810 (× 2)
- Rewrite the second fraction with denominator 10. 10 ÷ 10 = 1, so multiply the top and the bottom by 1.710 = 710 (× 1)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.8 > 7
- Write the answer using the original fractions. The first fraction is bigger.45 > 710
Check. A different route, as decimals: 45 = 0.8 and 710 = 0.7, so 0.8 > 0.7. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 7 and 2. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 7: 7, 14
Multiples of 2: 2, 4, 6, 8, 10, 12, 14
LCM = 14 - Rewrite the first fraction with denominator 14. Multiply the top and the bottom by the same number: 14 ÷ 7 = 2.37 = 614 (× 2)
- Rewrite the second fraction with denominator 14. 14 ÷ 2 = 7, so multiply the top and the bottom by 7.12 = 714 (× 7)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.6 < 7
- Write the answer using the original fractions. The second fraction is bigger.37 < 12
Check. A different route, as decimals: 37 = 0.4286 and 12 = 0.5, so 0.4286 < 0.5. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 5 and 10. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 5: 5, 10
Multiples of 10: 10
LCM = 10 - Rewrite the first fraction with denominator 10. Multiply the top and the bottom by the same number: 10 ÷ 5 = 2.35 = 610 (× 2)
- Rewrite the second fraction with denominator 10. 10 ÷ 10 = 1, so multiply the top and the bottom by 1.710 = 710 (× 1)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.6 < 7
- Write the answer using the original fractions. The second fraction is bigger.35 < 710
Check. A different route, as decimals: 35 = 0.6 and 710 = 0.7, so 0.6 < 0.7. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 9 and 2. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 9: 9, 18
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18
LCM = 18 - Rewrite the first fraction with denominator 18. Multiply the top and the bottom by the same number: 18 ÷ 9 = 2.49 = 818 (× 2)
- Rewrite the second fraction with denominator 18. 18 ÷ 2 = 9, so multiply the top and the bottom by 9.12 = 918 (× 9)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.8 < 9
- Write the answer using the original fractions. The second fraction is bigger.49 < 12
Check. A different route, as decimals: 49 = 0.4444 and 12 = 0.5, so 0.4444 < 0.5. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 8 and 3. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 8: 8, 16, 24
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24
LCM = 24 - Rewrite the first fraction with denominator 24. Multiply the top and the bottom by the same number: 24 ÷ 8 = 3.58 = 1524 (× 3)
- Rewrite the second fraction with denominator 24. 24 ÷ 3 = 8, so multiply the top and the bottom by 8.23 = 1624 (× 8)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.15 < 16
- Write the answer using the original fractions. The second fraction is bigger.58 < 23
Check. A different route, as decimals: 58 = 0.625 and 23 = 0.6667, so 0.625 < 0.6667. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 12 and 5. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 12: 12, 24, 36, 48, 60
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60
LCM = 60 - Rewrite the first fraction with denominator 60. Multiply the top and the bottom by the same number: 60 ÷ 12 = 5.712 = 3560 (× 5)
- Rewrite the second fraction with denominator 60. 60 ÷ 5 = 12, so multiply the top and the bottom by 12.35 = 3660 (× 12)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.35 < 36
- Write the answer using the original fractions. The second fraction is bigger.712 < 35
Check. A different route, as decimals: 712 = 0.5833 and 35 = 0.6, so 0.5833 < 0.6. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. To put fractions in order, give them all the same denominator, then order the numerators.
- Find the LCM of the denominators 2, 8, 4, 8. This is the common denominator.LCM = 8
- Rewrite every fraction with denominator 8. Multiply the top and bottom of each by the same number.12 = 48
38 = 38
14 = 28
58 = 58 - Put the numerators in order. The order is smallest first.2, 3, 4, 5
- Write the original fractions in that order. The question asks for the original fractions, not the converted ones.14 , 38 , 12 , 58
Check. As decimals: 14 = 0.25, 38 = 0.375, 12 = 0.5, 58 = 0.625.
Watch out. Always give your final answer using the original fractions, not the ones with the common denominator. And never order fractions by their denominators alone.
Method. To put fractions in order, give them all the same denominator, then order the numerators.
- Find the LCM of the denominators 3, 6, 2, 4. This is the common denominator.LCM = 12
- Rewrite every fraction with denominator 12. Multiply the top and bottom of each by the same number.23 = 812
56 = 1012
12 = 612
34 = 912 - Put the numerators in order. The order is smallest first.6, 8, 9, 10
- Write the original fractions in that order. The question asks for the original fractions, not the converted ones.12 , 23 , 34 , 56
Check. As decimals: 12 = 0.5, 23 = 0.6667, 34 = 0.75, 56 = 0.8333.
Watch out. Always give your final answer using the original fractions, not the ones with the common denominator. And never order fractions by their denominators alone.
Method. To put fractions in order, give them all the same denominator, then order the numerators.
- Find the LCM of the denominators 10, 5, 20, 2. This is the common denominator.LCM = 20
- Rewrite every fraction with denominator 20. Multiply the top and bottom of each by the same number.710 = 1420
35 = 1220
1320 = 1320
12 = 1020 - Put the numerators in order. The order is smallest first.10, 12, 13, 14
- Write the original fractions in that order. The question asks for the original fractions, not the converted ones.12 , 35 , 1320 , 710
Check. As decimals: 12 = 0.5, 35 = 0.6, 1320 = 0.65, 710 = 0.7.
Watch out. Always give your final answer using the original fractions, not the ones with the common denominator. And never order fractions by their denominators alone.
Method. To put fractions in order, give them all the same denominator, then order the numerators.
- Find the LCM of the denominators 5, 8, 4, 10. This is the common denominator.LCM = 40
- Rewrite every fraction with denominator 40. Multiply the top and bottom of each by the same number.45 = 3240
78 = 3540
34 = 3040
910 = 3640 - Put the numerators in order. The order is smallest first.30, 32, 35, 36
- Write the original fractions in that order. The question asks for the original fractions, not the converted ones.34 , 45 , 78 , 910
Check. As decimals: 34 = 0.75, 45 = 0.8, 78 = 0.875, 910 = 0.9.
Watch out. Always give your final answer using the original fractions, not the ones with the common denominator. And never order fractions by their denominators alone.
Method. Write both fractions with the same denominator, leaving room, then pick a numerator in between.
- Say what “between” means. We need a fraction that is bigger than one of them and smaller than the other.12 < ? < 34
- Try the lowest common denominator, 4. 12 = 24 and 34 stays the same. But there is no whole number between 2 and 3, so there is no room for another quarter.24 and 34: nothing in between
- Use a bigger common denominator, 8. A multiple of 4 gives more room. Double the top and the bottom of both fractions.12 = 48 and 34 = 68
- Choose a numerator between 4 and 6. The only whole number between 4 and 6 is 5.48 < 58 < 68
Check. 12 = 0.5, 58 = 0.625, 34 = 0.75, and 0.5 < 0.625 < 0.75. ✓
Watch out. If the lowest common denominator leaves no gap, use a bigger one. Other correct answers are possible, such as 3/5 or 2/3.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Decide what to compare. Who ran further is whoever has the bigger fraction of a kilometre. The distances are fractions, so compare them.Sam 34 km, Zara 710 km
- Find the lowest common multiple (LCM) of 4 and 10. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 4: 4, 8, 12, 16, 20
Multiples of 10: 10, 20
LCM = 20 - Rewrite the first fraction with denominator 20. Multiply the top and the bottom by the same number: 20 ÷ 4 = 5.34 = 1520 (× 5)
- Rewrite the second fraction with denominator 20. 20 ÷ 10 = 2, so multiply the top and the bottom by 2.710 = 1420 (× 2)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.15 > 14
- Write the answer using the original fractions. The first fraction is bigger.34 > 710
- Answer the question in words. Sam has the bigger fraction, so Sam ran further.Sam ran further.
Check. A different route, as decimals: 34 = 0.75 and 710 = 0.7, so 0.75 > 0.7. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. A bigger denominator means the whole is cut into more pieces, so each piece is smaller. Test Jo's claim with a common denominator.
- Say what Jo claims. Jo says 1/8 is bigger because 8 is bigger than 5. That treats the denominator like an ordinary number.Jo: 18 > 15
- Write both fractions with the same denominator. The LCM of 8 and 5 is 40. Multiply the top and the bottom of each fraction.18 = 540 (× 5)
15 = 840 (× 8) - Compare the numerators. The denominators match, so the bigger numerator is the bigger fraction.5 < 8, so 18 < 15
- Explain in words. If a cake is cut into 8 equal slices, each slice is smaller than if it is cut into only 5 slices.Jo is wrong: one eighth is smaller than one fifth.
Check. As decimals: 18 = 0.125 and 15 = 0.2. 0.125 < 0.2. ✓
Watch out. For unit fractions, the bigger the denominator, the smaller the fraction.
Method. To put fractions in order, give them all the same denominator, then order the numerators.
- Say what we are comparing. Each person ate a fraction of a pizza. The one who ate the biggest fraction ate the most and the one with the smallest fraction ate the least.Amy 23, Ben 58, Cal 712
- Find the LCM of the denominators 3, 8, 12. This is the common denominator.LCM = 24
- Rewrite every fraction with denominator 24. Multiply the top and bottom of each by the same number.23 = 1624
58 = 1524
712 = 1424 - Put the numerators in order. The order is smallest first.14, 15, 16
- Write the original fractions in that order. The question asks for the original fractions, not the converted ones.712 , 58 , 23
- Answer the question. The order above is smallest first (Cal, Ben, Amy), so Cal ate the least and Amy ate the most.Amy ate the most. Cal ate the least.
Check. As decimals: 712 = 0.5833, 58 = 0.625, 23 = 0.6667.
Watch out. Always give your final answer using the original fractions, not the ones with the common denominator. And never order fractions by their denominators alone.
Method. Rewrite the two fractions with denominator 12, then choose a numerator between them.
- Use the denominator 12. The question asks for a fraction with denominator 12, so rewrite the two given fractions in twelfths.12 = 612 (× 6)
23 = 812 (× 4) - Compare the numerators. We want a numerator between 6 and 8.6 < ? < 8
- Choose the numerator. The only whole number between 6 and 8 is 7.612 < 712 < 812
- Check the fraction is in twelfths. The question asks for denominator 12, and 12 is the bottom number.712
Check. 12 = 0.5, 712 = 0.583…, 23 = 0.667…, and 0.5 < 0.583 < 0.667. ✓
Watch out. Do not simplify the answer: the question asks for denominator 12.
Method. Use a common denominator with plenty of room, then choose numerators in between.
- Say what “between” means. We need three fractions that are bigger than one third and smaller than one half.13 < ? < 12
- Find a common denominator with room. The LCM of 3 and 2 is 6, but that leaves no gap (2/6 and 3/6). Use a bigger multiple of 6, such as 24.13 = 824 (× 8)
12 = 1224 (× 12) - Choose three numerators between 8 and 12. The whole numbers between 8 and 12 are 9, 10 and 11.924, 1024, 1124
- Simplify where possible. Divide the top and the bottom by the same number.924 = 38
1024 = 512
1124 cannot be simplified
Check. As decimals: 13 = 0.333, 38 = 0.375, 512 = 0.417, 1124 = 0.458, 12 = 0.5. All three are between 0.333 and 0.5. ✓
Watch out. Choose a common denominator that gives enough whole numbers between the two numerators.
Mixed numbers and improper fractions
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
A mixed number has a whole-number part and a fraction part. An improper fraction shows the same amount as a single fraction. For example, 7 quarters is 74 or 134.
Key ideas
- Improper → mixed: divide the numerator by the denominator. The answer is the whole number, the remainder is the new numerator.
- Mixed → improper: whole number × denominator, then add the numerator. Keep the same denominator.
Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Say what the fraction means. 175 means 17 pieces, each one 15 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.17 pieces of size 15
- Work out how many pieces make one whole. The bottom number tells us this: 5 pieces make 1 whole. So we want to know how many groups of 5 we can make from 17 pieces.5 pieces = 1 whole
- Divide 17 by 5 to find the whole number. Count how many whole groups of 5 fit into 17. 5 × 3 = 15, but 5 × 4 = 20 is too big.17 ÷ 5 = 3 remainder 2
- Find the pieces left over. After 3 whole ones we have used 15 pieces. The rest are the remainder.17 − 15 = 2
- Write the mixed number. The whole number is 3. The 2 leftover pieces of size 15 make the fraction 25. The bottom number stays 5 because the pieces are the same size.175 = 325
Check. Convert back: 3 × 5 + 2 = 15 + 2 = 17. As decimals, 17 ÷ 5 = 3.4 and 325 = 3.4. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.
- Say what the mixed number means. 437 means 4 whole ones and 3 extra pieces of size 17.4 wholes + 37
- Change the whole ones into pieces. Each whole one is 7 pieces of size 17, so 4 wholes is 4 groups of 7 pieces.4 × 7 = 28 pieces
- Add the extra pieces. Now add the 3 pieces from the fraction part.28 + 3 = 31 pieces
- Write the total over the same denominator. Every piece is still of size 17, so the denominator does not change.437 = 317
Check. Divide back: 31 ÷ 7 = 4 remainder 3, which is 437. ✓
Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.
Method. To compare numbers in different forms, write them in the same form first.
- Choose the form to use. One number is an improper fraction and one is a mixed number. We will change the mixed number into an improper fraction.114 and 212
- Change 212 to an improper fraction. Whole number × denominator, plus the numerator, over the same denominator.2 × 2 + 1 = 5, so 212 = 52
- Give both fractions the same denominator. The denominators are 4 and 2. The LCM is 4, so multiply the top and the bottom of the second fraction by 2.52 = 104
- Compare the numerators. The denominators match, so the bigger numerator is the bigger fraction.11 > 10, so 114 > 104
- Write the answer using the original numbers. Answer the question that was asked.114 is bigger than 212.
Check. A different route: change 114 to a mixed number. 11 ÷ 4 = 2 remainder 3, so 114 = 234. Both have 2 wholes, and 34 > 12, so 234 > 212. ✓
Watch out. Always write both numbers in the same form before you compare them.
Method. Count how many quarters there are in the total. An improper fraction with denominator 4 shows the number of quarters directly.
- Decide what to find. Each glass is one quarter of a litre. So the number of glasses is the number of quarters in the juice.Number of glasses = number of quarters in 234
- Change the mixed number to an improper fraction. An improper fraction counts all the pieces together. Whole number × denominator, plus the numerator.2 × 4 + 3 = 11, so 234 = 114
- Read the numerator. 114 means 11 pieces, each one quarter of a litre.11 quarters
- Answer the question in words. One glass for each quarter-litre.11 glasses
Check. 11 glasses × 14 litre = 114 litres = 234 litres, the amount we started with. ✓
Watch out. Write the answer as a whole number of glasses, not as a fraction.
Exercise 3D Mixed numbers and improper fractions
Non-calculatorMethod. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Say what the fraction means. 72 means 7 pieces, each one 12 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.7 pieces of size 12
- Work out how many pieces make one whole. The bottom number tells us this: 2 pieces make 1 whole. So we want to know how many groups of 2 we can make from 7 pieces.2 pieces = 1 whole
- Divide 7 by 2 to find the whole number. Count how many whole groups of 2 fit into 7. 2 × 3 = 6, but 2 × 4 = 8 is too big.7 ÷ 2 = 3 remainder 1
- Find the pieces left over. After 3 whole ones we have used 6 pieces. The rest are the remainder.7 − 6 = 1
- Write the mixed number. The whole number is 3. The 1 leftover piece of size 12 make the fraction 12. The bottom number stays 2 because the pieces are the same size.72 = 312
Check. Convert back: 3 × 2 + 1 = 6 + 1 = 7. As decimals, 7 ÷ 2 = 3.5 and 312 = 3.5. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Say what the fraction means. 114 means 11 pieces, each one 14 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.11 pieces of size 14
- Work out how many pieces make one whole. The bottom number tells us this: 4 pieces make 1 whole. So we want to know how many groups of 4 we can make from 11 pieces.4 pieces = 1 whole
- Divide 11 by 4 to find the whole number. Count how many whole groups of 4 fit into 11. 4 × 2 = 8, but 4 × 3 = 12 is too big.11 ÷ 4 = 2 remainder 3
- Find the pieces left over. After 2 whole ones we have used 8 pieces. The rest are the remainder.11 − 8 = 3
- Write the mixed number. The whole number is 2. The 3 leftover pieces of size 14 make the fraction 34. The bottom number stays 4 because the pieces are the same size.114 = 234
Check. Convert back: 2 × 4 + 3 = 8 + 3 = 11. As decimals, 11 ÷ 4 = 2.75 and 234 = 2.75. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Say what the fraction means. 133 means 13 pieces, each one 13 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.13 pieces of size 13
- Work out how many pieces make one whole. The bottom number tells us this: 3 pieces make 1 whole. So we want to know how many groups of 3 we can make from 13 pieces.3 pieces = 1 whole
- Divide 13 by 3 to find the whole number. Count how many whole groups of 3 fit into 13. 3 × 4 = 12, but 3 × 5 = 15 is too big.13 ÷ 3 = 4 remainder 1
- Find the pieces left over. After 4 whole ones we have used 12 pieces. The rest are the remainder.13 − 12 = 1
- Write the mixed number. The whole number is 4. The 1 leftover piece of size 13 make the fraction 13. The bottom number stays 3 because the pieces are the same size.133 = 413
Check. Convert back: 4 × 3 + 1 = 12 + 1 = 13. As decimals, 13 ÷ 3 = 4.3333 and 413 = 4.3333. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Say what the fraction means. 195 means 19 pieces, each one 15 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.19 pieces of size 15
- Work out how many pieces make one whole. The bottom number tells us this: 5 pieces make 1 whole. So we want to know how many groups of 5 we can make from 19 pieces.5 pieces = 1 whole
- Divide 19 by 5 to find the whole number. Count how many whole groups of 5 fit into 19. 5 × 3 = 15, but 5 × 4 = 20 is too big.19 ÷ 5 = 3 remainder 4
- Find the pieces left over. After 3 whole ones we have used 15 pieces. The rest are the remainder.19 − 15 = 4
- Write the mixed number. The whole number is 3. The 4 leftover pieces of size 15 make the fraction 45. The bottom number stays 5 because the pieces are the same size.195 = 345
Check. Convert back: 3 × 5 + 4 = 15 + 4 = 19. As decimals, 19 ÷ 5 = 3.8 and 345 = 3.8. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Say what the fraction means. 236 means 23 pieces, each one 16 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.23 pieces of size 16
- Work out how many pieces make one whole. The bottom number tells us this: 6 pieces make 1 whole. So we want to know how many groups of 6 we can make from 23 pieces.6 pieces = 1 whole
- Divide 23 by 6 to find the whole number. Count how many whole groups of 6 fit into 23. 6 × 3 = 18, but 6 × 4 = 24 is too big.23 ÷ 6 = 3 remainder 5
- Find the pieces left over. After 3 whole ones we have used 18 pieces. The rest are the remainder.23 − 18 = 5
- Write the mixed number. The whole number is 3. The 5 leftover pieces of size 16 make the fraction 56. The bottom number stays 6 because the pieces are the same size.236 = 356
Check. Convert back: 3 × 6 + 5 = 18 + 5 = 23. As decimals, 23 ÷ 6 = 3.8333 and 356 = 3.8333. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Say what the fraction means. 298 means 29 pieces, each one 18 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.29 pieces of size 18
- Work out how many pieces make one whole. The bottom number tells us this: 8 pieces make 1 whole. So we want to know how many groups of 8 we can make from 29 pieces.8 pieces = 1 whole
- Divide 29 by 8 to find the whole number. Count how many whole groups of 8 fit into 29. 8 × 3 = 24, but 8 × 4 = 32 is too big.29 ÷ 8 = 3 remainder 5
- Find the pieces left over. After 3 whole ones we have used 24 pieces. The rest are the remainder.29 − 24 = 5
- Write the mixed number. The whole number is 3. The 5 leftover pieces of size 18 make the fraction 58. The bottom number stays 8 because the pieces are the same size.298 = 358
Check. Convert back: 3 × 8 + 5 = 24 + 5 = 29. As decimals, 29 ÷ 8 = 3.625 and 358 = 3.625. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Say what the fraction means. 419 means 41 pieces, each one 19 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.41 pieces of size 19
- Work out how many pieces make one whole. The bottom number tells us this: 9 pieces make 1 whole. So we want to know how many groups of 9 we can make from 41 pieces.9 pieces = 1 whole
- Divide 41 by 9 to find the whole number. Count how many whole groups of 9 fit into 41. 9 × 4 = 36, but 9 × 5 = 45 is too big.41 ÷ 9 = 4 remainder 5
- Find the pieces left over. After 4 whole ones we have used 36 pieces. The rest are the remainder.41 − 36 = 5
- Write the mixed number. The whole number is 4. The 5 leftover pieces of size 19 make the fraction 59. The bottom number stays 9 because the pieces are the same size.419 = 459
Check. Convert back: 4 × 9 + 5 = 36 + 5 = 41. As decimals, 41 ÷ 9 = 4.5556 and 459 = 4.5556. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Say what the fraction means. 507 means 50 pieces, each one 17 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.50 pieces of size 17
- Work out how many pieces make one whole. The bottom number tells us this: 7 pieces make 1 whole. So we want to know how many groups of 7 we can make from 50 pieces.7 pieces = 1 whole
- Divide 50 by 7 to find the whole number. Count how many whole groups of 7 fit into 50. 7 × 7 = 49, but 7 × 8 = 56 is too big.50 ÷ 7 = 7 remainder 1
- Find the pieces left over. After 7 whole ones we have used 49 pieces. The rest are the remainder.50 − 49 = 1
- Write the mixed number. The whole number is 7. The 1 leftover piece of size 17 make the fraction 17. The bottom number stays 7 because the pieces are the same size.507 = 717
Check. Convert back: 7 × 7 + 1 = 49 + 1 = 50. As decimals, 50 ÷ 7 = 7.1429 and 717 = 7.1429. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.
- Say what the mixed number means. 112 means 1 whole ones and 1 extra piece of size 12.1 wholes + 12
- Change the whole ones into pieces. Each whole one is 2 pieces of size 12, so 1 wholes is 1 groups of 2 pieces.1 × 2 = 2 pieces
- Add the extra pieces. Now add the 1 pieces from the fraction part.2 + 1 = 3 pieces
- Write the total over the same denominator. Every piece is still of size 12, so the denominator does not change.112 = 32
Check. Divide back: 3 ÷ 2 = 1 remainder 1, which is 112. ✓
Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.
Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.
- Say what the mixed number means. 234 means 2 whole ones and 3 extra pieces of size 14.2 wholes + 34
- Change the whole ones into pieces. Each whole one is 4 pieces of size 14, so 2 wholes is 2 groups of 4 pieces.2 × 4 = 8 pieces
- Add the extra pieces. Now add the 3 pieces from the fraction part.8 + 3 = 11 pieces
- Write the total over the same denominator. Every piece is still of size 14, so the denominator does not change.234 = 114
Check. Divide back: 11 ÷ 4 = 2 remainder 3, which is 234. ✓
Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.
Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.
- Say what the mixed number means. 325 means 3 whole ones and 2 extra pieces of size 15.3 wholes + 25
- Change the whole ones into pieces. Each whole one is 5 pieces of size 15, so 3 wholes is 3 groups of 5 pieces.3 × 5 = 15 pieces
- Add the extra pieces. Now add the 2 pieces from the fraction part.15 + 2 = 17 pieces
- Write the total over the same denominator. Every piece is still of size 15, so the denominator does not change.325 = 175
Check. Divide back: 17 ÷ 5 = 3 remainder 2, which is 325. ✓
Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.
Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.
- Say what the mixed number means. 456 means 4 whole ones and 5 extra pieces of size 16.4 wholes + 56
- Change the whole ones into pieces. Each whole one is 6 pieces of size 16, so 4 wholes is 4 groups of 6 pieces.4 × 6 = 24 pieces
- Add the extra pieces. Now add the 5 pieces from the fraction part.24 + 5 = 29 pieces
- Write the total over the same denominator. Every piece is still of size 16, so the denominator does not change.456 = 296
Check. Divide back: 29 ÷ 6 = 4 remainder 5, which is 456. ✓
Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.
Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.
- Say what the mixed number means. 578 means 5 whole ones and 7 extra pieces of size 18.5 wholes + 78
- Change the whole ones into pieces. Each whole one is 8 pieces of size 18, so 5 wholes is 5 groups of 8 pieces.5 × 8 = 40 pieces
- Add the extra pieces. Now add the 7 pieces from the fraction part.40 + 7 = 47 pieces
- Write the total over the same denominator. Every piece is still of size 18, so the denominator does not change.578 = 478
Check. Divide back: 47 ÷ 8 = 5 remainder 7, which is 578. ✓
Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.
Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.
- Say what the mixed number means. 619 means 6 whole ones and 1 extra piece of size 19.6 wholes + 19
- Change the whole ones into pieces. Each whole one is 9 pieces of size 19, so 6 wholes is 6 groups of 9 pieces.6 × 9 = 54 pieces
- Add the extra pieces. Now add the 1 pieces from the fraction part.54 + 1 = 55 pieces
- Write the total over the same denominator. Every piece is still of size 19, so the denominator does not change.619 = 559
Check. Divide back: 55 ÷ 9 = 6 remainder 1, which is 619. ✓
Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.
Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.
- Say what the mixed number means. 10310 means 10 whole ones and 3 extra pieces of size 110.10 wholes + 310
- Change the whole ones into pieces. Each whole one is 10 pieces of size 110, so 10 wholes is 10 groups of 10 pieces.10 × 10 = 100 pieces
- Add the extra pieces. Now add the 3 pieces from the fraction part.100 + 3 = 103 pieces
- Write the total over the same denominator. Every piece is still of size 110, so the denominator does not change.10310 = 10310
Check. Divide back: 103 ÷ 10 = 10 remainder 3, which is 10310. ✓
Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.
Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.
- Say what the mixed number means. 847 means 8 whole ones and 4 extra pieces of size 17.8 wholes + 47
- Change the whole ones into pieces. Each whole one is 7 pieces of size 17, so 8 wholes is 8 groups of 7 pieces.8 × 7 = 56 pieces
- Add the extra pieces. Now add the 4 pieces from the fraction part.56 + 4 = 60 pieces
- Write the total over the same denominator. Every piece is still of size 17, so the denominator does not change.847 = 607
Check. Divide back: 60 ÷ 7 = 8 remainder 4, which is 847. ✓
Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.
Method. Count all the shaded parts. Each whole pie is split into the same number of equal parts.
- Count the parts in each whole pie. This is the denominator, because it tells us the size of each piece.4 parts, so each piece is 14
- Count all the shaded parts across all the pies. This is the numerator of the improper fraction.5 shaded parts
- Write the improper fraction. Shaded parts over parts in one pie.54
- Write it as a mixed number. 5 ÷ 4 = 1 remainder 1, so there is 1 complete pie and 1 extra part.54 = 114
Check. Convert back: 1 × 4 + 1 = 5. ✓
Watch out. The denominator is the number of parts in one pie, not the total number of shaded parts.
Method. Count all the shaded parts. Each whole pie is split into the same number of equal parts.
- Count the parts in each whole pie. This is the denominator, because it tells us the size of each piece.3 parts, so each piece is 13
- Count all the shaded parts across all the pies. This is the numerator of the improper fraction.7 shaded parts
- Write the improper fraction. Shaded parts over parts in one pie.73
- Write it as a mixed number. 7 ÷ 3 = 2 remainder 1, so there are 2 complete pies and 1 extra part.73 = 213
Check. Convert back: 2 × 3 + 1 = 7. ✓
Watch out. The denominator is the number of parts in one pie, not the total number of shaded parts.
Method. Count all the shaded parts. Each whole pie is split into the same number of equal parts.
- Count the parts in each whole pie. This is the denominator, because it tells us the size of each piece.6 parts, so each piece is 16
- Count all the shaded parts across all the pies. This is the numerator of the improper fraction.11 shaded parts
- Write the improper fraction. Shaded parts over parts in one pie.116
- Write it as a mixed number. 11 ÷ 6 = 1 remainder 5, so there is 1 complete pie and 5 extra parts.116 = 156
Check. Convert back: 1 × 6 + 5 = 11. ✓
Watch out. The denominator is the number of parts in one pie, not the total number of shaded parts.
Method. Count all the shaded parts. Each whole pie is split into the same number of equal parts.
- Count the parts in each whole pie. This is the denominator, because it tells us the size of each piece.5 parts, so each piece is 15
- Count all the shaded parts across all the pies. This is the numerator of the improper fraction.8 shaded parts
- Write the improper fraction. Shaded parts over parts in one pie.85
- Write it as a mixed number. 8 ÷ 5 = 1 remainder 3, so there is 1 complete pie and 3 extra parts.85 = 135
Check. Convert back: 1 × 5 + 3 = 8. ✓
Watch out. The denominator is the number of parts in one pie, not the total number of shaded parts.
Method. To put fractions in order, give them all the same denominator, then order the numerators.
- Change the mixed numbers to improper fractions. We can only compare numbers of the same type. A mixed number is a whole number plus a fraction. Whole × denominator + numerator, over the same denominator.123 = (1 × 3 + 2) over 3 = 53
112 = (1 × 2 + 1) over 2 = 32 - Find the LCM of the denominators 4, 3, 5, 2. This is the common denominator.LCM = 60
- Rewrite every fraction with denominator 60. Multiply the top and bottom of each by the same number.74 = 10560
123 = 10060
95 = 10860
112 = 9060 - Put the numerators in order. The order is smallest first.90, 100, 105, 108
- Write the original fractions in that order. The question asks for the original fractions, not the converted ones.112 , 123 , 74 , 95
Check. As decimals: 112 = 1.5, 123 = 1.6667, 74 = 1.75, 95 = 1.8.
Watch out. Always give your final answer using the original fractions, not the ones with the common denominator. And never order fractions by their denominators alone.
Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Decide the operation. The number of pizzas is the number of slices divided by the number of slices in one pizza.21 slices ÷ 8 slices per pizza
- Say what the fraction means. 218 means 21 pieces, each one 18 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.21 pieces of size 18
- Work out how many pieces make one whole. The bottom number tells us this: 8 pieces make 1 whole. So we want to know how many groups of 8 we can make from 21 pieces.8 pieces = 1 whole
- Divide 21 by 8 to find the whole number. Count how many whole groups of 8 fit into 21. 8 × 2 = 16, but 8 × 3 = 24 is too big.21 ÷ 8 = 2 remainder 5
- Find the pieces left over. After 2 whole ones we have used 16 pieces. The rest are the remainder.21 − 16 = 5
- Write the mixed number. The whole number is 2. The 5 leftover pieces of size 18 make the fraction 58. The bottom number stays 8 because the pieces are the same size.218 = 258
- Answer in words. A pizza has 8 slices, so 2 whole pizzas and 5 slices out of 8 is 5/8 of a pizza.258 pizzas
Check. Convert back: 2 × 8 + 5 = 16 + 5 = 21. As decimals, 21 ÷ 8 = 2.625 and 258 = 2.625. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. A mixed number is a whole number plus a fraction. Count all the pieces: whole number × denominator, add the numerator, keep the denominator.
- Decide what to count. A quarter-cup scoop measures one quarter of a cup. So the number of scoops is the number of quarters in the flour.Scoops = number of quarters in 314 cups
- Say what the mixed number means. 314 means 3 whole ones and 1 extra piece of size 14.3 wholes + 14
- Change the whole ones into pieces. Each whole one is 4 pieces of size 14, so 3 wholes is 3 groups of 4 pieces.3 × 4 = 12 pieces
- Add the extra pieces. Now add the 1 pieces from the fraction part.12 + 1 = 13 pieces
- Write the total over the same denominator. Every piece is still of size 14, so the denominator does not change.314 = 134
- Read the numerator. 134 is 13 pieces of size 14, so it is 13 quarters. That is 13 scoops.13 scoops
Check. 13 quarters = 13 ÷ 4 = 3.25 cups = 3 and a quarter cups. ✓
Watch out. Multiply the whole number by the denominator (not the numerator), then add the numerator.
Method. A mixed number has a whole-number part and a fraction part. Check what the statement says is true.
- Work out the value of 7/7. Any fraction with the top equal to the bottom is exactly 1 whole.77 = 7 ÷ 7 = 1
- Look at what a mixed number needs. A mixed number needs a whole number and a proper fraction. 1 has no fraction part.1 is a whole number only
- Decide. The statement says 7/7 is a mixed number.False. It is the whole number 1.
Check. If you divide 7 by 7 there is no remainder, so there is no fraction part. ✓
Watch out. Only a whole number and a proper fraction together make a mixed number.
Method. Change the mixed number to an improper fraction and compare.
- Multiply the whole number by the denominator. Each whole one is 3 thirds.5 × 3 = 15
- Add the numerator. Add the extra 2 thirds.15 + 2 = 17
- Compare. The result has denominator 3, the same as the original.523 = 173. True.
Check. Convert back: 17 ÷ 3 = 5 remainder 2, which is 523. ✓
Watch out. Whole number × denominator, then add the numerator.
Method. Change the improper fraction to a mixed number and compare.
- Divide the top by the bottom. This finds the whole number part and the remainder.11 ÷ 4 = 2 remainder 3
- Write the mixed number. The remainder 3 goes over the denominator 4.114 = 234
- Compare with the statement. The statement says 214, but the remainder is 3, not 1.234 is not 214. False.
Check. Convert the statement back: 214 = 94, which is not 114. ✓
Watch out. The remainder is the new numerator. Check it carefully.
Method. Write 3 and 4 as fractions with denominator 5, then see which numerators fit between.
- Write 3 as fifths. The whole number 3 is 3 wholes. Each whole is 5 fifths.3 = 155 (3 × 5 = 15)
- Write 4 as fifths. In the same way.4 = 205 (4 × 5 = 20)
- Find the numerators between 15 and 20. n5 must be between 3 and 4, but not equal to either, so n is more than 15 and less than 20.15 < n < 20
- List the whole-number values of n. Do not include 15 or 20.n = 16, 17, 18, 19
Check. 165 = 3.2 and 195 = 3.8, both between 3 and 4. 155 = 3 and 205 = 4 are not between, so they are excluded. ✓
Watch out. “Between” does not include the end values.
Adding and subtracting fractions
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
You can only add or subtract fractions when the parts are the same size, that is, when the denominators match.
Key ideas
- Same denominator: add (or subtract) the numerators, keep the denominator.
- Different denominators: change to equivalent fractions with a common denominator (use the LCM), then add or subtract.
- Mixed numbers: either add/subtract the whole numbers and fractions separately, or change to improper fractions first.
- Always simplify your answer, and write it as a mixed number if it is larger than 1.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).25 + 13
- Find a common denominator: the LCM of 5 and 3. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 5: 5, 10, 15
Multiples of 3: 3, 6, 9, 12, 15
LCM = 15 - Rewrite both fractions with denominator 15. Multiply the top and the bottom by the same number so the value does not change.25 = 615 (× 3)
13 = 515 (× 5) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.615 + 515 = 1115
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.11 and 15 have no common factor except 1, so 1115 is the answer.
Check. A different route, as decimals: 0.4 + 0.3333 = 0.7333, and our answer 1115 = 0.7333. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).56 − 38
- Find a common denominator: the LCM of 6 and 8. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 6: 6, 12, 18, 24
Multiples of 8: 8, 16, 24
LCM = 24 - Rewrite both fractions with denominator 24. Multiply the top and the bottom by the same number so the value does not change.56 = 2024 (× 4)
38 = 924 (× 3) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.2024 − 924 = 1124
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.11 and 24 have no common factor except 1, so 1124 is the answer.
Check. A different route, as decimals: 0.8333 − 0.375 = 0.4583, and our answer 1124 = 0.4583. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. Add mixed numbers by adding the whole numbers and the fractions separately, then combine the two results.
- Split each mixed number into a whole number and a fraction. A mixed number is a whole number plus a fraction, so we can deal with the two parts one at a time.213 = 2 + 13
134 = 1 + 34 - Add the whole numbers. Whole numbers are added in the normal way.2 + 1 = 3
- Find a common denominator for the fractions. We can only add pieces that are the same size. The LCM of 3 and 4 is the smallest number both divide into.Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
LCM = 12
13 = 412 (× 4) and 34 = 912 (× 3) - Add the fractions. Add the numerators and keep the denominator.412 + 912 = 1312
- Change the improper fraction to a mixed number. The fraction part is more than 1, so it contains one whole.13 ÷ 12 = 1 remainder 1, so 1312 = 1112
- Add the whole number and the mixed number. Put the two answers together.3 + 1112 = 4112
Check. A different route: change to improper fractions. 73 + 74 = 2812 + 2112 = 4912, and 49 ÷ 12 = 4 remainder 1, which is 4112. ✓
Watch out. If the fraction parts add up to more than 1, carry the extra whole one across to the whole numbers.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).314 − 123
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.314 = (3 × 4 + 1) over 4 = 134
123 = (1 × 3 + 2) over 3 = 53 - Find a common denominator: the LCM of 4 and 3. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 4: 4, 8, 12
Multiples of 3: 3, 6, 9, 12
LCM = 12 - Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.134 = 3912 (× 3)
53 = 2012 (× 4) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.3912 − 2012 = 1912
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.19 ÷ 12 = 1 remainder 7
1912 = 1712
Check. A different route, as decimals: 3.25 − 1.6667 = 1.5833, and our answer 1712 = 1.5833. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. When the fraction you are subtracting is bigger than the fraction you start with, borrow one whole from the whole number.
- Look at the fraction parts first. We need 16 − 23, but 23 (which is 46) is bigger than 16, so we cannot take it away yet.16 < 46
- Borrow one whole from the 5. Take 1 off the whole number 5 to get 4, and write that 1 as sixths so that it can join the fraction (1 = 6 sixths).516 = 4 + 66 + 16 = 476
- Write the second number with sixths. Both fractions need the same denominator: 6.223 = 246
- Subtract the fractions. Subtract the numerators and keep the denominator.76 − 46 = 36 = 12
- Subtract the whole numbers. Now the whole numbers are 4 and 2.4 − 2 = 2
- Put the answers together. Whole number first, then the fraction.212
Check. A different route: improper fractions. 316 − 166 = 156 = 52 = 212. ✓
Watch out. Do not subtract the smaller fraction from the larger one just to avoid the problem. The order matters: the second number is taken away from the first.
Method. The whole club is 1 (or 10 tenths). Add the fractions already used, then subtract the total from the whole.
- Decide what the whole is. Every member is in exactly one group, so the three fractions must add up to 1 whole club.football + netball + swimming = 1
- Find a common denominator. To add halves and fifths we need pieces of the same size. The LCM of 2 and 5 is 10.12 = 510 and 15 = 210
- Add football and netball. Add the numerators and keep the denominator.510 + 210 = 710
- Write 1 as tenths. A whole club is all 10 of the 10 equal parts.1 = 1010
- Subtract to find the swimmers. What is left of the club after football and netball.1010 − 710 = 310
Check. Add all three groups: 510 + 210 + 310 = 1010 = 1. ✓
Watch out. “The rest” means: take what has been used away from 1 whole.
Watch out
Do not add the denominators. 13 + 14 is not 27. The answer is 712.Exercise 3E Adding and subtracting fractions
Non-calculatorMethod. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).27 + 37
- Look at the denominators. They are both 7, so the pieces are the same size and we can add straight away.Denominators: 7 and 7
- Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.27 + 37 = 57
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.5 and 7 have no common factor except 1, so 57 is the answer.
Check. A different route, as decimals: 0.2857 + 0.4286 = 0.7143, and our answer 57 = 0.7143. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).59 + 29
- Look at the denominators. They are both 9, so the pieces are the same size and we can add straight away.Denominators: 9 and 9
- Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.59 + 29 = 79
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.7 and 9 have no common factor except 1, so 79 is the answer.
Check. A different route, as decimals: 0.5556 + 0.2222 = 0.7778, and our answer 79 = 0.7778. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).710 − 310
- Look at the denominators. They are both 10, so the pieces are the same size and we can subtract straight away.Denominators: 10 and 10
- Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.710 − 310 = 410
- Simplify the answer. 4 and 10 have a common factor of 2 (their HCF), so divide the top and the bottom by 2.4 ÷ 2 = 2 and 10 ÷ 2 = 5
410 = 25
Check. A different route, as decimals: 0.7 − 0.3 = 0.4, and our answer 25 = 0.4. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).1112 − 512
- Look at the denominators. They are both 12, so the pieces are the same size and we can subtract straight away.Denominators: 12 and 12
- Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.1112 − 512 = 612
- Simplify the answer. 6 and 12 have a common factor of 6 (their HCF), so divide the top and the bottom by 6.6 ÷ 6 = 1 and 12 ÷ 6 = 2
612 = 12
Check. A different route, as decimals: 0.9167 − 0.4167 = 0.5, and our answer 12 = 0.5. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).12 + 14
- Find a common denominator: the LCM of 2 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 2: 2, 4
Multiples of 4: 4
LCM = 4 - Rewrite both fractions with denominator 4. Multiply the top and the bottom by the same number so the value does not change.12 = 24 (× 2)
14 = 14 (× 1) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.24 + 14 = 34
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.3 and 4 have no common factor except 1, so 34 is the answer.
Check. A different route, as decimals: 0.5 + 0.25 = 0.75, and our answer 34 = 0.75. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).13 + 16
- Find a common denominator: the LCM of 3 and 6. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 3: 3, 6
Multiples of 6: 6
LCM = 6 - Rewrite both fractions with denominator 6. Multiply the top and the bottom by the same number so the value does not change.13 = 26 (× 2)
16 = 16 (× 1) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.26 + 16 = 36
- Simplify the answer. 3 and 6 have a common factor of 3 (their HCF), so divide the top and the bottom by 3.3 ÷ 3 = 1 and 6 ÷ 3 = 2
36 = 12
Check. A different route, as decimals: 0.3333 + 0.1667 = 0.5, and our answer 12 = 0.5. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).25 + 110
- Find a common denominator: the LCM of 5 and 10. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 5: 5, 10
Multiples of 10: 10
LCM = 10 - Rewrite both fractions with denominator 10. Multiply the top and the bottom by the same number so the value does not change.25 = 410 (× 2)
110 = 110 (× 1) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.410 + 110 = 510
- Simplify the answer. 5 and 10 have a common factor of 5 (their HCF), so divide the top and the bottom by 5.5 ÷ 5 = 1 and 10 ÷ 5 = 2
510 = 12
Check. A different route, as decimals: 0.4 + 0.1 = 0.5, and our answer 12 = 0.5. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).34 − 18
- Find a common denominator: the LCM of 4 and 8. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 4: 4, 8
Multiples of 8: 8
LCM = 8 - Rewrite both fractions with denominator 8. Multiply the top and the bottom by the same number so the value does not change.34 = 68 (× 2)
18 = 18 (× 1) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.68 − 18 = 58
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.5 and 8 have no common factor except 1, so 58 is the answer.
Check. A different route, as decimals: 0.75 − 0.125 = 0.625, and our answer 58 = 0.625. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).56 − 13
- Find a common denominator: the LCM of 6 and 3. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 6: 6
Multiples of 3: 3, 6
LCM = 6 - Rewrite both fractions with denominator 6. Multiply the top and the bottom by the same number so the value does not change.56 = 56 (× 1)
13 = 26 (× 2) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.56 − 26 = 36
- Simplify the answer. 3 and 6 have a common factor of 3 (their HCF), so divide the top and the bottom by 3.3 ÷ 3 = 1 and 6 ÷ 3 = 2
36 = 12
Check. A different route, as decimals: 0.8333 − 0.3333 = 0.5, and our answer 12 = 0.5. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).78 − 34
- Find a common denominator: the LCM of 8 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 8: 8
Multiples of 4: 4, 8
LCM = 8 - Rewrite both fractions with denominator 8. Multiply the top and the bottom by the same number so the value does not change.78 = 78 (× 1)
34 = 68 (× 2) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.78 − 68 = 18
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 8 have no common factor except 1, so 18 is the answer.
Check. A different route, as decimals: 0.875 − 0.75 = 0.125, and our answer 18 = 0.125. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).13 + 14
- Find a common denominator: the LCM of 3 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.13 = 412 (× 4)
14 = 312 (× 3) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.412 + 312 = 712
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.7 and 12 have no common factor except 1, so 712 is the answer.
Check. A different route, as decimals: 0.3333 + 0.25 = 0.5833, and our answer 712 = 0.5833. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).25 + 13
- Find a common denominator: the LCM of 5 and 3. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 5: 5, 10, 15
Multiples of 3: 3, 6, 9, 12, 15
LCM = 15 - Rewrite both fractions with denominator 15. Multiply the top and the bottom by the same number so the value does not change.25 = 615 (× 3)
13 = 515 (× 5) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.615 + 515 = 1115
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.11 and 15 have no common factor except 1, so 1115 is the answer.
Check. A different route, as decimals: 0.4 + 0.3333 = 0.7333, and our answer 1115 = 0.7333. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).34 − 25
- Find a common denominator: the LCM of 4 and 5. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 4: 4, 8, 12, 16, 20
Multiples of 5: 5, 10, 15, 20
LCM = 20 - Rewrite both fractions with denominator 20. Multiply the top and the bottom by the same number so the value does not change.34 = 1520 (× 5)
25 = 820 (× 4) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.1520 − 820 = 720
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.7 and 20 have no common factor except 1, so 720 is the answer.
Check. A different route, as decimals: 0.75 − 0.4 = 0.35, and our answer 720 = 0.35. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).78 − 13
- Find a common denominator: the LCM of 8 and 3. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 8: 8, 16, 24
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24
LCM = 24 - Rewrite both fractions with denominator 24. Multiply the top and the bottom by the same number so the value does not change.78 = 2124 (× 3)
13 = 824 (× 8) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.2124 − 824 = 1324
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.13 and 24 have no common factor except 1, so 1324 is the answer.
Check. A different route, as decimals: 0.875 − 0.3333 = 0.5417, and our answer 1324 = 0.5417. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).56 + 34
- Find a common denominator: the LCM of 6 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 6: 6, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.56 = 1012 (× 2)
34 = 912 (× 3) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.1012 + 912 = 1912
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.19 ÷ 12 = 1 remainder 7
1912 = 1712
Check. A different route, as decimals: 0.8333 + 0.75 = 1.5833, and our answer 1712 = 1.5833. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).45 − 310
- Find a common denominator: the LCM of 5 and 10. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 5: 5, 10
Multiples of 10: 10
LCM = 10 - Rewrite both fractions with denominator 10. Multiply the top and the bottom by the same number so the value does not change.45 = 810 (× 2)
310 = 310 (× 1) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.810 − 310 = 510
- Simplify the answer. 5 and 10 have a common factor of 5 (their HCF), so divide the top and the bottom by 5.5 ÷ 5 = 1 and 10 ÷ 5 = 2
510 = 12
Check. A different route, as decimals: 0.8 − 0.3 = 0.5, and our answer 12 = 0.5. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).112 + 214
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.112 = (1 × 2 + 1) over 2 = 32
214 = (2 × 4 + 1) over 4 = 94 - Find a common denominator: the LCM of 2 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 2: 2, 4
Multiples of 4: 4
LCM = 4 - Rewrite both fractions with denominator 4. Multiply the top and the bottom by the same number so the value does not change.32 = 64 (× 2)
94 = 94 (× 1) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.64 + 94 = 154
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.15 ÷ 4 = 3 remainder 3
154 = 334
Check. A different route, as decimals: 1.5 + 2.25 = 3.75, and our answer 334 = 3.75. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).325 + 112
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.325 = (3 × 5 + 2) over 5 = 175
112 = (1 × 2 + 1) over 2 = 32 - Find a common denominator: the LCM of 5 and 2. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 5: 5, 10
Multiples of 2: 2, 4, 6, 8, 10
LCM = 10 - Rewrite both fractions with denominator 10. Multiply the top and the bottom by the same number so the value does not change.175 = 3410 (× 2)
32 = 1510 (× 5) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.3410 + 1510 = 4910
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.49 ÷ 10 = 4 remainder 9
4910 = 4910
Check. A different route, as decimals: 3.4 + 1.5 = 4.9, and our answer 4910 = 4.9. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).223 + 134
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.223 = (2 × 3 + 2) over 3 = 83
134 = (1 × 4 + 3) over 4 = 74 - Find a common denominator: the LCM of 3 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.83 = 3212 (× 4)
74 = 2112 (× 3) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.3212 + 2112 = 5312
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.53 ÷ 12 = 4 remainder 5
5312 = 4512
Check. A different route, as decimals: 2.6667 + 1.75 = 4.4167, and our answer 4512 = 4.4167. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).412 − 113
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.412 = (4 × 2 + 1) over 2 = 92
113 = (1 × 3 + 1) over 3 = 43 - Find a common denominator: the LCM of 2 and 3. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 2: 2, 4, 6
Multiples of 3: 3, 6
LCM = 6 - Rewrite both fractions with denominator 6. Multiply the top and the bottom by the same number so the value does not change.92 = 276 (× 3)
43 = 86 (× 2) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.276 − 86 = 196
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.19 ÷ 6 = 3 remainder 1
196 = 316
Check. A different route, as decimals: 4.5 − 1.3333 = 3.1667, and our answer 316 = 3.1667. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).314 − 112
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.314 = (3 × 4 + 1) over 4 = 134
112 = (1 × 2 + 1) over 2 = 32 - Find a common denominator: the LCM of 4 and 2. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 4: 4
Multiples of 2: 2, 4
LCM = 4 - Rewrite both fractions with denominator 4. Multiply the top and the bottom by the same number so the value does not change.134 = 134 (× 1)
32 = 64 (× 2) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.134 − 64 = 74
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.7 ÷ 4 = 1 remainder 3
74 = 134
Check. A different route, as decimals: 3.25 − 1.5 = 1.75, and our answer 134 = 1.75. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).516 − 223
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.516 = (5 × 6 + 1) over 6 = 316
223 = (2 × 3 + 2) over 3 = 83 - Find a common denominator: the LCM of 6 and 3. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 6: 6
Multiples of 3: 3, 6
LCM = 6 - Rewrite both fractions with denominator 6. Multiply the top and the bottom by the same number so the value does not change.316 = 316 (× 1)
83 = 166 (× 2) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.316 − 166 = 156
- Simplify the answer. 15 and 6 have a common factor of 3 (their HCF), so divide the top and the bottom by 3.15 ÷ 3 = 5 and 6 ÷ 3 = 2
156 = 52 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.5 ÷ 2 = 2 remainder 1
52 = 212
Check. A different route, as decimals: 5.1667 − 2.6667 = 2.5, and our answer 212 = 2.5. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Decide the operation. “Altogether” means we add the two distances. The pieces must be the same size first, so look at the denominators.23 km + 34 km
- Find a common denominator: the LCM of 3 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.23 = 812 (× 4)
34 = 912 (× 3) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.812 + 912 = 1712
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.17 ÷ 12 = 1 remainder 5
1712 = 1512 - Answer in words with units. The distance is in kilometres.Maya walks 1512 km altogether.
Check. A different route, as decimals: 0.6667 + 0.75 = 1.4167, and our answer 1512 = 1.4167. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Decide the operation. “Left” means we take away the juice poured out from the juice we started with.56 litre − 14 litre
- Find a common denominator: the LCM of 6 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 6: 6, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.56 = 1012 (× 2)
14 = 312 (× 3) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.1012 − 312 = 712
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.7 and 12 have no common factor except 1, so 712 is the answer.
- Answer in words with units. The amount of juice is in litres.712 litre of juice is left.
Check. A different route, as decimals: 0.8333 − 0.25 = 0.5833, and our answer 712 = 0.5833. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Decide the operation. The piece that is left is the full length take away the piece cut off.312 m − 134 m
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.312 = (3 × 2 + 1) over 2 = 72
134 = (1 × 4 + 3) over 4 = 74 - Find a common denominator: the LCM of 2 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 2: 2, 4
Multiples of 4: 4
LCM = 4 - Rewrite both fractions with denominator 4. Multiply the top and the bottom by the same number so the value does not change.72 = 144 (× 2)
74 = 74 (× 1) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.144 − 74 = 74
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.7 ÷ 4 = 1 remainder 3
74 = 134 - Answer in words with units. Lengths are in metres.The piece that is left is 134 m long.
Check. A different route, as decimals: 3.5 − 1.75 = 1.75, and our answer 134 = 1.75. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To add fractions you need the same denominator. The denominators must not be added.
- Spot Dev's method. Dev added the tops (1 + 1 = 2) and added the bottoms (3 + 4 = 7).Dev: 13 + 14 = 1 + 13 + 4 = 27
- Explain why this is wrong. The denominator tells us the size of the pieces. A third and a quarter are different sizes, so we cannot just count the pieces. We must first make them the same size.Thirds and quarters are different sizes
- Find a common denominator. The LCM of 3 and 4 is 12.13 = 412 (× 4)
14 = 312 (× 3) - Add the numerators and keep the denominator. Now the pieces are the same size (twelfths).412 + 312 = 712
- Compare with Dev's answer. The correct answer is not the same as Dev's.712 ≠ 27
Check. As decimals: 13 + 14 = 0.333 + 0.25 = 0.583, and 712 = 0.583. But 27 = 0.286, which is less than one of the fractions being added, so it cannot be right. ✓
Watch out. Adding two positive numbers must give something bigger than both. Dev's answer is smaller than 1/3, which shows it is wrong.
Method. The whole school is 1 (or 10/10). Add the fractions already used, then subtract the total from the whole.
- Decide what the whole is. Every pupil walks, takes the bus or cycles, so the three fractions add up to 1 whole school.walk + bus + cycle = 1
- Find a common denominator. To add halves and fifths we need the same size pieces. The LCM of 2 and 5 is 10.12 = 510 and 15 = 210
- Add the walkers and the bus users. Add the numerators and keep the denominator.510 + 210 = 710
- Subtract from the whole. The rest cycle. Write 1 as tenths: 1 = 10/10.1010 − 710 = 310
Check. Add all three groups: 510 + 210 + 310 = 1010 = 1. ✓
Watch out. “The rest” means: take what has been used away from 1 whole.
Method. Call the unknown numerators a and b. Write an equation using a common denominator of 12, then test whole numbers.
- Write the two fractions with unknown tops. One has denominator 3 and the other has denominator 4. Let the numerators be a and b.a3 + b4 = 1712
- Use the common denominator 12. The first fraction needs × 4 and the second needs × 3.a3 = 4a12 and b4 = 3b12
- Write an equation for the numerators. The denominators match, so the numerators must add up to 17.4a + 3b = 17
- Try whole numbers for a. The numerators are positive whole numbers and 4a cannot be bigger than 17, so a can only be 1, 2, 3 or 4. For each a, work out 3b = 17 − 4a and see whether b is a whole number.a = 1: 3b = 13, no
a = 2: 3b = 9, so b = 3 ✓
a = 3: 3b = 5, no
a = 4: 3b = 1, no - Write the two fractions. a = 2 over 3 and b = 3 over 4.23 and 34
Check. 23 + 34 = 812 + 912 = 1712. ✓
Watch out. Always check your answer in the original problem, not just the equation.
Multiplying fractions
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Finding “a fraction of a fraction” means multiplying. In the diagram, the blue columns show 34 and the orange rows show 23. The dark overlap is 34 × 23: 6 squares out of 12.
Key ideas
- Multiply the numerators and multiply the denominators, then simplify.
- You can cancel first: divide a numerator and a denominator by a common factor before multiplying.
- Write whole numbers as fractions over 1, for example 4 = 41.
- For mixed numbers, change to improper fractions first.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.23 × 45
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 2, 4. Bottoms: 3, 5. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.2 × 4 = 8
3 × 5 = 15
815 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.8 and 15 have no common factor except 1, so 815 is the answer.
Check. A different route, as decimals: 0.6667 × 0.8 = 0.5333, and our answer 815 = 0.5333. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.38 × 49
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.38 × 49 = 12 × 13
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 3 = 6
16 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 6 have no common factor except 1, so 16 is the answer.
Check. A different route, as decimals: 0.375 × 0.4444 = 0.1667, and our answer 16 = 0.1667. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.212 × 115
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.212 = (2 × 2 + 1) over 2 = 52
115 = (1 × 5 + 1) over 5 = 65 - Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.52 × 65 = 11 × 31
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 3 = 3
1 × 1 = 1
31 - Write the answer. The bottom number is 1, so the answer is a whole number.31 = 3
Check. A different route, as decimals: 2.5 × 1.2 = 3, and our answer 3 = 3. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.5 × 29
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.5 = 51
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 5, 2. Bottoms: 1, 9. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.5 × 2 = 10
1 × 9 = 9
109 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.10 ÷ 9 = 1 remainder 1
109 = 119
Check. A different route, as decimals: 5 × 0.2222 = 1.1111, and our answer 119 = 1.1111. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Decide what to calculate. The area of a rectangle is length × width, so we multiply the two lengths.Area = 56 × 310
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.56 × 310 = 12 × 12
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 2 = 4
14 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 4 have no common factor except 1, so 14 is the answer.
- Write the answer with its units. We multiplied two lengths in metres, so the area is in square metres.Area = 14 m²
Check. A different route, as decimals: 0.8333 × 0.3 = 0.25, and our answer 14 = 0.25. ✓
Watch out. Area is in square units (m²), not m. And multiplying two fractions less than 1 gives a smaller answer.
Watch out
Multiplying does not always make a number bigger. Multiplying by a fraction less than 1 makes it smaller: 12 × 12 = 14.Exercise 3F Multiplying fractions
Non-calculatorMethod. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.12 × 13
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 1, 1. Bottoms: 2, 3. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 3 = 6
16 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 6 have no common factor except 1, so 16 is the answer.
Check. A different route, as decimals: 0.5 × 0.3333 = 0.1667, and our answer 16 = 0.1667. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.25 × 37
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 2, 3. Bottoms: 5, 7. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.2 × 3 = 6
5 × 7 = 35
635 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.6 and 35 have no common factor except 1, so 635 is the answer.
Check. A different route, as decimals: 0.4 × 0.4286 = 0.1714, and our answer 635 = 0.1714. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.34 × 29
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.34 × 29 = 12 × 13
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 3 = 6
16 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 6 have no common factor except 1, so 16 is the answer.
Check. A different route, as decimals: 0.75 × 0.2222 = 0.1667, and our answer 16 = 0.1667. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.45 × 38
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.45 × 38 = 15 × 32
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 3 = 3
5 × 2 = 10
310 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.3 and 10 have no common factor except 1, so 310 is the answer.
Check. A different route, as decimals: 0.8 × 0.375 = 0.3, and our answer 310 = 0.3. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.56 × 310
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.56 × 310 = 12 × 12
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 2 = 4
14 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 4 have no common factor except 1, so 14 is the answer.
Check. A different route, as decimals: 0.8333 × 0.3 = 0.25, and our answer 14 = 0.25. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.79 × 314
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.79 × 314 = 13 × 12
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
3 × 2 = 6
16 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 6 have no common factor except 1, so 16 is the answer.
Check. A different route, as decimals: 0.7778 × 0.2143 = 0.1667, and our answer 16 = 0.1667. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.3 × 25
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.3 = 31
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 3, 2. Bottoms: 1, 5. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 2 = 6
1 × 5 = 5
65 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.6 ÷ 5 = 1 remainder 1
65 = 115
Check. A different route, as decimals: 3 × 0.4 = 1.2, and our answer 115 = 1.2. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.4 × 38
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.4 = 41
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.41 × 38 = 11 × 32
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 3 = 3
1 × 2 = 2
32 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.3 ÷ 2 = 1 remainder 1
32 = 112
Check. A different route, as decimals: 4 × 0.375 = 1.5, and our answer 112 = 1.5. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.6 × 59
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.6 = 61
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.61 × 59 = 21 × 53
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.2 × 5 = 10
1 × 3 = 3
103 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.10 ÷ 3 = 3 remainder 1
103 = 313
Check. A different route, as decimals: 6 × 0.5556 = 3.3333, and our answer 313 = 3.3333. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.23 × 12
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.12 = 121
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.23 × 121 = 21 × 41
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.2 × 4 = 8
1 × 1 = 1
81 - Write the answer. The bottom number is 1, so the answer is a whole number.81 = 8
Check. A different route, as decimals: 0.6667 × 12 = 8, and our answer 8 = 8. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.57 × 14
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.14 = 141
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.57 × 141 = 51 × 21
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.5 × 2 = 10
1 × 1 = 1
101 - Write the answer. The bottom number is 1, so the answer is a whole number.101 = 10
Check. A different route, as decimals: 0.7143 × 14 = 10, and our answer 10 = 10. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.34 × 10
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.10 = 101
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.34 × 101 = 32 × 51
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 5 = 15
2 × 1 = 2
152 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.15 ÷ 2 = 7 remainder 1
152 = 712
Check. A different route, as decimals: 0.75 × 10 = 7.5, and our answer 712 = 7.5. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.112 × 23
- Change the mixed number to improper fraction. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.112 = (1 × 2 + 1) over 2 = 32
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.32 × 23 = 11 × 11
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
1 × 1 = 1
11 - Write the answer. The bottom number is 1, so the answer is a whole number.11 = 1
Check. A different route, as decimals: 1.5 × 0.6667 = 1, and our answer 1 = 1. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.214 × 113
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.214 = (2 × 4 + 1) over 4 = 94
113 = (1 × 3 + 1) over 3 = 43 - Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.94 × 43 = 31 × 11
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 1 = 3
1 × 1 = 1
31 - Write the answer. The bottom number is 1, so the answer is a whole number.31 = 3
Check. A different route, as decimals: 2.25 × 1.3333 = 3, and our answer 3 = 3. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.313 × 145
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.313 = (3 × 3 + 1) over 3 = 103
145 = (1 × 5 + 4) over 5 = 95 - Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.103 × 95 = 21 × 31
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.2 × 3 = 6
1 × 1 = 1
61 - Write the answer. The bottom number is 1, so the answer is a whole number.61 = 6
Check. A different route, as decimals: 3.3333 × 1.8 = 6, and our answer 6 = 6. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.225 × 114
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.225 = (2 × 5 + 2) over 5 = 125
114 = (1 × 4 + 1) over 4 = 54 - Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.125 × 54 = 31 × 11
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 1 = 3
1 × 1 = 1
31 - Write the answer. The bottom number is 1, so the answer is a whole number.31 = 3
Check. A different route, as decimals: 2.4 × 1.25 = 3, and our answer 3 = 3. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.134 × 227
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.134 = (1 × 4 + 3) over 4 = 74
227 = (2 × 7 + 2) over 7 = 167 - Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.74 × 167 = 11 × 41
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 4 = 4
1 × 1 = 1
41 - Write the answer. The bottom number is 1, so the answer is a whole number.41 = 4
Check. A different route, as decimals: 1.75 × 2.2857 = 4, and our answer 4 = 4. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. In an area model, the fraction of a fraction is the overlap of the two shadings.
- Count the squares in the whole grid. The grid is 4 columns by 2 rows. The total is the denominator of the answer.4 × 2 = 8 squares
- Count the squares shaded by the first fraction. 34 means 3 of the 4 columns are shaded.3 columns
- Count the squares shaded by the second fraction. 12 means 1 of the 2 rows is shaded.1 row
- Count the overlap (the dark squares). Where a shaded column and the shaded row cross, squares are shaded twice. Multiply columns by rows.3 × 1 = 3 dark squares
- Write the fraction. Dark squares over total squares.38
Check. Multiply the fractions: 12 × 34 = 38 (tops 1 × 3 = 3, bottoms 2 × 4 = 8). ✓
Watch out. The overlap is the answer, not the total shaded area.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Decide the operation. The area of a rectangle is length × width, so multiply the two sides.Area = 34 × 23
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.34 × 23 = 12 × 11
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 1 = 2
12 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 2 have no common factor except 1, so 12 is the answer.
- Write the answer with its units. Area is measured in square units, here square metres.Area = 12 m²
Check. A different route, as decimals: 0.75 × 0.6667 = 0.5, and our answer 12 = 0.5. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Decide the operation. “A fraction of” means multiply. The water used is one quarter of the water that is in the tank, which is three fifths of the whole tank.14 of 35 = 35 × 14
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.Tops: 3, 1. Bottoms: 5, 4. Nothing cancels here.
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.3 × 1 = 3
5 × 4 = 20
320 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.3 and 20 have no common factor except 1, so 320 is the answer.
- Interpret the answer. The product is a fraction of the whole tank, because the three fifths were already a fraction of the whole tank.320 of the whole tank is used.
Check. A different route, as decimals: 0.6 × 0.25 = 0.15, and our answer 320 = 0.15. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To show a statement is false you need just one counter-example: a case where it does not work.
- Say what Sam claims. Sam says the product of two numbers is always bigger than both of them.Claim: a × b > a and a × b > b
- Choose two small fractions. Fractions less than 1 will make the product smaller. Pick the simplest one.12 and 12
- Multiply them. Multiply the tops and the bottoms.12 × 12 = 1 × 12 × 2 = 14
- Compare the answer with the numbers. 14 is smaller than 12, so the answer is smaller than both numbers.14 < 12
- Conclude. One counter-example is enough to prove the claim is false.Sam is wrong.
Check. Think of it as “half of a half”. Half of something is less than it, so the answer 14 must be smaller than 12. ✓
Watch out. The claim is true for whole numbers bigger than 1, but not for fractions. One counter-example is enough to disprove it.
Method. Area of a triangle = ½ × base × height. Use improper fractions and multiply two at a time.
- Change the mixed numbers to improper fractions. Whole × denominator + numerator, over the same denominator.112 = 32 and 223 = 83
- Write the calculation. Multiply the three numbers.Area = 12 × 32 × 83
- Multiply the first two. Multiply the tops and the bottoms.12 × 32 = 34
- Multiply by the third and cancel. Cancel the 3s, and cancel 4 with 8 (both divide by 4).34 × 83 = 11 × 21 = 2
- Write the answer with its units. Area is in square centimetres.Area = 2 cm²
Check. As decimals: 0.5 × 1.5 × 2.667 = 2. ✓
Watch out. Do not forget the ½ in the triangle formula.
Method. To find a missing fraction in a multiplication, divide the answer by the known fraction.
- Write the problem as an equation. The missing fraction times 2/3 gives 1/6.23 × ? = 16
- Use the inverse operation. The inverse of multiplying is dividing. So the missing fraction is the answer divided by the known fraction.? = 16 ÷ 23
- Keep, change, flip. Dividing by a fraction is multiplying by its reciprocal.16 × 32 = 312
- Simplify. Divide the top and the bottom by 3.312 = 14
Check. Multiply back: 23 × 14 = 212 = 16. ✓
Watch out. Flip only the fraction you are dividing by (here 2/3), not the 1/6.
Dividing fractions
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Dividing asks “how many fit into?” How many halves are in 2? There are 4, so 2 ÷ 12 = 4.
Key ideas
- The reciprocal of a fraction is found by turning it upside down. The reciprocal of 23 is 32. The reciprocal of 5 is 15.
- To divide by a fraction, multiply by its reciprocal. (Keep, change, flip.)
- To divide by a whole number, treat it as a fraction over 1, or divide the numerator if it goes in.
- Change mixed numbers to improper fractions first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.34 ÷ 25
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.34 ÷ 25 = 34 × 52
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 5 = 15
4 × 2 = 8
158 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.15 ÷ 8 = 1 remainder 7
158 = 178
Check. Multiply back: 178 × 25 should give 34. 1.875 × 0.4 = 0.75 and 34 = 0.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.214 ÷ 112
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.214 = (2 × 4 + 1) over 4 = 94
112 = (1 × 2 + 1) over 2 = 32 - Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.94 ÷ 32 = 94 × 23
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.94 × 23 = 32 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 1 = 3
2 × 1 = 2
32 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.3 ÷ 2 = 1 remainder 1
32 = 112
Check. Multiply back: 112 × 112 should give 214. 1.5 × 1.5 = 2.25 and 214 = 2.25. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.6 ÷ 34
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.6 = 61
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.61 ÷ 34 = 61 × 43
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.61 × 43 = 21 × 41
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 4 = 8
1 × 1 = 1
81 - Write the answer. The bottom number is 1, so the answer is a whole number.81 = 8
Check. Multiply back: 8 × 34 should give 6. 8 × 0.75 = 6 and 6 = 6. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.56 ÷ 4
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.4 = 41
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.56 ÷ 41 = 56 × 14
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.5 × 1 = 5
6 × 4 = 24
524 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.5 and 24 have no common factor except 1, so 524 is the answer.
Check. Multiply back: 524 × 4 should give 56. 0.2083 × 4 = 0.8333 and 56 = 0.8333. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Decide the operation. “How many bottles fit” asks how many times the bottle size goes into the total, so we divide the total by the size of one bottle.513 ÷ 23
- Change the mixed number to improper fraction. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.513 = (5 × 3 + 1) over 3 = 163
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.163 ÷ 23 = 163 × 32
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.163 × 32 = 81 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.8 × 1 = 8
1 × 1 = 1
81 - Write the answer. The bottom number is 1, so the answer is a whole number.81 = 8
- Answer the question in words. The answer is a whole number of bottles.8 bottles
Check. Multiply back: 8 × 23 should give 513. 8 × 0.6667 = 5.3333 and 513 = 5.3333. ✓
Watch out. Divide the total by the size of one bottle, not the other way round.
Watch out
Only flip the second fraction (the one you divide by). Also 12 ÷ 14 = 2 but 14 ÷ 12 = 12: order matters when dividing.Exercise 3G Dividing fractions
Non-calculatorMethod. The reciprocal of a number is found by turning the fraction upside down (swap the top and the bottom).
- Say what a reciprocal is. The reciprocal of a number is 1 divided by that number. For a fraction it is the same fraction turned upside down.reciprocal of 23
- Flip the fraction. Swap the numerator and the denominator.23 becomes 32
- Write the answer. If the new fraction has 1 on the bottom, it is a whole number.32
Check. A number times its reciprocal is 1: 23 × 32 = 1. ✓
Watch out. Only flip the fraction. Do not change its sign or add anything. The reciprocal of 5 is 1/5, not −5.
Method. The reciprocal of a number is found by turning the fraction upside down (swap the top and the bottom).
- Say what a reciprocal is. The reciprocal of a number is 1 divided by that number. For a fraction it is the same fraction turned upside down.reciprocal of 58
- Flip the fraction. Swap the numerator and the denominator.58 becomes 85
- Write the answer. If the new fraction has 1 on the bottom, it is a whole number.85
Check. A number times its reciprocal is 1: 58 × 85 = 1. ✓
Watch out. Only flip the fraction. Do not change its sign or add anything. The reciprocal of 5 is 1/5, not −5.
Method. The reciprocal of a number is found by turning the fraction upside down (swap the top and the bottom).
- Say what a reciprocal is. The reciprocal of a number is 1 divided by that number. For a fraction it is the same fraction turned upside down.reciprocal of 4
- Write the whole number as a fraction over 1. Every whole number can be written over 1 without changing its value.4 = 41
- Flip the fraction. Swap the numerator and the denominator.41 becomes 14
- Write the answer. If the new fraction has 1 on the bottom, it is a whole number.14
Check. A number times its reciprocal is 1: 4 × 14 = 1. ✓
Watch out. Only flip the fraction. Do not change its sign or add anything. The reciprocal of 5 is 1/5, not −5.
Method. The reciprocal of a number is found by turning the fraction upside down (swap the top and the bottom).
- Say what a reciprocal is. The reciprocal of a number is 1 divided by that number. For a fraction it is the same fraction turned upside down.reciprocal of 17
- Flip the fraction. Swap the numerator and the denominator.17 becomes 71
- Write the answer. If the new fraction has 1 on the bottom, it is a whole number.7
Check. A number times its reciprocal is 1: 17 × 7 = 1. ✓
Watch out. Only flip the fraction. Do not change its sign or add anything. The reciprocal of 5 is 1/5, not −5.
Method. The reciprocal of a number is found by turning the fraction upside down (swap the top and the bottom).
- Say what a reciprocal is. The reciprocal of a number is 1 divided by that number. For a fraction it is the same fraction turned upside down.reciprocal of 312
- Change the mixed number to an improper fraction. We need a single top and a single bottom before we can flip it.312 = 72
- Flip the fraction. Swap the numerator and the denominator.72 becomes 27
- Write the answer. If the new fraction has 1 on the bottom, it is a whole number.27
Check. A number times its reciprocal is 1: 72 × 27 = 1. ✓
Watch out. Only flip the fraction. Do not change its sign or add anything. The reciprocal of 5 is 1/5, not −5.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.2 ÷ 12
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.2 = 21
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.21 ÷ 12 = 21 × 21
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 2 = 4
1 × 1 = 1
41 - Write the answer. The bottom number is 1, so the answer is a whole number.41 = 4
Check. Multiply back: 4 × 12 should give 2. 4 × 0.5 = 2 and 2 = 2. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.3 ÷ 14
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.3 = 31
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.31 ÷ 14 = 31 × 41
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 4 = 12
1 × 1 = 1
121 - Write the answer. The bottom number is 1, so the answer is a whole number.121 = 12
Check. Multiply back: 12 × 14 should give 3. 12 × 0.25 = 3 and 3 = 3. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.5 ÷ 13
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.5 = 51
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.51 ÷ 13 = 51 × 31
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.5 × 3 = 15
1 × 1 = 1
151 - Write the answer. The bottom number is 1, so the answer is a whole number.151 = 15
Check. Multiply back: 15 × 13 should give 5. 15 × 0.3333 = 5 and 5 = 5. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.4 ÷ 23
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.4 = 41
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.41 ÷ 23 = 41 × 32
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.41 × 32 = 21 × 31
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 3 = 6
1 × 1 = 1
61 - Write the answer. The bottom number is 1, so the answer is a whole number.61 = 6
Check. Multiply back: 6 × 23 should give 4. 6 × 0.6667 = 4 and 4 = 4. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.6 ÷ 34
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.6 = 61
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.61 ÷ 34 = 61 × 43
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.61 × 43 = 21 × 41
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 4 = 8
1 × 1 = 1
81 - Write the answer. The bottom number is 1, so the answer is a whole number.81 = 8
Check. Multiply back: 8 × 34 should give 6. 8 × 0.75 = 6 and 6 = 6. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.12 ÷ 14
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.12 ÷ 14 = 12 × 41
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.12 × 41 = 11 × 21
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 2 = 2
1 × 1 = 1
21 - Write the answer. The bottom number is 1, so the answer is a whole number.21 = 2
Check. Multiply back: 2 × 14 should give 12. 2 × 0.25 = 0.5 and 12 = 0.5. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.34 ÷ 12
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.34 ÷ 12 = 34 × 21
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.34 × 21 = 32 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 1 = 3
2 × 1 = 2
32 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.3 ÷ 2 = 1 remainder 1
32 = 112
Check. Multiply back: 112 × 12 should give 34. 1.5 × 0.5 = 0.75 and 34 = 0.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.23 ÷ 49
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.23 ÷ 49 = 23 × 94
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.23 × 94 = 11 × 32
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 3 = 3
1 × 2 = 2
32 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.3 ÷ 2 = 1 remainder 1
32 = 112
Check. Multiply back: 112 × 49 should give 23. 1.5 × 0.4444 = 0.6667 and 23 = 0.6667. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.25 ÷ 34
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.25 ÷ 34 = 25 × 43
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 4 = 8
5 × 3 = 15
815 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.8 and 15 have no common factor except 1, so 815 is the answer.
Check. Multiply back: 815 × 34 should give 25. 0.5333 × 0.75 = 0.4 and 25 = 0.4. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.78 ÷ 14
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.78 ÷ 14 = 78 × 41
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.78 × 41 = 72 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.7 × 1 = 7
2 × 1 = 2
72 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.7 ÷ 2 = 3 remainder 1
72 = 312
Check. Multiply back: 312 × 14 should give 78. 3.5 × 0.25 = 0.875 and 78 = 0.875. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.49 ÷ 23
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.49 ÷ 23 = 49 × 32
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.49 × 32 = 23 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 1 = 2
3 × 1 = 3
23 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.2 and 3 have no common factor except 1, so 23 is the answer.
Check. Multiply back: 23 × 23 should give 49. 0.6667 × 0.6667 = 0.4444 and 49 = 0.4444. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.34 ÷ 3
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.3 = 31
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.34 ÷ 31 = 34 × 13
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.34 × 13 = 14 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 1 = 1
4 × 1 = 4
14 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 4 have no common factor except 1, so 14 is the answer.
Check. Multiply back: 14 × 3 should give 34. 0.25 × 3 = 0.75 and 34 = 0.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.45 ÷ 2
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.2 = 21
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.45 ÷ 21 = 45 × 12
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.45 × 12 = 25 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 1 = 2
5 × 1 = 5
25 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.2 and 5 have no common factor except 1, so 25 is the answer.
Check. Multiply back: 25 × 2 should give 45. 0.4 × 2 = 0.8 and 45 = 0.8. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.67 ÷ 4
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.4 = 41
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.67 ÷ 41 = 67 × 14
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.67 × 14 = 37 × 12
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 1 = 3
7 × 2 = 14
314 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.3 and 14 have no common factor except 1, so 314 is the answer.
Check. Multiply back: 314 × 4 should give 67. 0.2143 × 4 = 0.8571 and 67 = 0.8571. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.56 ÷ 10
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.10 = 101
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.56 ÷ 101 = 56 × 110
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.56 × 110 = 16 × 12
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 1 = 1
6 × 2 = 12
112 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 12 have no common factor except 1, so 112 is the answer.
Check. Multiply back: 112 × 10 should give 56. 0.0833 × 10 = 0.8333 and 56 = 0.8333. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.212 ÷ 114
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.212 = (2 × 2 + 1) over 2 = 52
114 = (1 × 4 + 1) over 4 = 54 - Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.52 ÷ 54 = 52 × 45
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.52 × 45 = 11 × 21
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 2 = 2
1 × 1 = 1
21 - Write the answer. The bottom number is 1, so the answer is a whole number.21 = 2
Check. Multiply back: 2 × 114 should give 212. 2 × 1.25 = 2.5 and 212 = 2.5. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.313 ÷ 23
- Change the mixed number to improper fraction. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.313 = (3 × 3 + 1) over 3 = 103
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.103 ÷ 23 = 103 × 32
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.103 × 32 = 51 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.5 × 1 = 5
1 × 1 = 1
51 - Write the answer. The bottom number is 1, so the answer is a whole number.51 = 5
Check. Multiply back: 5 × 23 should give 313. 5 × 0.6667 = 3.3333 and 313 = 3.3333. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.134 ÷ 213
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.134 = (1 × 4 + 3) over 4 = 74
213 = (2 × 3 + 1) over 3 = 73 - Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.74 ÷ 73 = 74 × 37
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.74 × 37 = 14 × 31
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 3 = 3
4 × 1 = 4
34 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.3 and 4 have no common factor except 1, so 34 is the answer.
Check. Multiply back: 34 × 213 should give 134. 0.75 × 2.3333 = 1.75 and 134 = 1.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.412 ÷ 118
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.412 = (4 × 2 + 1) over 2 = 92
118 = (1 × 8 + 1) over 8 = 98 - Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.92 ÷ 98 = 92 × 89
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.92 × 89 = 11 × 41
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 4 = 4
1 × 1 = 1
41 - Write the answer. The bottom number is 1, so the answer is a whole number.41 = 4
Check. Multiply back: 4 × 118 should give 412. 4 × 1.125 = 4.5 and 412 = 4.5. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Decide the operation. “How many pieces can be cut” asks how many times the piece length goes into the ribbon length, so divide the total length by the length of one piece.412 m ÷ 34 m
- Change the mixed number to improper fraction. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.412 = (4 × 2 + 1) over 2 = 92
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.92 ÷ 34 = 92 × 43
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.92 × 43 = 31 × 21
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 2 = 6
1 × 1 = 1
61 - Write the answer. The bottom number is 1, so the answer is a whole number.61 = 6
- Answer in words. The answer is a whole number of pieces and there is nothing left over.6 pieces can be cut.
Check. Multiply back: 6 × 34 should give 412. 6 × 0.75 = 4.5 and 412 = 4.5. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Decide the operation. “Shared equally between 3 people” means divide by 3.35 ÷ 3
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.3 = 31
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.35 ÷ 31 = 35 × 13
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.35 × 13 = 15 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 1 = 1
5 × 1 = 5
15 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 5 have no common factor except 1, so 15 is the answer.
- Answer in words. Each person gets this fraction of the whole pizza.Each person gets 15 of the pizza.
Check. Multiply back: 15 × 3 should give 35. 0.2 × 3 = 0.6 and 35 = 0.6. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. Work out both divisions and compare the answers. If division were commutative they would be the same.
- Work out the first division. Keep, change, flip: turn the second fraction upside down and multiply.12 ÷ 14 = 12 × 41 = 42 = 2
- Work out the second division. Same method, but the numbers are the other way round.14 ÷ 12 = 14 × 21 = 24 = 12
- Compare the answers. The first answer is 2 and the second is a half.2 ≠ 12
- Write the conclusion. Different answers mean the two divisions are not the same.No, they are not the same.
Check. Think of it in words. How many quarters fit into a half? Two. How many halves fit into a quarter? Only half of one. Two and a half are not the same. ✓
Watch out. Division is not commutative: the order matters (unlike multiplication).
Method. Work from left to right, because multiplying and dividing have the same priority in BIDMAS.
- Say the order of working. There are no brackets or powers. Multiplication and division have equal priority, so we work from left to right.23 × 34 ÷ 12: multiply first, then divide
- Multiply first. Cancel the 3s, then multiply the tops and the bottoms.23 × 34 = 24 = 12
- Then divide. Keep, change, flip.12 ÷ 12 = 12 × 21
- Multiply and simplify. Multiply the tops and the bottoms.22 = 1
Check. Any number divided by itself is 1, and the first step gave 1/2 divided by 1/2. ✓
Watch out. Do not do the division first. Work left to right.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Decide the operation. Area of a rectangle = length × width, so width = area ÷ length.Width = 34 ÷ 56
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.34 ÷ 56 = 34 × 65
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.34 × 65 = 32 × 35
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 3 = 9
2 × 5 = 10
910 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.9 and 10 have no common factor except 1, so 910 is the answer.
- Write the answer with its units. The width is a length, so it is in metres.Width = 910 m
Check. Multiply back: 910 × 56 should give 34. 0.9 × 0.8333 = 0.75 and 34 = 0.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Fractions of amounts
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
To find a fraction of an amount, divide by the denominator, then multiply by the numerator. A bar model makes this clear.
Key ideas
- Fraction of an amount: divide by the denominator, multiply by the numerator.
- One quantity as a fraction of another: write the first over the second, then simplify (same units).
- Finding the whole: if you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 35 of 40 means. The denominator 5 says the amount is split into 5 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 40 into 5 equal parts. Take 3 parts.
- Divide 40 by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).40 ÷ 5 = 8
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 8 × 3 gives the size of 3 parts.8 × 3 = 24
- Write the answer. Give the answer with its units.35 of 40 = 24
Check. A different route: multiply first, then divide. 40 × 3 = 120, and 120 ÷ 5 = 24. The answers match. ✓ Also the other 2 parts make 16, and 24 + 16 = 40.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. A fraction compares a part with a whole, so both numbers must be measured in the same units.Part = 15, whole = 40
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1540
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 15: 1, 3, 5, 15
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
HCF = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value the same but gives smaller numbers.15 ÷ 5 = 3 and 40 ÷ 5 = 8
1540 = 38
Check. 15 ÷ 40 = 0.375 and 38 = 0.375. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Write down what we know. The fraction 38 means the whole is split into 8 equal parts, and 3 of those parts together make 27.3 parts = 27
- Find one part: divide by 3. If 3 parts make 27, then one part is 27 shared into 3.27 ÷ 3 = 9
- Find the whole: multiply by 8. The whole is all 8 parts, so multiply the size of one part by 8.9 × 8 = 72
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 72.
Check. Work forwards: 38 of 72 = 72 ÷ 8 × 3 = 9 × 3 = 27. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. Work out what fraction of his original money is left at the end, then work backwards from the £12.
- Find the fraction left after the game. He spent one third, so two thirds stay in his pocket.1 − 13 = 23
- Find the fraction left after the food. He spent half of what was left. Half of 23 is 23 ÷ 2, so the same amount remains.12 × 23 = 26 = 13
- Link this to the £12. He has 13 of his starting money left, and this is £12.13 of the start = £12, so 1 part = £12
- Find the whole by multiplying by the denominator. The whole is 3 thirds, so multiply one third by 3.£12 × 3 = £36
- Write the answer. Give the starting amount.Ben started with £36.
Check. Work forwards: £36. Game: £36 ÷ 3 = £12, so £24 is left. Food: half of £24 = £12, so £12 is left. This matches. ✓
Watch out. “Half of what is left” is half of the remaining money, not half of the original.
Method. Do the steps one at a time. The second fraction is a fraction of the money that is left after the first purchase.
- Find the cost of the coat. 27 of £84: divide by the denominator 7 to get one part, then multiply by the numerator 2.£84 ÷ 7 = £12
£12 × 2 = £24 - Find the money left after the coat. Take the cost away from the starting amount.£84 − £24 = £60
- Find the cost of the bag. It is 12 of the money that is left, which is £60 (not £84).£60 ÷ 2 = £30
- Find the money left at the end. Take the cost of the bag away from what she had before buying it.£60 − £30 = £30
- Write the answer. Answer the question that was asked.Zara has £30 left.
Check. A different route: after the coat she has 57 of her money, and half of that is 514. 514 of £84 = 84 ÷ 14 × 5 = 6 × 5 = £30. ✓
Watch out. “Of what is left” means use the new, smaller amount.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Change the hours into minutes. To find a fraction of an amount it helps to use a unit that divides easily. There are 60 minutes in 1 hour.3 × 60 = 180 minutes
- Say what 25 of 180 minutes means. The denominator 5 says the amount is split into 5 equal parts. The numerator 2 says we take 2 of those parts. The word "of" tells us we are finding a share of an amount.Split 180 minutes into 5 equal parts. Take 2 parts.
- Divide 180 minutes by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).180 minutes ÷ 5 = 36 minutes
- Multiply by the numerator 2. The numerator 2 tells us how many parts we take. Here 36 minutes × 2 gives the size of 2 parts.36 minutes × 2 = 72 minutes
- Write the answer. Give the answer with its units.25 of 180 minutes = 72 minutes
- Change to hours and minutes (to see if it is sensible). 72 minutes is more than 60, so it is 1 hour and some minutes.72 minutes = 1 hour 12 minutes
Check. A different route: multiply first, then divide. 180 × 2 = 360, and 360 ÷ 5 = 72. The answers match. ✓ Also the other 3 parts make 108 minutes, and 72 + 108 = 180.
Watch out. Change to a smaller unit first. Do not try to work with “two fifths of 3” when the answer is needed in minutes.
Exercise 3H Fractions of amounts
Non-calculatorMethod. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 12 of 18 means. The denominator 2 says the amount is split into 2 equal parts. The numerator 1 says we take 1 of those parts. The word "of" tells us we are finding a share of an amount.Split 18 into 2 equal parts. Take 1 part.
- Divide 18 by the denominator 2. Sharing into 2 equal parts finds the size of one part (this is 12 of the amount).18 ÷ 2 = 9
- Take the number of parts the numerator asks for. The numerator is 1, so we take just one part. One part is already our answer.12 of 18 = 9
- Write the answer. Give the answer with its units.12 of 18 = 9
Check. A different route: multiply first, then divide. 18 × 1 = 18, and 18 ÷ 2 = 9. The answers match. ✓ Also the other 1 part make 9, and 9 + 9 = 18.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 15 of 35 means. The denominator 5 says the amount is split into 5 equal parts. The numerator 1 says we take 1 of those parts. The word "of" tells us we are finding a share of an amount.Split 35 into 5 equal parts. Take 1 part.
- Divide 35 by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).35 ÷ 5 = 7
- Take the number of parts the numerator asks for. The numerator is 1, so we take just one part. One part is already our answer.15 of 35 = 7
- Write the answer. Give the answer with its units.15 of 35 = 7
Check. A different route: multiply first, then divide. 35 × 1 = 35, and 35 ÷ 5 = 7. The answers match. ✓ Also the other 4 parts make 28, and 7 + 28 = 35.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 18 of 64 means. The denominator 8 says the amount is split into 8 equal parts. The numerator 1 says we take 1 of those parts. The word "of" tells us we are finding a share of an amount.Split 64 into 8 equal parts. Take 1 part.
- Divide 64 by the denominator 8. Sharing into 8 equal parts finds the size of one part (this is 18 of the amount).64 ÷ 8 = 8
- Take the number of parts the numerator asks for. The numerator is 1, so we take just one part. One part is already our answer.18 of 64 = 8
- Write the answer. Give the answer with its units.18 of 64 = 8
Check. A different route: multiply first, then divide. 64 × 1 = 64, and 64 ÷ 8 = 8. The answers match. ✓ Also the other 7 parts make 56, and 8 + 56 = 64.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 110 of 250 means. The denominator 10 says the amount is split into 10 equal parts. The numerator 1 says we take 1 of those parts. The word "of" tells us we are finding a share of an amount.Split 250 into 10 equal parts. Take 1 part.
- Divide 250 by the denominator 10. Sharing into 10 equal parts finds the size of one part (this is 110 of the amount).250 ÷ 10 = 25
- Take the number of parts the numerator asks for. The numerator is 1, so we take just one part. One part is already our answer.110 of 250 = 25
- Write the answer. Give the answer with its units.110 of 250 = 25
Check. A different route: multiply first, then divide. 250 × 1 = 250, and 250 ÷ 10 = 25. The answers match. ✓ Also the other 9 parts make 225, and 25 + 225 = 250.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 34 of 20 means. The denominator 4 says the amount is split into 4 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 20 into 4 equal parts. Take 3 parts.
- Divide 20 by the denominator 4. Sharing into 4 equal parts finds the size of one part (this is 14 of the amount).20 ÷ 4 = 5
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 5 × 3 gives the size of 3 parts.5 × 3 = 15
- Write the answer. Give the answer with its units.34 of 20 = 15
Check. A different route: multiply first, then divide. 20 × 3 = 60, and 60 ÷ 4 = 15. The answers match. ✓ Also the other 1 part make 5, and 15 + 5 = 20.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 25 of 45 means. The denominator 5 says the amount is split into 5 equal parts. The numerator 2 says we take 2 of those parts. The word "of" tells us we are finding a share of an amount.Split 45 into 5 equal parts. Take 2 parts.
- Divide 45 by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).45 ÷ 5 = 9
- Multiply by the numerator 2. The numerator 2 tells us how many parts we take. Here 9 × 2 gives the size of 2 parts.9 × 2 = 18
- Write the answer. Give the answer with its units.25 of 45 = 18
Check. A different route: multiply first, then divide. 45 × 2 = 90, and 90 ÷ 5 = 18. The answers match. ✓ Also the other 3 parts make 27, and 18 + 27 = 45.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 56 of 42 means. The denominator 6 says the amount is split into 6 equal parts. The numerator 5 says we take 5 of those parts. The word "of" tells us we are finding a share of an amount.Split 42 into 6 equal parts. Take 5 parts.
- Divide 42 by the denominator 6. Sharing into 6 equal parts finds the size of one part (this is 16 of the amount).42 ÷ 6 = 7
- Multiply by the numerator 5. The numerator 5 tells us how many parts we take. Here 7 × 5 gives the size of 5 parts.7 × 5 = 35
- Write the answer. Give the answer with its units.56 of 42 = 35
Check. A different route: multiply first, then divide. 42 × 5 = 210, and 210 ÷ 6 = 35. The answers match. ✓ Also the other 1 part make 7, and 35 + 7 = 42.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 710 of 120 means. The denominator 10 says the amount is split into 10 equal parts. The numerator 7 says we take 7 of those parts. The word "of" tells us we are finding a share of an amount.Split 120 into 10 equal parts. Take 7 parts.
- Divide 120 by the denominator 10. Sharing into 10 equal parts finds the size of one part (this is 110 of the amount).120 ÷ 10 = 12
- Multiply by the numerator 7. The numerator 7 tells us how many parts we take. Here 12 × 7 gives the size of 7 parts.12 × 7 = 84
- Write the answer. Give the answer with its units.710 of 120 = 84
Check. A different route: multiply first, then divide. 120 × 7 = 840, and 840 ÷ 10 = 84. The answers match. ✓ Also the other 3 parts make 36, and 84 + 36 = 120.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 38 of 96 means. The denominator 8 says the amount is split into 8 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 96 into 8 equal parts. Take 3 parts.
- Divide 96 by the denominator 8. Sharing into 8 equal parts finds the size of one part (this is 18 of the amount).96 ÷ 8 = 12
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 12 × 3 gives the size of 3 parts.12 × 3 = 36
- Write the answer. Give the answer with its units.38 of 96 = 36
Check. A different route: multiply first, then divide. 96 × 3 = 288, and 288 ÷ 8 = 36. The answers match. ✓ Also the other 5 parts make 60, and 36 + 60 = 96.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 49 of 72 means. The denominator 9 says the amount is split into 9 equal parts. The numerator 4 says we take 4 of those parts. The word "of" tells us we are finding a share of an amount.Split 72 into 9 equal parts. Take 4 parts.
- Divide 72 by the denominator 9. Sharing into 9 equal parts finds the size of one part (this is 19 of the amount).72 ÷ 9 = 8
- Multiply by the numerator 4. The numerator 4 tells us how many parts we take. Here 8 × 4 gives the size of 4 parts.8 × 4 = 32
- Write the answer. Give the answer with its units.49 of 72 = 32
Check. A different route: multiply first, then divide. 72 × 4 = 288, and 288 ÷ 9 = 32. The answers match. ✓ Also the other 5 parts make 40, and 32 + 40 = 72.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 23 of 45 means. The denominator 3 says the amount is split into 3 equal parts. The numerator 2 says we take 2 of those parts. The word "of" tells us we are finding a share of an amount.Split 45 into 3 equal parts. Take 2 parts.
- Divide 45 by the denominator 3. Sharing into 3 equal parts finds the size of one part (this is 13 of the amount).45 ÷ 3 = 15
- Multiply by the numerator 2. The numerator 2 tells us how many parts we take. Here 15 × 2 gives the size of 2 parts.15 × 2 = 30
- Write the answer. Give the answer with its units.23 of 45 = 30
- Add the units. The amount is money, so write the pound sign.£30
Check. A different route: multiply first, then divide. 45 × 2 = 90, and 90 ÷ 3 = 30. The answers match. ✓ Also the other 1 part make 15, and 30 + 15 = 45.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Change the kilograms into grams. Fractions of an amount are easier with a smaller unit that divides by 5. There are 1000 g in 1 kg.2 kg = 2 × 1000 = 2000 g
- Say what 35 of 2000 g means. The denominator 5 says the amount is split into 5 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 2000 g into 5 equal parts. Take 3 parts.
- Divide 2000 g by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).2000 g ÷ 5 = 400 g
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 400 g × 3 gives the size of 3 parts.400 g × 3 = 1200 g
- Write the answer. Give the answer with its units.35 of 2000 g = 1200 g
- Change back to kilograms (if you like). The question does not say which unit to use, so both answers are correct.1200 g = 1.2 kg
Check. A different route: multiply first, then divide. 2000 × 3 = 6000, and 6000 ÷ 5 = 1200. The answers match. ✓ Also the other 2 parts make 800 g, and 1200 + 800 = 2000.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 58 of 4 m means. The denominator 8 says the amount is split into 8 equal parts. The numerator 5 says we take 5 of those parts. The word "of" tells us we are finding a share of an amount.Split 4 m into 8 equal parts. Take 5 parts.
- Divide 4 m by the denominator 8. Sharing into 8 equal parts finds the size of one part (this is 18 of the amount).4 m ÷ 8 = 0.5 m
- Multiply by the numerator 5. The numerator 5 tells us how many parts we take. Here 0.5 m × 5 gives the size of 5 parts.0.5 m × 5 = 2.5 m
- Write the answer. Give the answer with its units.58 of 4 m = 2.5 m
Check. A different route: multiply first, then divide. 4 × 5 = 20, and 20 ÷ 8 = 2.5. The answers match. ✓ Also the other 3 parts make 1.5 m, and 2.5 + 1.5 = 4.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Change the hour into minutes. There are 60 minutes in 1 hour. 60 divides by 12 exactly, which makes the working easy.1 hour = 60 minutes
- Say what 712 of 60 minutes means. The denominator 12 says the amount is split into 12 equal parts. The numerator 7 says we take 7 of those parts. The word "of" tells us we are finding a share of an amount.Split 60 minutes into 12 equal parts. Take 7 parts.
- Divide 60 minutes by the denominator 12. Sharing into 12 equal parts finds the size of one part (this is 112 of the amount).60 minutes ÷ 12 = 5 minutes
- Multiply by the numerator 7. The numerator 7 tells us how many parts we take. Here 5 minutes × 7 gives the size of 7 parts.5 minutes × 7 = 35 minutes
- Write the answer. Give the answer with its units.712 of 60 minutes = 35 minutes
Check. A different route: multiply first, then divide. 60 × 7 = 420, and 420 ÷ 12 = 35. The answers match. ✓ Also the other 5 parts make 25 minutes, and 35 + 25 = 60.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 34 of 2 litres means. The denominator 4 says the amount is split into 4 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 2 litres into 4 equal parts. Take 3 parts.
- Divide 2 litres by the denominator 4. Sharing into 4 equal parts finds the size of one part (this is 14 of the amount).2 litres ÷ 4 = 0.5 litres
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 0.5 litres × 3 gives the size of 3 parts.0.5 litres × 3 = 1.5 litres
- Write the answer. Give the answer with its units.34 of 2 litres = 1.5 litres
Check. A different route: multiply first, then divide. 2 × 3 = 6, and 6 ÷ 4 = 1.5. The answers match. ✓ Also the other 1 part make 0.5 litres, and 1.5 + 0.5 = 2.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Change the pounds into pence. It is easier to divide 350 by 5 than to divide £3.50. There are 100p in £1.£3.50 = 350p
- Say what 25 of 350p means. The denominator 5 says the amount is split into 5 equal parts. The numerator 2 says we take 2 of those parts. The word "of" tells us we are finding a share of an amount.Split 350p into 5 equal parts. Take 2 parts.
- Divide 350p by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).350p ÷ 5 = 70p
- Multiply by the numerator 2. The numerator 2 tells us how many parts we take. Here 70p × 2 gives the size of 2 parts.70p × 2 = 140p
- Write the answer. Give the answer with its units.25 of 350p = 140p
- Change back to pounds. 100p = £1, so move the decimal point two places.140p = £1.40
Check. A different route: multiply first, then divide. 350 × 2 = 700, and 700 ÷ 5 = 140. The answers match. ✓ Also the other 3 parts make 210p, and 140 + 210 = 350.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Write down what we know. The fraction 14 means the whole is split into 4 equal parts, and 1 of those parts together make 12.1 part = 12
- Find one part: divide by 1. If 1 parts make 12, then one part is 12 shared into 1.12 ÷ 1 = 12
- Find the whole: multiply by 4. The whole is all 4 parts, so multiply the size of one part by 4.12 × 4 = 48
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 48.
Check. Work forwards: 14 of 48 = 48 ÷ 4 × 1 = 12 × 1 = 12. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Write down what we know. The fraction 25 means the whole is split into 5 equal parts, and 2 of those parts together make 18.2 parts = 18
- Find one part: divide by 2. If 2 parts make 18, then one part is 18 shared into 2.18 ÷ 2 = 9
- Find the whole: multiply by 5. The whole is all 5 parts, so multiply the size of one part by 5.9 × 5 = 45
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 45.
Check. Work forwards: 25 of 45 = 45 ÷ 5 × 2 = 9 × 2 = 18. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Write down what we know. The fraction 37 means the whole is split into 7 equal parts, and 3 of those parts together make 21.3 parts = 21
- Find one part: divide by 3. If 3 parts make 21, then one part is 21 shared into 3.21 ÷ 3 = 7
- Find the whole: multiply by 7. The whole is all 7 parts, so multiply the size of one part by 7.7 × 7 = 49
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 49.
Check. Work forwards: 37 of 49 = 49 ÷ 7 × 3 = 7 × 3 = 21. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Write down what we know. The fraction 56 means the whole is split into 6 equal parts, and 5 of those parts together make 35.5 parts = 35
- Find one part: divide by 5. If 5 parts make 35, then one part is 35 shared into 5.35 ÷ 5 = 7
- Find the whole: multiply by 6. The whole is all 6 parts, so multiply the size of one part by 6.7 × 6 = 42
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 42.
Check. Work forwards: 56 of 42 = 42 ÷ 6 × 5 = 7 × 5 = 35. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. Both amounts are numbers of items, so the units already match.Part = 15, whole = 40
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1540
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 15: 1, 3, 5, 15
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
HCF = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value the same but gives smaller numbers.15 ÷ 5 = 3 and 40 ÷ 5 = 8
1540 = 38
Check. 15 ÷ 40 = 0.375 and 38 = 0.375. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. Both amounts are numbers of items, so the units already match.Part = 12, whole = 18
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1218
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
HCF = 6 - Divide the top and the bottom by 6. Dividing both by the same number keeps the value the same but gives smaller numbers.12 ÷ 6 = 2 and 18 ÷ 6 = 3
1218 = 23
Check. 12 ÷ 18 = 0.6667 and 23 = 0.6667. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. Both amounts are in pounds, so the units match.Part = 4, whole = 20
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).420
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 4: 1, 2, 4
Factors of 20: 1, 2, 4, 5, 10, 20
HCF = 4 - Divide the top and the bottom by 4. Dividing both by the same number keeps the value the same but gives smaller numbers.4 ÷ 4 = 1 and 20 ÷ 4 = 5
420 = 15
Check. 4 ÷ 20 = 0.2 and 15 = 0.2. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. The amounts are in different units: 30 minutes and 2 hours. Change the hours into minutes: 2 × 60 = 120 minutes.Part = 30 min, whole = 120 min
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).30120
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
HCF = 30 - Divide the top and the bottom by 30. Dividing both by the same number keeps the value the same but gives smaller numbers.30 ÷ 30 = 1 and 120 ÷ 30 = 4
30120 = 14
Check. 30 ÷ 120 = 0.25 and 14 = 0.25. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 38 of 840 means. The denominator 8 says the amount is split into 8 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 840 into 8 equal parts. Take 3 parts.
- Divide 840 by the denominator 8. Sharing into 8 equal parts finds the size of one part (this is 18 of the amount).840 ÷ 8 = 105
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 105 × 3 gives the size of 3 parts.105 × 3 = 315
- Write the answer. Give the answer with its units.38 of 840 = 315
- Answer in words. The amount we found is the number of pupils who walk.315 pupils walk to school.
Check. A different route: multiply first, then divide. 840 × 3 = 2520, and 2520 ÷ 8 = 315. The answers match. ✓ Also the other 5 parts make 525, and 315 + 525 = 840.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. The pupils who do not walk are the rest of the school. Either subtract the walkers from the total, or find the fraction who do not walk.
- Use the answer to part (a). 315 pupils walk and there are 840 pupils altogether.840 pupils − 315 walkers
- Subtract. The pupils who do not walk are what is left.840 − 315 = 525
- Write the answer. Say what the number means.525 pupils do not walk.
Check. A different route: the fraction who do not walk is 1 − 38 = 58. 840 ÷ 8 = 105, and 105 × 5 = 525. ✓
Watch out. Make sure you answer the question that is asked: this part asks for the pupils who do not walk.
Method. Find the number of red and blue counters, add them, then subtract from the total to find the green ones.
- Find the red counters. 13 of 60: divide by the denominator 3. The numerator is 1, so one part is the answer.60 ÷ 3 = 20 red
- Find the blue counters. 14 of 60: divide by 4.60 ÷ 4 = 15 blue
- Add the red and blue counters. These are the counters that are not green.20 + 15 = 35
- Subtract from the total. The rest are green.60 − 35 = 25 green
Check. A different route: the fraction that is red or blue is 13 + 14 = 412 + 312 = 712. So green is 512, and 60 ÷ 12 × 5 = 25. ✓
Watch out. The green counters are what is left over. Do not find a fraction of the wrong total.
Method. The reduction is a fraction of the price. Find the reduction and take it away from the original price.
- Find the reduction. The price is reduced by 14 of £80. Divide by the denominator 4.£80 ÷ 4 = £20
- Subtract the reduction from the original price. The sale price is the original price take away the reduction.£80 − £20 = £60
- Write the answer. Say what the number means.The sale price is £60.
Check. A different route: if 14 is taken off, 34 is left. 34 of £80 = 80 ÷ 4 × 3 = 20 × 3 = £60. ✓
Watch out. Do not stop at the reduction (£20). The question asks for the sale price.
Method. Do the two steps one at a time, and use the amount left after the first step.
- Find the cost of the book. 13 of £60: divide by 3.60 ÷ 3 = £20
- Find the money left after the book. Take the cost away from the starting amount.60 − 20 = £40
- Find the cost of lunch. It is 14 of what is left, which is £40 (not £60).40 ÷ 4 = £10
- Find the money left at the end. Take the cost of lunch away from what she had before lunch.40 − 10 = £30
Check. A different route: after the book she has 23 of her money. After lunch she has 34 of that: 34 × 23 = 12. Half of £60 = £30. ✓
Watch out. “Of what is left” means use the remaining amount, not the starting amount.
Method. Work out both amounts and compare.
- Work out 23 of 30. Divide by the denominator 3, then multiply by the numerator 2.30 ÷ 3 = 10
10 × 2 = 20 - Work out 34 of 24. Divide by the denominator 4, then multiply by the numerator 3.24 ÷ 4 = 6
6 × 3 = 18 - Compare the two answers. Both are numbers, so the bigger number is the bigger amount.20 > 18
- Write the conclusion. Answer the question that was asked.23 of 30 is bigger.
Check. A different route: 23 × 30 = 603 = 20 and 34 × 24 = 724 = 18. The answers match. ✓
Watch out. Do not compare the fractions 23 and 34 alone. The amounts they are taken from are different (30 and 24).
Method. Track the fraction of the money that is left after each step, then work backwards from the £12.
- Find the fraction left after the game. Ben spent 13, so 23 is left.1 − 13 = 23
- Find the fraction left after the food. He spent half of what was left, so half of 23 remains.12 × 23 = 26 = 13
- Use the £12. He has 13 of his money left, and this is £12.13 of the start = £12
- Find the whole. The whole is 3 thirds, so multiply by 3.3 × £12 = £36
Check. Work forwards from £36. Game: £36 ÷ 3 = £12 spent, so £24 left. Food: half of £24 = £12 spent, so £12 left. This matches. ✓
Watch out. “Half of the rest” means half of what is left after the game.
Method. Find the fraction left after both gifts, then work backwards from the 21 marbles.
- Add the fractions given away. Use a common denominator of 15 (the LCM of 5 and 3).15 + 13 = 315 + 515 = 815
- Find the fraction left. Take the fraction given away from 1 whole (15/15).1515 − 815 = 715
- Use the 21 marbles. 715 of the marbles is 21, so one fifteenth is 21 ÷ 7.1 part = 21 ÷ 7 = 3
- Find the whole. The whole is 15 parts.15 × 3 = 45 marbles
Check. Work forwards from 45. Bo gets 45 ÷ 5 = 9 and Cy gets 45 ÷ 3 = 15, so 45 − 9 − 15 = 21 are left. ✓
Watch out. Add the fractions given away first. Do not take the fractions from 21.
Fractions, decimals and percentages
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Fractions, decimals and percentages are three ways to write the same number. These are the equivalents worth knowing by heart.
| Fraction | Decimal | Percentage |
|---|---|---|
| 12 | 0.5 | 50% |
| 14 | 0.25 | 25% |
| 34 | 0.75 | 75% |
| 15 | 0.2 | 20% |
| 110 | 0.1 | 10% |
| 18 | 0.125 | 12.5% |
| 13 | 0.333… (0.3) | 33.3…% |
Key ideas
- Fraction → decimal: divide the numerator by the denominator.
- Decimal → fraction: use place value (tenths, hundredths, thousandths), then simplify.
- Fraction → percentage: make the denominator 100 (or convert to a decimal and multiply by 100).
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.38 = 3 ÷ 8
- Set up the division 3 ÷ 8. 3 is smaller than 8, so the answer is less than 1. Write 3 as 3.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.3.000… ÷ 8
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.8 into 3 goes 0 times, remainder 3
8 into 30 goes 3 times, remainder 6
8 into 60 goes 7 times, remainder 4
8 into 40 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.3 ÷ 8 = 0.375
Check. Multiply back: 0.375 × 8 = 3, which equals 3. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.65 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.65 = 65100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 65: 1, 5, 13, 65
HCF of 65 and 100 = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value and gives smaller numbers.65 ÷ 5 = 13 and 100 ÷ 5 = 20
65100 = 1320
Check. Divide the top by the bottom: 1320 gives 65 ÷ 100 = 0.65. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.35% = 35 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.7 ÷ 20 = 0.35
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.35 × 100 = 35
- Write the percentage. Add the % sign.720 = 35%
Check. A different route: 720 = 35100 (multiply top and bottom by 5), which is 35%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.04 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.04 = 4100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 4: 1, 2, 4
HCF of 4 and 100 = 4 - Divide the top and the bottom by 4. Dividing both by the same number keeps the value and gives smaller numbers.4 ÷ 4 = 1 and 100 ÷ 4 = 25
4100 = 125
Check. Divide the top by the bottom: 125 gives 4 ÷ 100 = 0.04. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.22.5% = 22.5 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.9 ÷ 40 = 0.225
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.225 × 100 = 22.5
- Write the percentage. Add the % sign.940 = 22.5%
Check. A different route: 22.5% of 100 is 22.5, and 940 of 100 = 100 ÷ 40 × 9 = 2.5 × 9 = 22.5. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To compare numbers in different forms, change them all to the same form. Percentages are often the easiest because they are all out of 100.
- Choose one form. We have a fraction, a decimal and a percentage. Percentages are all “out of 100”, so they are easy to compare.Change everything to a percentage.
- Change 34 to a percentage. Multiply the top and the bottom by 25 to make the denominator 100.34 = 75100 = 75%
- Change 0.7 to a percentage. Multiply a decimal by 100 to get a percentage.0.7 × 100 = 70%
- Keep 68% as it is. It is already a percentage.68%
- Put the percentages in order. Smallest first: compare 75, 70 and 68.68%, 70%, 75%
- Write the original numbers in that order. The question asks for the original values, not the percentages.68%, 0.7, 34
Check. A different route: change everything to decimals. 68% = 0.68, 0.7 stays 0.7, 34 = 0.75. Order: 0.68, 0.7, 0.75. ✓
Watch out. 0.7 is 70%, not 7%. Line the decimals up by place value before comparing.
Exercise 3I Fractions, decimals and percentages
Non-calculatorMethod. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.12 = 1 ÷ 2
- Set up the division 1 ÷ 2. 1 is smaller than 2, so the answer is less than 1. Write 1 as 1.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.1.000… ÷ 2
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.2 into 1 goes 0 times, remainder 1
2 into 10 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.1 ÷ 2 = 0.5
Check. Multiply back: 0.5 × 2 = 1, which equals 1. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.34 = 3 ÷ 4
- Set up the division 3 ÷ 4. 3 is smaller than 4, so the answer is less than 1. Write 3 as 3.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.3.000… ÷ 4
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.4 into 3 goes 0 times, remainder 3
4 into 30 goes 7 times, remainder 2
4 into 20 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.3 ÷ 4 = 0.75
Check. Multiply back: 0.75 × 4 = 3, which equals 3. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.15 = 1 ÷ 5
- Set up the division 1 ÷ 5. 1 is smaller than 5, so the answer is less than 1. Write 1 as 1.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.1.000… ÷ 5
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.5 into 1 goes 0 times, remainder 1
5 into 10 goes 2 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.1 ÷ 5 = 0.2
Check. Multiply back: 0.2 × 5 = 1, which equals 1. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.710 = 7 ÷ 10
- Set up the division 7 ÷ 10. 7 is smaller than 10, so the answer is less than 1. Write 7 as 7.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.7.000… ÷ 10
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.10 into 7 goes 0 times, remainder 7
10 into 70 goes 7 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.7 ÷ 10 = 0.7
Check. Multiply back: 0.7 × 10 = 7, which equals 7. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.38 = 3 ÷ 8
- Set up the division 3 ÷ 8. 3 is smaller than 8, so the answer is less than 1. Write 3 as 3.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.3.000… ÷ 8
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.8 into 3 goes 0 times, remainder 3
8 into 30 goes 3 times, remainder 6
8 into 60 goes 7 times, remainder 4
8 into 40 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.3 ÷ 8 = 0.375
Check. Multiply back: 0.375 × 8 = 3, which equals 3. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.920 = 9 ÷ 20
- Set up the division 9 ÷ 20. 9 is smaller than 20, so the answer is less than 1. Write 9 as 9.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.9.000… ÷ 20
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.20 into 9 goes 0 times, remainder 9
20 into 90 goes 4 times, remainder 10
20 into 100 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.9 ÷ 20 = 0.45
Check. Multiply back: 0.45 × 20 = 9, which equals 9. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.5 has 1 decimal place, so it is in tenths.
- Write the digits as a fraction over 10, 100 or 1000. 1 decimal place means the denominator is 1 followed by 1 zero.0.5 = 510
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 5: 1, 5
HCF of 5 and 10 = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value and gives smaller numbers.5 ÷ 5 = 1 and 10 ÷ 5 = 2
510 = 12
Check. Divide the top by the bottom: 12 gives 5 ÷ 10 = 0.5. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.2 has 1 decimal place, so it is in tenths.
- Write the digits as a fraction over 10, 100 or 1000. 1 decimal place means the denominator is 1 followed by 1 zero.0.2 = 210
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 2: 1, 2
HCF of 2 and 10 = 2 - Divide the top and the bottom by 2. Dividing both by the same number keeps the value and gives smaller numbers.2 ÷ 2 = 1 and 10 ÷ 2 = 5
210 = 15
Check. Divide the top by the bottom: 15 gives 2 ÷ 10 = 0.2. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.75 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.75 = 75100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 75: 1, 3, 5, 15, 25, 75
HCF of 75 and 100 = 25 - Divide the top and the bottom by 25. Dividing both by the same number keeps the value and gives smaller numbers.75 ÷ 25 = 3 and 100 ÷ 25 = 4
75100 = 34
Check. Divide the top by the bottom: 34 gives 75 ÷ 100 = 0.75. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.35 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.35 = 35100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 35: 1, 5, 7, 35
HCF of 35 and 100 = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value and gives smaller numbers.35 ÷ 5 = 7 and 100 ÷ 5 = 20
35100 = 720
Check. Divide the top by the bottom: 720 gives 35 ÷ 100 = 0.35. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.08 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.08 = 8100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 8: 1, 2, 4, 8
HCF of 8 and 100 = 4 - Divide the top and the bottom by 4. Dividing both by the same number keeps the value and gives smaller numbers.8 ÷ 4 = 2 and 100 ÷ 4 = 25
8100 = 225
Check. Divide the top by the bottom: 225 gives 8 ÷ 100 = 0.08. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.125 has 3 decimal places, so it is in thousandths.
- Write the digits as a fraction over 10, 100 or 1000. 3 decimal places means the denominator is 1 followed by 3 zeros.0.125 = 1251000
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 125: 1, 5, 25, 125
HCF of 125 and 1000 = 125 - Divide the top and the bottom by 125. Dividing both by the same number keeps the value and gives smaller numbers.125 ÷ 125 = 1 and 1000 ÷ 125 = 8
1251000 = 18
Check. Divide the top by the bottom: 18 gives 125 ÷ 1000 = 0.125. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.25% = 25 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.1 ÷ 4 = 0.25
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.25 × 100 = 25
- Write the percentage. Add the % sign.14 = 25%
Check. A different route: 14 = 25100 (multiply top and bottom by 25), which is 25%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.60% = 60 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.3 ÷ 5 = 0.6
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.6 × 100 = 60
- Write the percentage. Add the % sign.35 = 60%
Check. A different route: 35 = 60100 (multiply top and bottom by 20), which is 60%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.35% = 35 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.7 ÷ 20 = 0.35
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.35 × 100 = 35
- Write the percentage. Add the % sign.720 = 35%
Check. A different route: 720 = 35100 (multiply top and bottom by 5), which is 35%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.90% = 90 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.9 ÷ 10 = 0.9
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.9 × 100 = 90
- Write the percentage. Add the % sign.910 = 90%
Check. A different route: 910 = 90100 (multiply top and bottom by 10), which is 90%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.12% = 12 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.3 ÷ 25 = 0.12
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.12 × 100 = 12
- Write the percentage. Add the % sign.325 = 12%
Check. A different route: 325 = 12100 (multiply top and bottom by 4), which is 12%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.62.5% = 62.5 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.5 ÷ 8 = 0.625
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.625 × 100 = 62.5
- Write the percentage. Add the % sign.58 = 62.5%
Check. A different route: 62.5% of 100 is 62.5, and 58 of 100 = 100 ÷ 8 × 5 = 12.5 × 5 = 62.5. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.20% = 20100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 20 - Divide the top and the bottom by 20. The same number is divided into both, so the value does not change.20 ÷ 20 = 1 and 100 ÷ 20 = 5
20100 = 15
Check. 15 = 0.2 and 20% = 0.2. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.45% = 45100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 5 - Divide the top and the bottom by 5. The same number is divided into both, so the value does not change.45 ÷ 5 = 9 and 100 ÷ 5 = 20
45100 = 920
Check. 920 = 0.45 and 45% = 0.45. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.12% = 12100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 4 - Divide the top and the bottom by 4. The same number is divided into both, so the value does not change.12 ÷ 4 = 3 and 100 ÷ 4 = 25
12100 = 325
Check. 325 = 0.12 and 12% = 0.12. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.75% = 75100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 75: 1, 3, 5, 15, 25, 75
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 25 - Divide the top and the bottom by 25. The same number is divided into both, so the value does not change.75 ÷ 25 = 3 and 100 ÷ 25 = 4
75100 = 34
Check. 34 = 0.75 and 75% = 0.75. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.5% = 5100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 5: 1, 5
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 5 - Divide the top and the bottom by 5. The same number is divided into both, so the value does not change.5 ÷ 5 = 1 and 100 ÷ 5 = 20
5100 = 120
Check. 120 = 0.05 and 5% = 0.05. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.60% = 60100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 20 - Divide the top and the bottom by 20. The same number is divided into both, so the value does not change.60 ÷ 20 = 3 and 100 ÷ 20 = 5
60100 = 35
Check. 35 = 0.6 and 60% = 0.6. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
| Fraction | Decimal | Percentage |
|---|---|---|
| 35 | ? | ? |
| ? | 0.4 | ? |
| ? | ? | 85% |
Method. Convert between the three forms: fraction → decimal by dividing, decimal → fraction by place value, and decimal → percentage by multiplying by 100.
- Row 1: fraction to decimal. The fraction bar means divide: 3 ÷ 5.35 = 3 ÷ 5 = 0.6
- Row 1: decimal to percentage. Multiply by 100 (percent means out of 100).0.6 × 100 = 60%
- Row 2: decimal to fraction. 0.4 has one decimal place, so it is tenths. Then simplify by dividing the top and bottom by 2.0.4 = 410 = 25
- Row 2: decimal to percentage. Multiply by 100.0.4 × 100 = 40%
- Row 3: percentage to fraction. Percent means out of 100, so write 85 over 100, then simplify by dividing top and bottom by 5.85% = 85100 = 1720
- Row 3: percentage to decimal. Divide by 100.85 ÷ 100 = 0.85
Check. Check each row says the same number in three ways: 35 = 0.6 = 60%, 25 = 0.4 = 40%, 1720 = 0.85 = 85%. ✓
Watch out. A percentage is always out of 100. Do not mix up the divide-by-100 and multiply-by-100 steps.
Method. To compare numbers in mixed forms, change everything to the same form. Percentages are easiest here.
- Choose one form. Percentages are all out of 100, so they are easy to compare.Change everything to a percentage.
- Change 35 to a percentage. Multiply the top and the bottom by 20 to make the denominator 100.35 = 60100 = 60%
- Change 0.55 to a percentage. Multiply by 100.0.55 × 100 = 55%
- Change 58 to a percentage. Divide 5 by 8, then multiply by 100.5 ÷ 8 = 0.625
0.625 × 100 = 62.5% - Put the percentages in order. 58% needs no change. Smallest first: 55, 58, 60, 62.5.55%, 58%, 60%, 62.5%
- Write the original values in that order. The question asks for the original values.0.55, 58%, 35, 58
Check. A different route: change to decimals. 35 = 0.6, 0.55, 58% = 0.58, 58 = 0.625. Order: 0.55, 0.58, 0.6, 0.625. Same order. ✓
Watch out. Compare the decimals digit by digit: 0.55 < 0.58 < 0.6 < 0.625.
Method. Convert both scores to percentages to compare them.
- Choose the form. Raj's score is a percentage, so change Priya's fraction into a percentage as well.Priya 1720, Raj 82%
- Change Priya's score to a percentage. Multiply the top and the bottom by 5 to make the denominator 100.1720 = 85100 = 85%
- Compare. Raj scored 82%.85% > 82%
- Write the conclusion. The bigger score is the better score.Priya did better.
Check. A different route: change Raj's score to a fraction over 20 as a decimal. 82% = 0.82, and 1720 = 0.85. 0.85 > 0.82. ✓
Watch out. Compare numbers only after writing them in the same form.
Method. Divide 1 by 3 and look at the decimal. Then compare it with 0.3.
- Divide 1 by 3. The fraction bar means divide. Add zeros and divide step by step.3 into 1 = 0 remainder 1
3 into 10 = 3 remainder 1
3 into 10 = 3 remainder 1
The same remainder keeps coming back - Say what this means. The remainder is 1 every time, so the digit 3 repeats for ever.1 ÷ 3 = 0.3333…
- Compare with 0.3. 0.3 is exactly three tenths. Use a common denominator of 30 to compare with one third.0.3 = 310 = 930
13 = 1030 - Conclude. 9 is smaller than 10, so 0.3 is smaller than one third. Sara is not quite right.1/3 = 0.3 (recurring), which is slightly more than 0.3
Check. Multiply 0.3 by 3: 0.3 × 3 = 0.9, not 1. But one third × 3 = 1. So 0.3 cannot be exactly one third. ✓
Watch out. A recurring decimal never ends. Writing it as 0.3 is only an approximation.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Decide what is asked. To write a mass as a fraction of 1 kg we write the decimal as a fraction of 1.0.375 kg out of 1 kg
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.375 has 3 decimal places, so it is in thousandths.
- Write the digits as a fraction over 10, 100 or 1000. 3 decimal places means the denominator is 1 followed by 3 zeros.0.375 = 3751000
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 375: 1, 3, 5, 15, 25, 75, 125, 375
HCF of 375 and 1000 = 125 - Divide the top and the bottom by 125. Dividing both by the same number keeps the value and gives smaller numbers.375 ÷ 125 = 3 and 1000 ÷ 125 = 8
3751000 = 38 - Write the answer with its units. The fraction is a fraction of one kilogram.38 kg
Check. Divide the top by the bottom: 38 gives 375 ÷ 1000 = 0.375. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Chapter summary
| Skill | Method |
|---|---|
| Equivalent fractions | Multiply or divide top and bottom by the same number |
| Simplify | Divide top and bottom by their HCF |
| Compare / order | Common denominator (LCM), then compare numerators |
| Mixed ↔ improper | Divide with remainder / (whole × denominator) + numerator |
| Add / subtract | Common denominator, then add or subtract numerators only |
| Multiply | Top × top, bottom × bottom (cancel first) |
| Divide | Keep, change, flip: multiply by the reciprocal |
| Fraction of an amount | Divide by denominator, multiply by numerator |
| Find the whole | Divide by numerator, multiply by denominator |
| Fraction → decimal / % | Numerator ÷ denominator / make denominator 100 |
Can you…?
- I can name the numerator and denominator and say what they mean.
- I can find equivalent fractions and simplify fractions.
- I can order fractions using a common denominator.
- I can change between mixed numbers and improper fractions.
- I can add and subtract fractions and mixed numbers.
- I can multiply and divide fractions and mixed numbers.
- I can find a fraction of an amount and find the whole from a fraction.
- I can change between fractions, decimals and percentages.
Key terms
- Common denominator
- A denominator that is the same for two or more fractions; the LCM of the denominators is the smallest one.
- Denominator
- The bottom number of a fraction: the number of equal parts the whole is split into.
- Equivalent fractions
- Fractions that have the same value, for example 1/2 and 2/4.
- Improper fraction
- A fraction whose numerator is greater than or equal to its denominator.
- Mixed number
- A number made of a whole number and a proper fraction.
- Numerator
- The top number of a fraction: how many parts are taken.
- Proper fraction
- A fraction less than 1: the numerator is smaller than the denominator.
- Reciprocal
- A fraction turned upside down. A number multiplied by its reciprocal is 1.
- Simplest form
- A fraction where the numerator and denominator have no common factor except 1.
- Unit fraction
- A fraction with a numerator of 1.
Exercise 3J Check-up (non-calculator)
Non-calculatorMethod. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.1218
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 3, 6
HCF = 6 - Divide the top by 6. Whatever we do to the top we must also do to the bottom.12 ÷ 6 = 2
- Divide the bottom by 6. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.18 ÷ 6 = 3
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.1218 = 23
Check. A different route: divide in small steps (÷ 2, then ÷ 3). 1218 → 69 → 23. We reach the same fraction. As decimals, 1218 = 0.6667 and 23 = 0.6667. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).34 + 23
- Find a common denominator: the LCM of 4 and 3. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 4: 4, 8, 12
Multiples of 3: 3, 6, 9, 12
LCM = 12 - Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.34 = 912 (× 3)
23 = 812 (× 4) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.912 + 812 = 1712
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.17 ÷ 12 = 1 remainder 5
1712 = 1512
Check. A different route, as decimals: 0.75 + 0.6667 = 1.4167, and our answer 1512 = 1.4167. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).78 − 14
- Find a common denominator: the LCM of 8 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 8: 8
Multiples of 4: 4, 8
LCM = 8 - Rewrite both fractions with denominator 8. Multiply the top and the bottom by the same number so the value does not change.78 = 78 (× 1)
14 = 28 (× 2) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.78 − 28 = 58
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.5 and 8 have no common factor except 1, so 58 is the answer.
Check. A different route, as decimals: 0.875 − 0.25 = 0.625, and our answer 58 = 0.625. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.25 × 34
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.25 × 34 = 15 × 32
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 3 = 3
5 × 2 = 10
310 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.3 and 10 have no common factor except 1, so 310 is the answer.
Check. A different route, as decimals: 0.4 × 0.75 = 0.3, and our answer 310 = 0.3. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.56 ÷ 512
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.56 ÷ 512 = 56 × 125
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.56 × 125 = 11 × 21
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 2 = 2
1 × 1 = 1
21 - Write the answer. The bottom number is 1, so the answer is a whole number.21 = 2
Check. Multiply back: 2 × 512 should give 56. 2 × 0.4167 = 0.8333 and 56 = 0.8333. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. Divide the top by the bottom: the answer is the whole number, the remainder goes over the same bottom number.
- Say what the fraction means. 236 means 23 pieces, each one 16 of a whole. It is called improper because the top is bigger than the bottom, so it is more than one whole.23 pieces of size 16
- Work out how many pieces make one whole. The bottom number tells us this: 6 pieces make 1 whole. So we want to know how many groups of 6 we can make from 23 pieces.6 pieces = 1 whole
- Divide 23 by 6 to find the whole number. Count how many whole groups of 6 fit into 23. 6 × 3 = 18, but 6 × 4 = 24 is too big.23 ÷ 6 = 3 remainder 5
- Find the pieces left over. After 3 whole ones we have used 18 pieces. The rest are the remainder.23 − 18 = 5
- Write the mixed number. The whole number is 3. The 5 leftover pieces of size 16 make the fraction 56. The bottom number stays 6 because the pieces are the same size.236 = 356
Check. Convert back: 3 × 6 + 5 = 18 + 5 = 23. As decimals, 23 ÷ 6 = 3.8333 and 356 = 3.8333. ✓
Watch out. Keep the original denominator. Do not use the remainder as the denominator.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 59 of 72 means. The denominator 9 says the amount is split into 9 equal parts. The numerator 5 says we take 5 of those parts. The word "of" tells us we are finding a share of an amount.Split 72 into 9 equal parts. Take 5 parts.
- Divide 72 by the denominator 9. Sharing into 9 equal parts finds the size of one part (this is 19 of the amount).72 ÷ 9 = 8
- Multiply by the numerator 5. The numerator 5 tells us how many parts we take. Here 8 × 5 gives the size of 5 parts.8 × 5 = 40
- Write the answer. Give the answer with its units.59 of 72 = 40
Check. A different route: multiply first, then divide. 72 × 5 = 360, and 360 ÷ 9 = 40. The answers match. ✓ Also the other 4 parts make 32, and 40 + 32 = 72.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.
- Find the lowest common multiple (LCM) of 8 and 12. The LCM is the smallest number that both denominators divide into. It will be our common denominator.Multiples of 8: 8, 16, 24
Multiples of 12: 12, 24
LCM = 24 - Rewrite the first fraction with denominator 24. Multiply the top and the bottom by the same number: 24 ÷ 8 = 3.58 = 1524 (× 3)
- Rewrite the second fraction with denominator 24. 24 ÷ 12 = 2, so multiply the top and the bottom by 2.712 = 1424 (× 2)
- Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.15 > 14
- Write the answer using the original fractions. The first fraction is bigger.58 > 712
Check. A different route, as decimals: 58 = 0.625 and 712 = 0.5833, so 0.625 > 0.5833. ✓
Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).212 + 123
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.212 = (2 × 2 + 1) over 2 = 52
123 = (1 × 3 + 2) over 3 = 53 - Find a common denominator: the LCM of 2 and 3. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 2: 2, 4, 6
Multiples of 3: 3, 6
LCM = 6 - Rewrite both fractions with denominator 6. Multiply the top and the bottom by the same number so the value does not change.52 = 156 (× 3)
53 = 106 (× 2) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.156 + 106 = 256
- Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.25 ÷ 6 = 4 remainder 1
256 = 416
Check. A different route, as decimals: 2.5 + 1.6667 = 4.1667, and our answer 416 = 4.1667. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.45 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.45 = 45100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 45: 1, 3, 5, 9, 15, 45
HCF of 45 and 100 = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value and gives smaller numbers.45 ÷ 5 = 9 and 100 ÷ 5 = 20
45100 = 920
Check. Divide the top by the bottom: 920 gives 45 ÷ 100 = 0.45. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Total 40 marks. Allow about 40 minutes. Non-calculator.
Exercise 3K Exam-style questions
Non-calculatorMethod. The shape has 10 equal parts, but the fraction has denominator 5. Make an equivalent fraction with denominator 10.
- Compare the denominators. The shape is split into 10 equal parts, but the fraction is in fifths. We need the fraction in tenths.35 = ?10
- Find the multiplier. 10 ÷ 5 = 2, so the bottom was multiplied by 2.10 ÷ 5 = 2
- Multiply the top by the same number. To keep the same value, the top must be multiplied by 2 too.3 × 2 = 6, so 35 = 610
- Shade the parts. The top number 6 tells us how many of the 10 parts to shade.Shade 6 of the 10 parts.
Check. 6 shaded out of 10 is 610. Divide the top and the bottom by 2: 610 = 35. ✓
Watch out. Do not shade 3 parts. The fraction must be written in tenths first.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.1824
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 3, 6
HCF = 6 - Divide the top by 6. Whatever we do to the top we must also do to the bottom.18 ÷ 6 = 3
- Divide the bottom by 6. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.24 ÷ 6 = 4
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.1824 = 34
Check. A different route: divide in small steps (÷ 2, then ÷ 3). 1824 → 912 → 34. We reach the same fraction. As decimals, 1824 = 0.75 and 34 = 0.75. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. To add fractions, give them the same denominator, then add the numerators and keep the denominator.
- Read the question and decide the plan. We can only add pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).23 + 14
- Find a common denominator: the LCM of 3 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.23 = 812 (× 4)
14 = 312 (× 3) - Add the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.812 + 312 = 1112
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.11 and 12 have no common factor except 1, so 1112 is the answer.
Check. A different route, as decimals: 0.6667 + 0.25 = 0.9167, and our answer 1112 = 0.9167. ✓
Watch out. Never add the denominators. The denominator stays the same once the pieces are the same size.
Method. To find a fraction of an amount: divide by the denominator to find one part, then multiply by the numerator.
- Say what 35 of 40 means. The denominator 5 says the amount is split into 5 equal parts. The numerator 3 says we take 3 of those parts. The word "of" tells us we are finding a share of an amount.Split 40 into 5 equal parts. Take 3 parts.
- Divide 40 by the denominator 5. Sharing into 5 equal parts finds the size of one part (this is 15 of the amount).40 ÷ 5 = 8
- Multiply by the numerator 3. The numerator 3 tells us how many parts we take. Here 8 × 3 gives the size of 3 parts.8 × 3 = 24
- Write the answer. Give the answer with its units.35 of 40 = 24
Check. A different route: multiply first, then divide. 40 × 3 = 120, and 120 ÷ 5 = 24. The answers match. ✓ Also the other 2 parts make 16, and 24 + 16 = 40.
Watch out. Divide by the denominator first, then multiply by the numerator. Do not multiply by the denominator.
Method. To subtract fractions, give them the same denominator, then subtract the numerators and keep the denominator.
- Read the question and decide the plan. We can only subtract pieces that are the same size. So first look at the type of numbers and at the denominators (the bottom numbers).56 − 34
- Find a common denominator: the LCM of 6 and 4. We can only add or subtract pieces of the same size, so the denominators must match. The lowest common multiple (LCM) is the smallest number that both denominators divide into.Multiples of 6: 6, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Rewrite both fractions with denominator 12. Multiply the top and the bottom by the same number so the value does not change.56 = 1012 (× 2)
34 = 912 (× 3) - Subtract the numerators and keep the denominator. The denominator only tells us the size of each piece, so it stays the same.1012 − 912 = 112
- Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 12 have no common factor except 1, so 112 is the answer.
Check. A different route, as decimals: 0.8333 − 0.75 = 0.0833, and our answer 112 = 0.0833. ✓
Watch out. Subtract in the right order: the first fraction minus the second.
35 58 712[3 marks]
Method. To put fractions in order, give them all the same denominator, then order the numerators.
- Find the LCM of the denominators 5, 8, 12. This is the common denominator.LCM = 120
- Rewrite every fraction with denominator 120. Multiply the top and bottom of each by the same number.35 = 72120
58 = 75120
712 = 70120 - Put the numerators in order. The order is smallest first.70, 72, 75
- Write the original fractions in that order. The question asks for the original fractions, not the converted ones.712 , 35 , 58
Check. As decimals: 712 = 0.5833, 35 = 0.6, 58 = 0.625.
Watch out. Always give your final answer using the original fractions, not the ones with the common denominator. And never order fractions by their denominators alone.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.214 ÷ 34
- Change the mixed number to improper fraction. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.214 = (2 × 4 + 1) over 4 = 94
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.94 ÷ 34 = 94 × 43
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.94 × 43 = 31 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 1 = 3
1 × 1 = 1
31 - Write the answer. The bottom number is 1, so the answer is a whole number.31 = 3
Check. Multiply back: 3 × 34 should give 214. 3 × 0.75 = 2.25 and 214 = 2.25. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To multiply fractions, multiply the numerators together and the denominators together (cancel first to keep the numbers small).
- Read the question and decide the plan. Multiplying fractions does not need a common denominator. First look at the type of numbers we have.38 × 49
- Cancel common factors first. A number on the top of one fraction and a number on the bottom of the other can be divided by a common factor. This keeps the numbers small and gives an answer already in its simplest form.38 × 49 = 12 × 13
- Multiply the numerators, then the denominators. Multiplying fractions is just multiplying across: top × top and bottom × bottom.1 × 1 = 1
2 × 3 = 6
16 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 6 have no common factor except 1, so 16 is the answer.
Check. A different route, as decimals: 0.375 × 0.4444 = 0.1667, and our answer 16 = 0.1667. ✓
Watch out. Do not find a common denominator when multiplying. That is only for adding and subtracting.
Method. Work out the correct answer, then compare with Dev. Dev's method is a common mistake.
- Find what Dev did. Dev added the tops (2 + 1 = 3) and the bottoms (3 + 5 = 8) to get 3/8.Dev: 23 + 15 = 2 + 13 + 5 = 38
- Find a common denominator. Fractions can only be added when the pieces are the same size. The LCM of 3 and 5 is 15.23 = 1015 (× 5)
15 = 315 (× 3) - Add the numerators and keep the denominator. The pieces are now the same size.1015 + 315 = 1315
- Compare with Dev's answer. The correct answer is 1315, which is about 0.87. Dev's answer 38 is about 0.38.1315 ≠ 38
- Write the conclusion. The answers are different, so Dev is wrong.Dev is not correct.
Check. As decimals: 23 + 15 = 0.667 + 0.2 = 0.867, and 1315 = 0.867. Dev's 38 = 0.375, which is less than 23 alone, so it cannot be right. ✓
Watch out. Never add the denominators. The correct method needs a common denominator first.
Method. Do it in two steps, using the amount left after the first step.
- Find the cost of the book. 38 of £48: divide by the denominator 8, then multiply by the numerator 3.£48 ÷ 8 = £6
£6 × 3 = £18 - Find the money left after the book. Take the cost away from the starting amount.£48 − £18 = £30
- Find the cost of lunch. It is 13 of the money that is left, which is £30 (not £48).£30 ÷ 3 = £10
- Find the money left at the end. Take the cost of lunch away from what she had before lunch.£30 − £10 = £20
Check. A different route: after the book she has 58 of her money. After lunch she has 23 of that: 23 × 58 = 512. 48 ÷ 12 = 4 and 4 × 5 = £20. ✓
Watch out. “Of the money that is left” means use the new, smaller amount.
Method. If you know a fraction of the whole, divide by the numerator to find one part, then multiply by the denominator to find the whole.
- Say what the question gives us. The boys are a fraction of the whole club. We know the number of boys, so we can find the size of the whole club.25 of the club = 18 boys
- Write down what we know. The fraction 25 means the whole is split into 5 equal parts, and 2 of those parts together make 18.2 parts = 18
- Find one part: divide by 2. If 2 parts make 18, then one part is 18 shared into 2.18 ÷ 2 = 9
- Find the whole: multiply by 5. The whole is all 5 parts, so multiply the size of one part by 5.9 × 5 = 45
- Write the answer. This is the whole amount (the number the fraction was taken from).The whole is 45.
- Answer in words. The whole is the number of members in the club.There are 45 members in the club.
Check. Work forwards: 25 of 45 = 45 ÷ 5 × 2 = 9 × 2 = 18. ✓
Watch out. Go up to the whole by multiplying by the denominator. A common mistake is to find the fraction of the part instead.
Method. Find the sugar for one biscuit, then multiply for 20 biscuits.
- Find the sugar for one biscuit. 6 biscuits use 3/4 cup, so share the 3/4 cup between 6 biscuits: divide by 6.34 ÷ 6 = 34 × 16 = 324 = 18 cup
- Multiply for 20 biscuits. Each biscuit needs the same amount.18 × 20 = 208
- Simplify. Divide the top and the bottom by 4.208 = 52
- Write as a mixed number. 5 ÷ 2 = 2 remainder 1.52 = 212 cups
Check. A different route: 20 biscuits = 6 + 6 + 6 + 2 biscuits. Three lots of 6 need 3 × 34 = 94 = 214 cups. The 2 extra biscuits need 26 of 34 = 14 cup. 214 + 14 = 212. ✓
Watch out. Find the amount for 1 first. This is called the unitary method.
Method. Add the fractions used, then subtract the total from the whole tank.
- Find a common denominator. To add fifths and quarters we need pieces of the same size. The LCM of 5 and 4 is 20.25 = 820 (× 4)
14 = 520 (× 5) - Add the fractions used. Add the numerators and keep the denominator.820 + 520 = 1320
- Write 1 whole tank as twentieths. A full tank is all 20 of the 20 equal parts.1 = 2020
- Subtract the fuel used from the whole tank. What is left in the tank.2020 − 1320 = 720
Check. 1320 used + 720 left = 2020 = 1 whole tank. ✓
Watch out. “What is left” means subtract from 1 whole.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.35 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.35 = 35100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 35: 1, 5, 7, 35
HCF of 35 and 100 = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value and gives smaller numbers.35 ÷ 5 = 7 and 100 ÷ 5 = 20
35100 = 720
Check. Divide the top by the bottom: 720 gives 35 ÷ 100 = 0.35. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.38 = 3 ÷ 8
- Set up the division 3 ÷ 8. 3 is smaller than 8, so the answer is less than 1. Write 3 as 3.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.3.000… ÷ 8
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.8 into 3 goes 0 times, remainder 3
8 into 30 goes 3 times, remainder 6
8 into 60 goes 7 times, remainder 4
8 into 40 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.3 ÷ 8 = 0.375
Check. Multiply back: 0.375 × 8 = 3, which equals 3. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Decide the operation. Area of a rectangle = length × width, so width = area ÷ length.Width = 34 ÷ 56
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.34 ÷ 56 = 34 × 65
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.34 × 65 = 32 × 35
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 3 = 9
2 × 5 = 10
910 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.9 and 10 have no common factor except 1, so 910 is the answer.
- Write the answer with its units. The width is a length, so it is in metres.Width = 910 m
Check. Multiply back: 910 × 56 should give 34. 0.9 × 0.8333 = 0.75 and 34 = 0.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.