GCSE Maths: Understanding fractions (Foundation)
Part of the GCSE Maths fractions chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Understanding fractions
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.