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Primary Maths · Year 5Chapter 6 · Fractions
Year 5 · Chapter 6

Year 5 Equivalent Fractions and Mixed Numbers: worked examples and questions with answers

By the end of this chapter you will be able to

  • Find equivalent fractions with bars, number lines and by multiplying or dividing the top and the bottom by the same number, including tenths and hundredths
  • Simplify a fraction by dividing the top and the bottom by a common factor
  • Compare and order fractions (and mixed numbers) by changing them to a common denominator
  • Change improper fractions into mixed numbers and mixed numbers into improper fractions, and see a fraction as a division
  • Add and subtract fractions whose denominators are the same or multiples of one another
  • Add and subtract mixed numbers, with and without crossing a whole number
  • Multiply fractions and mixed numbers by whole numbers, and find fractions of amounts
  • Solve word problems with bar models and explain your reasoning

You should already know

  • Fractions of shapes, sets and amounts, and equivalent fractions in Year 4 (Year 4 Chapter 5)
  • Adding and subtracting fractions with the same denominator
  • Times tables and division facts to 12 × 12
  • Factors and multiples (Year 5 Chapter 3)

Where this leads

  • Decimals and percentages: tenths and hundredths as decimals (Chapter 7)
  • Measures with fractions and decimals (Chapter 8)
  • Adding and subtracting fractions with any denominators, and multiplying and dividing fractions in Year 6

National Curriculum for England, Key Stage 2 mathematics, Year 5. Content: Compare and order fractions whose denominators are all multiples of the same number, Identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths, Recognise mixed numbers and improper fractions and convert from one form to the other, and write mathematical statements > 1 as a mixed number (e.g. 2/5 + 4/5 = 6/5 = 1 1/5), Add and subtract fractions with the same denominator and denominators that are multiples of the same number, Multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
FWhat fraction of the bar is shaded?
2
FWork out 25 of 30.
3
FWrite the missing number.   12 = ?8
4
FWork out 59 + 39.
5
FWhat number is the arrow pointing to? Write it as a mixed number.
0123▼
6.1

Equivalent fractions

Year 5
Identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Equivalent fractionsComing soon

Equivalent fractions are different fractions that are worth the same amount. Half a pizza is the same amount of pizza as two quarters or four eighths. Look at the bars. They are all the same length. In each bar the shaded part ends at the same place, so 12, 24 and 48 are equal.

122448
The dashed line shows that all three shaded parts end in the same place: 12 = 24 = 48.

Why does it work? When you cut every part of a bar into 2 smaller equal parts, there are twice as many parts (the bottom number doubles) and you shade twice as many parts (the top number doubles). The amount shaded has not changed. So the rule is: multiply the top and the bottom by the same number. You can also divide the top and the bottom by the same number, as long as both numbers divide exactly.

0123014601812
On number lines from 0 to 1 the same point is 23, 46 and 812. Equivalent fractions live at the same point.

Tenths and hundredths. A hundred square is cut into 100 small squares. Each row is one tenth, and each row has 10 small squares. So 1 tenth is the same as 10 hundredths. This means 310 = 30100, and 12 = 510 = 50100. (In Chapter 7 you will write these as decimals: 0.3 and 0.5.)

3 rows shaded = 310. That is 30 small squares = 30100.

Key ideas

  • Equivalent fractions show the same amount.
  • To make an equivalent fraction, multiply the top and the bottom by the same number: 23 = 2 × 43 × 4 = 812.
  • You can also divide the top and the bottom by the same number: 69 = 23.
  • 1 tenth = 10 hundredths, so 710 = 70100.
  • Equivalent fractions are at the same point on a number line.
Worked example 6.1.1
Find the missing number.   23 = ?12

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Equivalent fractions are made by multiplying the top and the bottom by the same number.

  1. Look at the bottom numbers. What do we multiply 3 by to get 12? This tells us how the fraction has been changed. We must do the same to the other number.
    3 × 4 = 12, so the multiplier is 4.
  2. Do the same to the top number. To keep the fraction the same size, the top and the bottom must both be multiplied by 4.
    2 × 4 = 8
  3. Write the equivalent fractions. Now both numbers are filled in.
    23 = 812
    23812

Check. 23 and 812 both simplify to 23. ✓

Watch out. Whatever you do to the top, do to the bottom. Never add or subtract the same number.

Worked example 6.1.2
Find the missing number.   34 = 9?

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Equivalent fractions are made by multiplying the top and the bottom by the same number.

  1. Look at the top numbers. What do we multiply 3 by to get 9? This tells us how the fraction has been changed. We must do the same to the other number.
    3 × 3 = 9, so the multiplier is 3.
  2. Do the same to the bottom number. To keep the fraction the same size, the top and the bottom must both be multiplied by 3.
    4 × 3 = 12
  3. Write the equivalent fractions. Now both numbers are filled in.
    34 = 912
    34912

Check. 34 and 912 both simplify to 34. ✓

Watch out. Whatever you do to the top, do to the bottom. Never add or subtract the same number.

Worked example 6.1.3
Find the missing number.   1216 = ?4

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Equivalent fractions are made by divideing the top and the bottom by the same number.

  1. Look at the bottom numbers. What do we divide 16 by to get 4? This tells us how the fraction has been changed. We must do the same to the other number.
    16 ÷ 4 = 4, so the divider is 4.
  2. Do the same to the top number. To keep the fraction the same size, the top and the bottom must both be divided by 4.
    12 ÷ 4 = 3
  3. Write the equivalent fractions. Now both numbers are filled in.
    1216 = 34
    121634

Check. 1216 and 34 both simplify to 34. ✓

Watch out. Whatever you do to the top, do to the bottom. Never add or subtract the same number.

Worked example 6.1.4
Find the missing number.   310 = ?100

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Equivalent fractions are made by multiplying the top and the bottom by the same number.

  1. Look at the bottom numbers. What do we multiply 10 by to get 100? This tells us how the fraction has been changed. We must do the same to the other number.
    10 × 10 = 100, so the multiplier is 10.
  2. Do the same to the top number. To keep the fraction the same size, the top and the bottom must both be multiplied by 10.
    3 × 10 = 30
  3. Write the equivalent fractions. Now both numbers are filled in.
    310 = 30100

Check. 310 and 30100 both simplify to 310. ✓

Watch out. Whatever you do to the top, do to the bottom. Never add or subtract the same number.

Worked example 6.1.5
Find the missing number.   40100 = ?10

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Equivalent fractions are made by divideing the top and the bottom by the same number.

  1. Look at the bottom numbers. What do we divide 100 by to get 10? This tells us how the fraction has been changed. We must do the same to the other number.
    100 ÷ 10 = 10, so the divider is 10.
  2. Do the same to the top number. To keep the fraction the same size, the top and the bottom must both be divided by 10.
    40 ÷ 10 = 4
  3. Write the equivalent fractions. Now both numbers are filled in.
    40100 = 410

Check. 40100 and 410 both simplify to 25. ✓

Watch out. Whatever you do to the top, do to the bottom. Never add or subtract the same number.

Worked example 6.1.6
Find the missing number.   56 = 35?

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Equivalent fractions are made by multiplying the top and the bottom by the same number.

  1. Look at the top numbers. What do we multiply 5 by to get 35? This tells us how the fraction has been changed. We must do the same to the other number.
    5 × 7 = 35, so the multiplier is 7.
  2. Do the same to the bottom number. To keep the fraction the same size, the top and the bottom must both be multiplied by 7.
    6 × 7 = 42
  3. Write the equivalent fractions. Now both numbers are filled in.
    56 = 3542

Check. 56 and 3542 both simplify to 56. ✓

Watch out. Whatever you do to the top, do to the bottom. Never add or subtract the same number.

Worked example 6.1.7
Write three fractions that are equivalent to 25.

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Multiply the top and the bottom of the fraction by the same number each time. Every answer is an equivalent fraction.

  1. Multiply the top and the bottom by 2. Multiplying both numbers by 2 cuts every part into 2 smaller equal parts, so the amount stays the same.
    25 = 2 × 25 × 2 = 410
  2. Multiply the top and the bottom by 3. Multiplying both numbers by 3 cuts every part into 3 smaller equal parts, so the amount stays the same.
    25 = 2 × 35 × 3 = 615
  3. Multiply the top and the bottom by 4. Multiplying both numbers by 4 cuts every part into 4 smaller equal parts, so the amount stays the same.
    25 = 2 × 45 × 4 = 820
  4. Write your list. These fractions are all equivalent to the first one.
    410, 615, 820
    25410615820

Check. Each one simplifies back to 25. ✓

Watch out. Multiply the top and the bottom. Adding the same number to both changes the value.

Worked example 6.1.8
Are 69 and 46 equivalent fractions?

Key words. Equivalent fractions are different fractions that are worth the same amount, like 12 and 24. The numerator is the top number. The denominator is the bottom number.

Method. Write both fractions in their simplest form. If the simplest forms are the same, the fractions are equivalent.

  1. Write 69 in its simplest form. Divide the top and the bottom by their highest common factor. If both are already as small as possible, leave them.
    69 = 6 ÷ 39 ÷ 3 = 23
  2. Write 46 in its simplest form. Do the same for the other fraction.
    46 = 4 ÷ 26 ÷ 2 = 23
  3. Compare the simplest forms. Equivalent fractions have the same simplest form.
    23 and 23 are the same, so the fractions are equivalent.

Check. As a picture: both bars shade the same amount. ✓

Watch out. Do not just look at the numbers. 4/6 and 6/9 look different but are equivalent.

Watch out

Never add the same number to the top and the bottom. 12 is not equal to 23 (that is adding 1 to both). Always multiply or divide the top and the bottom by the same number.

Exercise 6A  Equivalent fractions

Non-calculator
1
FLook at each pair of bars. Write the missing number.
(a)
13?6
13 = ?6
(b)
34?12
34 = ?12
(c)
2?410
2? = 410
2
FWrite the missing numerators.
(a)13 = ?6
(b)25 = ?10
(c)34 = ?12
(d)56 = ?18
(e)710 = ?100
(f)49 = ?27
3
FWrite the missing denominators.
(a)23 = 8?
(b)35 = 12?
(c)58 = 15?
(d)710 = 70?
(e)14 = 5?
(f)37 = 21?
4
FDivide the top and the bottom to find the missing number.
(a)68 = ?4
(b)1015 = ?3
(c)1216 = ?4
(d)40100 = ?10
(e)912 = 3?
(f)2050 = 2?
5
FLook at the hundred square.
(a)Write the shaded part as a fraction with a denominator of 100.
(b)How many tenths are shaded?   30100 = ?10
6
FComplete each line.
(a)710 = ?100
(b)60100 = ?10
(c)12 = ?10
(d)35 = ?10
(e)910 = ?100
(f)14 = ?100
7
FWrite three equivalent fractions for each fraction by multiplying the top and the bottom by 2, 3 and 4.
(a)14
(b)23
(c)35
8
FThe three number lines show the same point. Write the two missing numbers.
01?01?01?
(a)23 = ?6
(b)23 = ?12
9
FAre these fractions equivalent? Write yes or no.
(a)34 and 912
(b)25 and 415
(c)56 and 1012
(d)38 and 612
10
RWhich is the odd one out? Explain how you know.    24   36   410   510
11
FWhich of these fractions are equal to 34?    68   912   812   1216   1524
12
PSElla eats 48 of a pizza. Sam eats 24 of an identical pizza. Who eats more? Explain.
13
RLeo says, “23 = 45, because I added 2 to the top and to the bottom.” What has Leo done wrong? What should he have written?
14
RAlways, sometimes or never?   “Adding 1 to the top and to the bottom of a fraction gives an equivalent fraction.”
15
RChallenge! Find a fraction that is equivalent to 23 where the top number and the bottom number add up to 35.
6.2

Simplifying fractions

Year 5
Identify, name and write equivalent fractions of a given fraction (simplest form), Use common factors

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Simplifying fractionsComing soon

The fraction 812 is the same amount as 23, but 23 uses smaller numbers. Writing a fraction with smaller numbers is called simplifying. We simplify by dividing the top and the bottom by the same number. The number we divide by must be a common factor: it must go into both the top and the bottom exactly.

The bar is cut into 12 parts and 8 are shaded: 812. Put the parts in groups of 4 (the thick lines). There are 3 groups and 2 of them are shaded: 23.

The best number to divide by is the highest common factor (HCF): the biggest number that goes into both. List the factors of the top and of the bottom. For 8 and 12: factors of 8 are 1, 2, 4, 8 and factors of 12 are 1, 2, 3, 4, 6, 12. The biggest in both lists is 4. So 812 = 8 ÷ 412 ÷ 4 = 23. A fraction is in its simplest form when the only common factor is 1. If you cannot spot the biggest factor, divide in small steps (by 2, then by 2 again...) until nothing else goes into both numbers.

Key ideas

  • Simplify a fraction: divide the top and the bottom by the same common factor.
  • The highest common factor does it in one step. Dividing in small steps works too.
  • A fraction is in its simplest form when the only common factor of the top and bottom is 1.
  • Simplifying does not change the value: 812 = 23.
  • In a story, write part over whole first. 12 of 30 children is 1230 = 25.
Worked example 6.2.1
Write 68 in its simplest form.

Key words. To simplify a fraction is to write it with smaller numbers. A common factor is a number that divides exactly into both the top and the bottom. The simplest form is when the only common factor is 1.

Method. Divide the top and the bottom by a common factor. The biggest common factor (the highest common factor, HCF) does it in one go.

  1. List the factors of 6 and of 8. A factor divides exactly into the number. We need the factors both numbers share (the common factors).
    Factors of 6: 1, 2, 3, 6
    Factors of 8: 1, 2, 4, 8
    Common factors: 1, 2
  2. Pick the biggest common factor: 2. Divide the top and the bottom by 2. Dividing both numbers by the same number does not change the value of the fraction. It just writes it with smaller numbers.
    68 = 6 ÷ 28 ÷ 2 = 34
  3. Check it is in its simplest form. If 3 and 4 still shared a factor, we could go on.
    The only common factor of 3 and 4 is 1, so 34 is the simplest form.

Check. Multiply back: 3 × 2 = 6 and 4 × 2 = 8. ✓

Watch out. Divide the top and the bottom. Subtracting the same number from both changes the fraction.

Worked example 6.2.2
Write 1218 in its simplest form.

Key words. To simplify a fraction is to write it with smaller numbers. A common factor is a number that divides exactly into both the top and the bottom. The simplest form is when the only common factor is 1.

Method. Divide the top and the bottom by a common factor. The biggest common factor (the highest common factor, HCF) does it in one go.

  1. List the factors of 12 and of 18. A factor divides exactly into the number. We need the factors both numbers share (the common factors).
    Factors of 12: 1, 2, 3, 4, 6, 12
    Factors of 18: 1, 2, 3, 6, 9, 18
    Common factors: 1, 2, 3, 6
  2. Pick the biggest common factor: 6. Divide the top and the bottom by 6. Dividing both numbers by the same number does not change the value of the fraction. It just writes it with smaller numbers.
    1218 = 12 ÷ 618 ÷ 6 = 23
  3. Check it is in its simplest form. If 2 and 3 still shared a factor, we could go on.
    The only common factor of 2 and 3 is 1, so 23 is the simplest form.

Check. Multiply back: 2 × 6 = 12 and 3 × 6 = 18. ✓

Watch out. Divide the top and the bottom. Subtracting the same number from both changes the fraction.

Worked example 6.2.3
Write 1525 in its simplest form.

Key words. To simplify a fraction is to write it with smaller numbers. A common factor is a number that divides exactly into both the top and the bottom. The simplest form is when the only common factor is 1.

Method. Divide the top and the bottom by a common factor. The biggest common factor (the highest common factor, HCF) does it in one go.

  1. List the factors of 15 and of 25. A factor divides exactly into the number. We need the factors both numbers share (the common factors).
    Factors of 15: 1, 3, 5, 15
    Factors of 25: 1, 5, 25
    Common factors: 1, 5
  2. Pick the biggest common factor: 5. Divide the top and the bottom by 5. Dividing both numbers by the same number does not change the value of the fraction. It just writes it with smaller numbers.
    1525 = 15 ÷ 525 ÷ 5 = 35
  3. Check it is in its simplest form. If 3 and 5 still shared a factor, we could go on.
    The only common factor of 3 and 5 is 1, so 35 is the simplest form.

Check. Multiply back: 3 × 5 = 15 and 5 × 5 = 25. ✓

Watch out. Divide the top and the bottom. Subtracting the same number from both changes the fraction.

Worked example 6.2.4
Write 40100 in its simplest form.

Key words. To simplify a fraction is to write it with smaller numbers. A common factor is a number that divides exactly into both the top and the bottom. The simplest form is when the only common factor is 1.

Method. Divide the top and the bottom by a common factor. The biggest common factor (the highest common factor, HCF) does it in one go.

  1. List the factors of 40 and of 100. A factor divides exactly into the number. We need the factors both numbers share (the common factors).
    Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
    Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
    Common factors: 1, 2, 4, 5, 10, 20
  2. Pick the biggest common factor: 20. Divide the top and the bottom by 20. Dividing both numbers by the same number does not change the value of the fraction. It just writes it with smaller numbers.
    40100 = 40 ÷ 20100 ÷ 20 = 25
  3. Check it is in its simplest form. If 2 and 5 still shared a factor, we could go on.
    The only common factor of 2 and 5 is 1, so 25 is the simplest form.

Check. Multiply back: 2 × 20 = 40 and 5 × 20 = 100. ✓

Watch out. Divide the top and the bottom. Subtracting the same number from both changes the fraction.

Worked example 6.2.5
Write 1824 in its simplest form.

Key words. To simplify a fraction is to write it with smaller numbers. A common factor is a number that divides exactly into both the top and the bottom. The simplest form is when the only common factor is 1.

Method. Divide the top and the bottom by a common factor. The biggest common factor (the highest common factor, HCF) does it in one go.

  1. List the factors of 18 and of 24. A factor divides exactly into the number. We need the factors both numbers share (the common factors).
    Factors of 18: 1, 2, 3, 6, 9, 18
    Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
    Common factors: 1, 2, 3, 6
  2. Pick the biggest common factor: 6. Divide the top and the bottom by 6. Dividing both numbers by the same number does not change the value of the fraction. It just writes it with smaller numbers.
    1824 = 18 ÷ 624 ÷ 6 = 34
  3. Check it is in its simplest form. If 3 and 4 still shared a factor, we could go on.
    The only common factor of 3 and 4 is 1, so 34 is the simplest form.

Check. Multiply back: 3 × 6 = 18 and 4 × 6 = 24. ✓

Watch out. Divide the top and the bottom. Subtracting the same number from both changes the fraction.

Worked example 6.2.6
Is 712 in its simplest form? Explain.

Key words. To simplify a fraction is to write it with smaller numbers. A common factor is a number that divides exactly into both the top and the bottom. The simplest form is when the only common factor is 1.

Method. A fraction is in its simplest form when the only number that divides into both the top and the bottom is 1.

  1. List the factors of 7 and of 12. A factor divides exactly into the number. We need the factors both numbers share (the common factors).
    Factors of 7: 1, 7
    Factors of 12: 1, 2, 3, 4, 6, 12
    Common factors: 1
  2. Look for a common factor bigger than 1. If the only common factor is 1, we cannot make the numbers smaller.
    The only common factor is 1, so 712 is already in its simplest form.
  3. Write the answer.
    712

Check. There is no whole number bigger than 1 that goes into both numbers. ✓

Watch out. Do not simplify just because the numbers are odd or even. Always look for a common factor.

Worked example 6.2.7
In Class 5 there are 30 children. 12 of them walk to school. What fraction walk to school? Give your answer in its simplest form.

Key words. To simplify a fraction is to write it with smaller numbers. A common factor is a number that divides exactly into both the top and the bottom. The simplest form is when the only common factor is 1.

Method. Divide the top and the bottom by a common factor. The biggest common factor (the highest common factor, HCF) does it in one go.

  1. Write the fraction: the part over the whole. 12 of the 30 children walk to school. The part goes on top and the whole goes on the bottom.
    1230
  2. List the factors of 12 and of 30. A factor divides exactly into the number. We need the factors both numbers share (the common factors).
    Factors of 12: 1, 2, 3, 4, 6, 12
    Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
    Common factors: 1, 2, 3, 6
  3. Pick the biggest common factor: 6. Divide the top and the bottom by 6. Dividing both numbers by the same number does not change the value of the fraction. It just writes it with smaller numbers.
    1230 = 12 ÷ 630 ÷ 6 = 25
  4. Check it is in its simplest form. If 2 and 5 still shared a factor, we could go on.
    The only common factor of 2 and 5 is 1, so 25 is the simplest form.

Check. Multiply back: 2 × 6 = 12 and 5 × 6 = 30. ✓

Watch out. Divide the top and the bottom. Subtracting the same number from both changes the fraction.

Worked example 6.2.8
Simplify 4864 by dividing in small steps.

Key words. To simplify a fraction is to write it with smaller numbers. A common factor is a number that divides exactly into both the top and the bottom. The simplest form is when the only common factor is 1.

Method. Divide the top and the bottom by any common factor you can see. Keep going until no common factor is left.

  1. Both numbers can be divided by 2. 48 and 64 are both in the 2 times table, so 2 is a common factor.
    4864 = 48 ÷ 264 ÷ 2 = 2432
  2. Both numbers can be divided by 2. 24 and 32 are both in the 2 times table, so 2 is a common factor.
    2432 = 24 ÷ 232 ÷ 2 = 1216
  3. Both numbers can be divided by 2. 12 and 16 are both in the 2 times table, so 2 is a common factor.
    1216 = 12 ÷ 216 ÷ 2 = 68
  4. Both numbers can be divided by 2. 6 and 8 are both in the 2 times table, so 2 is a common factor.
    68 = 6 ÷ 28 ÷ 2 = 34
  5. Check there is no common factor left. Stop when the only number that goes into both is 1.
    The only common factor of 3 and 4 is 1, so 34 is the simplest form.

Check. 34 = 4864: 3 × 16 = 48 and 4 × 16 = 64. ✓

Watch out. You do not need the biggest common factor straight away. Small steps get there too.

Watch out

Divide the top and the bottom. If you take the same number away from both (1015 → 510) the fraction changes. Also check your answer: 612 simplifies to 12, not to 36 (which can still be simplified).

Exercise 6B  Simplifying fractions

Non-calculator
1
FSimplify each fraction.
(a)24
(b)48
(c)312
(d)26
(e)510
(f)410
2
FSimplify each fraction.
(a)39
(b)515
(c)1025
(d)615
(e)1421
(f)820
3
FWrite each fraction in its simplest form.
(a)812
(b)915
(c)1624
(d)1830
(e)2032
(f)2436
4
FWrite each hundredth in its simplest form.
(a)25100
(b)60100
(c)45100
(d)75100
(e)8100
(f)36100
5
FWhat fraction of the bar is shaded? Write it in its simplest form.
6
FIs each fraction in its simplest form? Write yes, or write no and give the simplest form.
(a)49
(b)610
(c)715
(d)912
(e)1120
7
PSWrite each answer as a fraction in its simplest form.
(a)12 of the 30 pupils in a class walk to school. What fraction walk?
(b)Mia reads 18 of the 24 pages of a comic. What fraction has she read?
(c)There are 45 stickers in a pack. 15 are blue. What fraction are blue?
(d)A ribbon is 50 cm long. Amira cuts off 20 cm. What fraction of the ribbon is cut off?
8
FWhich of these fractions are equal to 23?    812   1015   610   915   1421
9
PSA fraction simplifies to 35. Its denominator is 35. What is its numerator?
10
RTom simplifies 1015 by taking 5 away from the top and from the bottom. He gets 510. What has he done wrong? What is the correct answer?
11
RPriya simplifies 2436 and gets 1218. Is her answer in its simplest form? Explain.
12
RAlways, sometimes or never?
(a)A fraction with an even top number and an even bottom number can be simplified.
(b)A proper fraction with a prime number on the bottom can be simplified.
13
RChallenge! Write all the fractions with a denominator of 12 that are bigger than 0 and smaller than 1 and are in their simplest form.
6.3

Comparing and ordering fractions

Year 5
Compare and order fractions whose denominators are all multiples of the same number, Fraction walls and common denominators

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Comparing and ordering fractionsComing soon

Which is bigger, 34 or 58? A fraction wall helps. Each row is one whole cut into equal bricks. Bricks in the same column line up, so you can see which is longer.

11212141414141818181818181818
The three quarters (blue) is longer than the five eighths (orange), so 34 > 58.

There are three useful rules.

1. Same denominator: the bigger numerator is the bigger fraction. 58 > 38. 2. Same numerator: the bigger denominator means smaller parts, so the fraction is smaller. 18 < 14. 3. Different denominators: change the fractions to a common denominator. When one denominator is a multiple of the other (like 4 and 8) use the bigger one: 34 = 68, so 68 > 58.

3458
The dashed lines show where each shaded part ends. The longer shading is the bigger fraction.

To order a list, change every fraction to the same denominator, then order the numerators. Remember to write the original fractions in your answer. You can also compare with 12: a fraction is more than a half when its top number is more than half of its bottom number. For mixed numbers and improper fractions, change them all to improper fractions first.

Key ideas

  • Same denominator: the bigger numerator is the bigger fraction.
  • Same numerator: the smaller denominator is the bigger fraction (halves are bigger than quarters).
  • Different denominators: change to a common denominator (a multiple of both), then compare the numerators.
  • To order, change every fraction, order the numerators, then write the original fractions.
  • Mixed numbers: compare the whole numbers first, or write everything as improper fractions.
Worked example 6.3.1
Write < or > in the box.   34  [ ? ]  58

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Make the denominators the same. Then the fraction with the bigger numerator is the bigger fraction.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 8, so 8 will do.
    8 ÷ 4 = 2, so 4 goes into 8
  2. Change both fractions to the common denominator. Multiply the top and the bottom by the same number each time, so the value does not change.
    34 = 3 × 24 × 2 = 68
    58
  3. Compare the top numbers. With the same bottom number, the fraction with the bigger top number is bigger.
    6 > 5
  4. Write the answer. Use the original fractions in your answer.
    34 > 58
3458
The picture shows the same: the bar that is shaded further is the bigger fraction.

Watch out. A bigger denominator does not mean a bigger fraction: 1/8 is smaller than 1/4.

Worked example 6.3.2
Write < or > in the box.   710  [ ? ]  35

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Make the denominators the same. Then the fraction with the bigger numerator is the bigger fraction.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 10, so 10 will do.
    10 ÷ 5 = 2, so 5 goes into 10
  2. Change both fractions to the common denominator. Multiply the top and the bottom by the same number each time, so the value does not change.
    710
    35 = 3 × 25 × 2 = 610
  3. Compare the top numbers. With the same bottom number, the fraction with the bigger top number is bigger.
    7 > 6
  4. Write the answer. Use the original fractions in your answer.
    710 > 35
71035
The picture shows the same: the bar that is shaded further is the bigger fraction.

Watch out. A bigger denominator does not mean a bigger fraction: 1/8 is smaller than 1/4.

Worked example 6.3.3
Write < or > in the box.   23  [ ? ]  712

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Make the denominators the same. Then the fraction with the bigger numerator is the bigger fraction.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 12, so 12 will do.
    12 ÷ 3 = 4, so 3 goes into 12
  2. Change both fractions to the common denominator. Multiply the top and the bottom by the same number each time, so the value does not change.
    23 = 2 × 43 × 4 = 812
    712
  3. Compare the top numbers. With the same bottom number, the fraction with the bigger top number is bigger.
    8 > 7
  4. Write the answer. Use the original fractions in your answer.
    23 > 712
23712
The picture shows the same: the bar that is shaded further is the bigger fraction.

Watch out. A bigger denominator does not mean a bigger fraction: 1/8 is smaller than 1/4.

Worked example 6.3.4
Put these fractions in order, smallest first.   12   38   34   58

Key words. To order fractions is to put them in a line from the smallest to the largest (ascending) or from the largest to the smallest (descending). Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Change every fraction to the same denominator. Then order the numerators.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 8, so 8 will do.
    8 ÷ 2 = 4, so 2 goes into 8
    8 ÷ 4 = 2, so 4 goes into 8
  2. Change every fraction to the common denominator. Multiply the top and the bottom by the same number each time.
    12 = 1 × 42 × 4 = 48
    38 (already in eighths)
    34 = 3 × 24 × 2 = 68
    58 (already in eighths)
  3. Put the top numbers in order. With the same bottom number, the bigger the top number, the bigger the fraction. We want smallest to largest.
    3, 4, 5, 6 (out of 8)
  4. Write the original fractions in that order. The answer uses the fractions from the question, not the changed ones.
    38,  12,  58,  34
    38125834

Check. Read the numerators again: 3, 4, 5, 6 go up each time. ✓

Watch out. Write down the original fractions in your final answer.

Worked example 6.3.5
Put these fractions in order, largest first.   23   56   712

Key words. To order fractions is to put them in a line from the smallest to the largest (ascending) or from the largest to the smallest (descending). Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Change every fraction to the same denominator. Then order the numerators.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 12, so 12 will do.
    12 ÷ 3 = 4, so 3 goes into 12
    12 ÷ 6 = 2, so 6 goes into 12
  2. Change every fraction to the common denominator. Multiply the top and the bottom by the same number each time.
    23 = 2 × 43 × 4 = 812
    56 = 5 × 26 × 2 = 1012
    712 (already in twelfths)
  3. Put the top numbers in order. With the same bottom number, the bigger the top number, the bigger the fraction. We want largest to smallest.
    10, 8, 7 (out of 12)
  4. Write the original fractions in that order. The answer uses the fractions from the question, not the changed ones.
    56,  23,  712
    5623712

Check. Read the numerators again: 10, 8, 7 go down each time. ✓

Watch out. Write down the original fractions in your final answer.

Worked example 6.3.6
Put these in order, smallest first.   134   2   158   118

Key words. To order fractions is to put them in a line from the smallest to the largest (ascending) or from the largest to the smallest (descending). Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Change every fraction to the same denominator. Then order the numerators.

  1. Write the mixed numbers as improper fractions. This makes every number a fraction, so we can compare them.
    134 = 1 × 4 + 34 = 74
    2 = 21
    158 = 1 × 8 + 58 = 138
  2. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 8, so 8 will do.
    8 ÷ 4 = 2, so 4 goes into 8
  3. Change every fraction to the common denominator. Multiply the top and the bottom by the same number each time.
    74 = 7 × 24 × 2 = 148
    21 = 2 × 81 × 8 = 168
    138 (already in eighths)
    118 (already in eighths)
  4. Put the top numbers in order. With the same bottom number, the bigger the top number, the bigger the fraction. We want smallest to largest.
    11, 13, 14, 16 (out of 8)
  5. Write the original fractions in that order. The answer uses the fractions from the question, not the changed ones.
    118,  158,  134,  2

Check. Read the numerators again: 11, 13, 14, 16 go up each time. ✓

Watch out. Write down the original fractions in your final answer.

Worked example 6.3.7
Write a fraction that is between 12 and 34.

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Write both fractions with the same denominator. Then pick a numerator between the two numerators.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 4, so 4 will do.
    4 ÷ 2 = 2, so 2 goes into 4
  2. Write both fractions with the common denominator. Then we can see the numerators.
    12 = 1 × 22 × 2 = 24
    34
  3. There is no whole number between the numerators. Double the denominator. Cutting every part in half makes room.
    24 = 48 and 34 = 68
  4. Choose a numerator between them. Any whole number between 4 and 6 works.
    4 < 5 < 6, so use 5.
  5. Write the answer.
    12 < 58 < 34

Check. 58 is bigger than 12 and smaller than 34 (see the numerators). ✓

Watch out. If the numerators are next to each other, double the denominator to make room.

Worked example 6.3.8
Which whole number can go in the box?   35 < ?10 < 45

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. Change the fractions to the same denominator (10). Then find the numerators that fit between.

  1. Write the two end fractions with denominator 10. The box fraction has a bottom number of 10, so we change the others to match.
    35 = 3 × 25 × 2 = 610
    45 = 4 × 25 × 2 = 810
  2. Look at the numerators. The box must be bigger than the first numerator and smaller than the second.
    6 < ? < 8
  3. List the numbers that fit. Only whole numbers go in the box.
    7

Check. Try it: 710 is between 35 and 45. ✓

Watch out. The numbers at the ends do not count: the box has to be strictly between them.

Worked example 6.3.9
Is 35 greater than 58? Prove it.

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. To compare fractions that have different denominators, change them to the same denominator. Then compare the numerators.

  1. Find a common denominator. We need a number that all the denominators go into. List the multiples of each one until you see a number in every list.
    Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40
    Multiples of 8: 8, 16, 24, 32, 40
    The first number in every list is 40.
  2. Change both fractions to the common denominator. Multiply the top and the bottom by the same number each time.
    35 = 3 × 85 × 8 = 2440
    58 = 5 × 58 × 5 = 2540
  3. Compare the top numbers. With the same denominator, the bigger numerator is the bigger fraction.
    24 < 25, so 2440 < 2540
  4. Write a sentence to prove it. In a “prove it” question the last line must say what you have shown.
    No. 35 = 2440 and 58 = 2540, and 24 is less than 25. So 35 is less than 58.
3558

Check. The bars: the 58 bar is longer than the 35 bar. ✓

Watch out. Do not guess from the numbers. Show the common denominator and the two new fractions.

Watch out

A bigger denominator does not make a bigger fraction. 18 is smaller than 14, because eighths are smaller pieces. Also do not forget to put the original fractions in an ordering answer, not the changed ones.

Exercise 6C  Comparing and ordering fractions

Non-calculator
1
FWrite < or > in each box. The denominators are the same.
(a)38  ?  58
(b)710  ?  410
(c)512  ?  912
(d)67  ?  27
2
FWrite < or > in each box. The numerators are the same.
(a)13  ?  15
(b)34  ?  38
(c)25  ?  23
(d)56  ?  512
3
FUse the fraction wall to compare.
11212141414141818181818181818
(a)12  ?  38
(b)34  ?  58
(c)14  ?  38
(d)78  ?  34
4
FWrite < or > in each box.
(a)23  ?  56
(b)34  ?  58
(c)710  ?  35
(d)512  ?  13
(e)45  ?  710
(f)12  ?  512
5
PSWho has more? Explain how you know.
(a)Omar drinks 58 of a litre of juice. Nia drinks 34 of a litre. Who drinks more?
(b)Ravi walks 310 of a mile. Mei walks 15 of a mile. Who walks further?
6
FWrite each set in order, smallest first.
(a)310  12  25
(b)34  58  12  1116
(c)712  23  12  56
(d)45  710  35  910
7
FWrite each set in order, largest first.
(a)56  23  34  712
(b)38  12  14
8
FLook at the number line. Each step is one twelfth.
01ABC
(a)Write the fraction for A, for B and for C. Give each as twelfths.
(b)Which letter shows the largest fraction?
9
FWhich whole numbers can go in each box?
(a)35 < ?10 < 45
(b)12 < ?8 < 34
(c)13 < ?12 < 12
(d)25 < ?20 < 12
10
FWhich of these fractions are greater than 12?    38   58   410   712   612
11
FWrite < or > in each box.
(a)134  ?  158
(b)212  ?  2410
(c)94  ?  212
(d)178  ?  2
12
FWrite these in order, smallest first.    112   54   178   118
13
RZainab says, “58 is greater than 23.” Is she correct? Prove it.
14
RLeo says, “16 is greater than 14 because 6 is greater than 4.” What mistake has Leo made?
15
RAlways, sometimes or never?
(a)When two fractions have the same denominator, the one with the bigger numerator is the bigger fraction.
(b)A fraction with a bigger denominator is a smaller fraction.
16
RChallenge! Write a fraction that is between 25 and 12.   (Hint: tenths are not enough.)
6.4

Improper fractions and mixed numbers

Year 5
Recognise mixed numbers and improper fractions and convert from one form to the other, Write mathematical statements greater than 1 as a mixed number, A fraction is a division

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Improper fractions and mixed numbersComing soon

When the top number is bigger than (or equal to) the bottom number, we have an improper fraction. It is worth 1 whole or more. The picture shows 114: 11 quarters. Two bars are full (8 quarters) and 3 quarters are in the third bar. So 114 = 2 and 34 = 234. A mixed number has a whole number part and a fraction part.

11 quarters = 2 whole bars and 3 more quarters: 114 = 234.

Improper to mixed: divide the top by the bottom. The answer is the whole number and the remainder is the new top number. 114: 11 ÷ 4 = 2 remainder 3, so 234. Mixed to improper: multiply the whole number by the bottom number, then add the top number. 234: 2 × 4 + 3 = 11, so 114.

014243415464742941041143A
On a number line, A is at 114 = 234. After 2 you count 3 more quarter-steps.

A fraction is a division. The line in a fraction means divide. If 3 cakes are shared equally between 4 people, each person gets 3 ÷ 4 = 34 of a cake. And 7 ÷ 4 = 74 = 134: if 7 cakes are shared between 4 people, each person gets 1 whole cake and 3 quarters of a cake.

Sharing 3 bars between 4 people: each bar is cut into 4 quarters and each person gets one quarter of each bar. That is 3 quarters, 34.

Key ideas

  • An improper fraction has a top number that is bigger than or equal to the bottom number: 114.
  • A mixed number is a whole number and a fraction: 234.
  • Improper → mixed: divide the top by the bottom. Whole number = answer, new top = remainder.
  • Mixed → improper: whole × bottom + top, over the same bottom number.
  • The fraction line means divide: 3 ÷ 4 = 34.
Worked example 6.4.1
What improper fraction and what mixed number does the picture show?

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal parts in one whole for the bottom number. Count all the shaded parts for the top number. The full whole ones give the mixed number.

  1. Look at how each bar is cut. Every bar is cut into the same number of equal parts. That number is the denominator.
    Each bar has 4 equal parts, so the denominator is 4.
  2. Count all the shaded parts. The total number of shaded parts is the numerator.
    2 full bars of 4 parts and 3 more parts: 2 × 4 + 3 = 11
  3. Write the improper fraction and the mixed number. There are 2 whole bars shaded and 34 of another.
    114 = 234

Check. 2 × 4 + 3 = 11. ✓

Watch out. Count every shaded part in all the shapes, not just the last one.

Worked example 6.4.2
Write 175 as a mixed number.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

  1. Divide 17 by 5. The fraction line means divide. The answer tells us how many whole ones we can make.
    17 ÷ 5 = 3 remainder 2
  2. The whole number is 3. 3 lots of 5 fifths make 3 whole ones: 3 × 5 = 15.
    17 − 15 = 2 fifths are left over.
  3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 5.
    3 and 25 = 325

Check. Change back: 3 × 5 + 2 = 17 = 17. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

Worked example 6.4.3
Write 228 as a mixed number in its simplest form.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. An improper fraction has more parts than one whole. Divide the top by the bottom to find how many whole ones it holds.

  1. Divide 22 by 8. The fraction line means divide. The answer tells us how many whole ones we can make.
    22 ÷ 8 = 2 remainder 6
  2. The whole number is 2. 2 lots of 8 eighths make 2 whole ones: 2 × 8 = 16.
    22 − 16 = 6 eighths are left over.
  3. Write the leftover parts as a fraction. The remainder is the numerator. The denominator stays 8.
    2 and 68 = 268
  4. Simplify the fraction part. 6 and 8 can both be divided by 2.
    68 = 34, so 268 = 234

Check. Change back: 2 × 8 + 6 = 22 = 22. ✓

Watch out. The leftover (remainder) goes on top of the same denominator. Do not put it over the whole number.

Worked example 6.4.4
Write 325 as an improper fraction.

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count how many parts there are altogether. A whole number is worth the denominator in parts.

  1. Change the 3 wholes into fifths. One whole is 55, so 3 wholes are 3 lots of 5.
    3 × 5 = 15, so 3 = 155
  2. Add the 2 extra fifths. The fraction part is already in the same size parts.
    15 + 2 = 17
  3. Write the improper fraction. The denominator stays the same.
    325 = 175

Check. Change back: 17 ÷ 5 = 3 remainder 2, so 175 = 325. ✓

Watch out. Multiply the whole number by the denominator, then add the numerator. Do not just write the digits next to each other.

Worked example 6.4.5
What number is the arrow pointing to? Write it as a mixed number and as an improper fraction.
1234▼

Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Count the equal steps from the nearest whole number (or from 0).

  1. Find the whole numbers either side. The point is between two whole numbers.
    It is between 2 and 3.
  2. Count the small steps. Each step is 14. There are 4 equal steps between each pair of whole numbers.
    From 2 it is 3 steps along, so it is 3 quarters more than 2.
  3. Write it as a mixed number and as an improper fraction. 2 wholes are 8 quarters. Add 3 more.
    234 = 114

Check. 2 × 4 + 3 = 11. ✓

Watch out. Count the steps between the numbers, not the tick marks. There is one more tick mark than there are steps.

Worked example 6.4.6
Work out 25 + 45. Write the answer as a mixed number.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

  1. Check the denominators. To add fractions, the parts must be the same size.
    Both denominators are 5, so we can add straight away.
  2. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
    25 + 45 = 2 + 45 = 65
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 65 means 6 ÷ 5.
    65 = 115
  4. Write the answer.
    25 + 45 = 115

Check. Take away: 115 − 45 = 25. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Worked example 6.4.7
Seven pizzas are shared equally between 4 friends. How much pizza does each friend get?

Key words. The line in a fraction means divide. So 34 means 3 ÷ 4.

Method. The fraction line means divide. So a ÷ b is the fraction a over b. If the top is bigger, change it to a mixed number.

  1. Write the division as a fraction. A fraction line means divide, so the top number is the number being shared.
    7 ÷ 4 = 74
  2. Change it to a mixed number. 7 wholes shared between 4: each gets 1 whole, and 3 wholes are left to share.
    7 ÷ 4 = 1 remainder 3
  3. Write the left-over part as a fraction. The 3 left over are shared between 4, so each gets 34.
    1 and 34 = 134
  4. Write the answer.
    134

Check. Multiply back: 134 × 4 = 7. ✓

Watch out. Put the number you are dividing (the one being shared) on the top.

Worked example 6.4.8
Write < or >.   175 [ ? ] 315

Key words. To compare fractions is to say which is bigger. < means “is less than” and > means “is greater than”. A common denominator is a bottom number that both denominators go into.

Method. The denominators are the same, so compare the numerators.

  1. Write the mixed numbers as improper fractions. This makes every number a fraction, so we can compare them.
    315 = 3 × 5 + 15 = 165
  2. Look at the denominators. If the bottom numbers are the same, the parts are the same size.
    Both are 5, so we can compare the top numbers.
  3. Compare the top numbers. With the same bottom number, the fraction with the bigger top number is bigger.
    17 > 16
  4. Write the answer. Use the original fractions in your answer.
    175 > 315

Check. Look at the changed fractions again: 175 and 165 have the same denominator, so the bigger numerator is the bigger fraction. ✓

Watch out. A bigger denominator does not mean a bigger fraction: 1/8 is smaller than 1/4.

Watch out

Write the remainder over the old bottom number: 17 ÷ 5 = 3 remainder 2 gives 325, not 323. And 325 does not mean 3 × 2 over 5. It means 3 + 25. To change to an improper fraction multiply the whole number by the bottom: 3 × 5 + 2 = 17.

Exercise 6D  Improper fractions and mixed numbers

Non-calculator
1
FLook at each picture. Write the improper fraction and the mixed number.
(a)
(b)
(c)
2
FWrite each improper fraction as a mixed number (or a whole number).
(a)74
(b)95
(c)133
(d)176
(e)228
(f)124
3
FWrite each mixed number as an improper fraction.
(a)123
(b)234
(c)315
(d)458
(e)2710
(f)512
4
FWrite the number shown by each letter as a mixed number.
0123ABC
(a)Letter A
(b)Letter B
(c)Letter C
5
FWrite the number in each box on the number line as a mixed number.
1?2??3
(a)Box 1
(b)Box 2
(c)Box 3
6
FFind each missing number.
(a)175 = 3 ?5
(b)234 = ? 34
(c)238 = ?8
(d)?6 = 416
7
FWork out each addition. Write each answer as a mixed number in its simplest form.
(a)35 + 45
(b)34 + 34
(c)56 + 56
(d)710 + 910
8
FWrite < or > in each box.
(a)175  ?  315
(b)234  ?  104
(c)53  ?  113
9
FWrite each division as a fraction. If it is more than 1, write it as a mixed number too.
(a)3 ÷ 4
(b)5 ÷ 8
(c)7 ÷ 4
(d)11 ÷ 3
10
PSMei shares 3 cakes equally between 4 friends. How much cake does each friend get?
11
PSNine pancakes are shared equally between 4 children. How many pancakes does each child get? Write your answer as a mixed number.
12
PS5 litres of water are poured equally into 3 jugs. How much water is in each jug?
13
PSA baker cuts 412 pies into halves. How many halves are there?
14
RWhich is the odd one out? Explain how you know.    74   134   148   178
15
RNoah says, “325 as an improper fraction is 65.” What has Noah done wrong? What is the correct answer?
16
RAlways, sometimes or never?
(a)An improper fraction is greater than 1.
(b)A mixed number is greater than 1.
17
RChallenge! I am an improper fraction with a denominator of 6. I am between 2 and 3. My numerator is a multiple of 5. What fraction am I? Write me as an improper fraction and as a mixed number.
6.5

Adding and subtracting fractions

Year 5
Add and subtract fractions with the same denominator and denominators that are multiples of the same number, Write statements greater than 1 as a mixed number

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Adding and subtracting fractionsComing soon

When the denominators are the same, the parts are the same size, so we just add the numerators. 38 + 28: 3 eighths and 2 eighths make 5 eighths, so the answer is 58. The denominator stays the same, because we are still counting eighths.

38 + 28 = 58. The blue parts and the orange parts are all eighths.

If the answer is more than 1, write it as a mixed number: 25 + 45 = 65 = 115. Take away works the same way: 56 − 16 = 46 = 23 (simplify at the end).

25 + 45 fills one whole bar and 1 fifth of another: 115.

What if the denominators are different, like 12 + 14? The parts are different sizes, so we must change one fraction first. Use a common denominator: a number that both denominators go into. Here 4 is a multiple of 2, so change halves to quarters: 12 = 24. Now 24 + 14 = 34. To take a fraction away from a whole, write the whole as a fraction: 1 = 88, so 1 − 38 = 58.

122434
12 = 24, and adding one more quarter makes 34.

Key ideas

  • Same denominator: add (or subtract) the numerators and keep the denominator.
  • Different denominators: change to a common denominator first (use the bigger denominator when it is a multiple of the other).
  • If the answer is an improper fraction, write it as a mixed number: 65 = 115.
  • Always simplify at the end: 46 = 23.
  • From a whole: write the whole as a fraction first: 1 = 88.
Worked example 6.5.1
Work out 38 + 28.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

  1. Check the denominators. To add fractions, the parts must be the same size.
    Both denominators are 8, so we can add straight away.
  2. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
    38 + 28 = 3 + 28 = 58
  3. Write the answer. It is already in its simplest form.
    58

Check. Take away: 58 − 28 = 38. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Worked example 6.5.2
Work out 34 + 34.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

  1. Check the denominators. To add fractions, the parts must be the same size.
    Both denominators are 4, so we can add straight away.
  2. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
    34 + 34 = 3 + 34 = 64
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 64 means 6 ÷ 4.
    64 = 124
  4. Simplify the fraction part. 2 and 4 can both be divided by 2.
    24 = 2 ÷ 24 ÷ 2 = 12
    So the answer is 112.
  5. Write the answer.
    34 + 34 = 112

Check. Take away: 112 − 34 = 34. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Worked example 6.5.3
Work out 12 + 14.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 4, so 4 will do.
    4 ÷ 2 = 2, so 2 goes into 4
  2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
    12 = 1 × 22 × 2 = 24
  3. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
    24 + 14 = 2 + 14 = 34
  4. Write the answer.
    12 + 14 = 34

Check. Take away: 34 − 14 = 12. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Worked example 6.5.4
Work out 23 + 16.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 6, so 6 will do.
    6 ÷ 3 = 2, so 3 goes into 6
  2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
    23 = 2 × 23 × 2 = 46
  3. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
    46 + 16 = 4 + 16 = 56
  4. Write the answer.
    23 + 16 = 56

Check. Take away: 56 − 16 = 23. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Worked example 6.5.5
Work out 712 + 14.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then add the numerators and keep the denominator.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 12, so 12 will do.
    12 ÷ 4 = 3, so 4 goes into 12
  2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
    14 = 1 × 34 × 3 = 312
  3. Add the numerators. We are adding parts of the same size, so we add the top numbers. The bottom number stays the same.
    712 + 312 = 7 + 312 = 1012
  4. Simplify the fraction part. 10 and 12 can both be divided by 2.
    1012 = 10 ÷ 212 ÷ 2 = 56
  5. Write the answer.
    712 + 14 = 56

Check. Take away: 56 − 14 = 712. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Worked example 6.5.6
Work out 35 − 110.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then subtract the numerators and keep the denominator.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 10, so 10 will do.
    10 ÷ 5 = 2, so 5 goes into 10
  2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
    35 = 3 × 25 × 2 = 610
  3. Subtract the numerators. We are taking parts of the same size away, so subtract the top numbers. The bottom number stays the same.
    610 − 110 = 6 − 110 = 510
  4. Simplify the fraction part. 5 and 10 can both be divided by 5.
    510 = 5 ÷ 510 ÷ 5 = 12
  5. Write the answer.
    35 − 110 = 12

Check. Add back: 12 + 110 = 35. ✓

Watch out. Subtract the numerators only. The denominator stays the same.

Worked example 6.5.7
Work out 56 − 13.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Then subtract the numerators and keep the denominator.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 6, so 6 will do.
    6 ÷ 3 = 2, so 3 goes into 6
  2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
    13 = 1 × 23 × 2 = 26
  3. Subtract the numerators. We are taking parts of the same size away, so subtract the top numbers. The bottom number stays the same.
    56 − 26 = 5 − 26 = 36
  4. Simplify the fraction part. 3 and 6 can both be divided by 3.
    36 = 3 ÷ 36 ÷ 3 = 12
  5. Write the answer.
    56 − 13 = 12

Check. Add back: 12 + 13 = 56. ✓

Watch out. Subtract the numerators only. The denominator stays the same.

Worked example 6.5.8
Work out 1 − 38.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into. Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Make the denominators the same, then subtract. If the fraction you are taking away is bigger, change both numbers to improper fractions first.

  1. The first fraction part (0) is smaller than the second (3), so we cannot take them away. Change both numbers to improper fractions. Each whole is worth 8 eighths, so the numbers become fractions over 8.
    108 = 1 × 8 + 08 = 88
    38 = 0 × 8 + 38 = 38
  2. Subtract the numerators. Now both numbers are fractions with the same denominator.
    88 − 38 = 8 − 38 = 58
  3. Write the answer. It is already in its simplest form.
    58

Check. Add back: 58 + 38 = 1. ✓

Watch out. Never take the smaller number from the bigger one in each column (3⅕ − 1⅗ is not 2⅖). Change to improper fractions when you need to.

Watch out

Add the numerators only. Do not add the denominators: 14 + 14 = 24, not 28. Two quarters is half a bar, not a smaller piece. Also, you cannot add halves and quarters until you have changed them to the same kind of part.

Exercise 6E  Adding and subtracting fractions

Non-calculator
1
FWork out each addition. Give each answer in its simplest form.
(a)27 + 37
(b)19 + 49
(c)512 + 312
(d)310 + 410
2
FWork out each subtraction. Give each answer in its simplest form.
(a)79 − 49
(b)810 − 310
(c)56 − 16
(d)1112 − 712
3
FWork out each addition. Write the answers as mixed numbers.
(a)34 + 34
(b)45 + 35
(c)58 + 78
(d)710 + 910
4
FLook at each picture. Write the calculation and work out the answer.
(a)
Blue part + orange part = ?
(b)
Seven tenths are shaded. The crossed parts are taken away. How much is left?
5
FWork out each addition. Give each answer in its simplest form.
(a)12 + 14
(b)13 + 16
(c)310 + 15
(d)512 + 13
6
FWork out each subtraction. Give each answer in its simplest form.
(a)34 − 18
(b)56 − 13
(c)710 − 25
(d)35 − 110
7
FWork out each subtraction from a whole.
(a)1 − 14
(b)1 − 35
(c)1 − 712
(d)2 − 34
8
FFind each missing fraction.
(a)25 + ?5 = 45
(b)14 + ?8 = 58
(c)?4 − 14 = 12
(d)78 − ?8 = 38
(e)1 − ?7 = 27
9
PSElla eats one quarter of a pie. Omar eats three eighths of the same pie.
(a)What fraction of the pie do they eat altogether?
(b)What fraction of the pie is left?
10
PSSolve each problem.
(a)Ben runs 34 km on Monday and 58 km on Tuesday. How far does he run altogether?
(b)Zainab uses 23 m of ribbon and then 16 m. How much ribbon does she use?
11
RAmy says, “12 + 13 = 25.” Explain her mistake and work out the correct answer.
12
RJack says, “34 + 34 = 68.” What is wrong with his answer? What is the correct answer?
13
RWhich two fractions in this list add up to 1? Find two pairs.    14   38   58   12   34
14
RAlways, sometimes or never?
(a)When you add two proper fractions the answer is less than 1.
(b)When you take a fraction away from 1 the answer is less than 1.
15
RSofia says, “12 + 13 + 16 = 1.” Show that she is correct.
16
RChallenge! Find the missing fraction.   13 + ? + 112 = 34
6.6

Adding and subtracting mixed numbers

Year 5
Add and subtract mixed numbers, with and without crossing a whole number, Write statements greater than 1 as a mixed number

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Adding and subtracting mixed numbersComing soon

To add mixed numbers, add the whole numbers first, then add the fractions. Put the answers together. 215 + 125: wholes 2 + 1 = 3, fractions 15 + 25 = 35. The answer is 335.

215 (blue) and 125 (orange) together fill 3 whole bars and 3 fifths.

Crossing a whole. If the fractions add to 1 or more, you must carry a whole. 234 + 134: wholes 2 + 1 = 3, fractions 34 + 34 = 64 = 124 = 112. Now add the extra whole: 3 + 112 = 412. If the denominators are different, change them to a common denominator first.

2214224234331432433444144244345S+1+ 3/4
Start at 234, add 1 to reach 334, then add 34 more: you land on 424 = 412.

Subtracting. If the fraction of the first number is big enough, subtract the wholes and the fractions separately: 456 − 116 = 3 and 46 = 346 = 323. If the fraction you are taking away is bigger, as in 315 − 135 (1 fifth is not enough to take away 3 fifths), change both numbers to improper fractions: 165 − 85 = 85 = 135. To take a mixed number from a whole number, such as 4 − 138, write 4 as 328 and 138 as 118.

Key ideas

  • Add: add the wholes, add the fractions (same denominator first), then put them together.
  • If the fractions add to 1 or more, change the fraction part to a whole and a fraction and add the whole.
  • Subtract: if the first fraction is big enough, subtract wholes and fractions separately.
  • If the fraction you take away is bigger, change both numbers to improper fractions, subtract, then change back to a mixed number.
  • Always simplify the fraction at the end.
Worked example 6.6.1
Work out 215 + 125.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Add the whole numbers, then add the numerators. Change the answer to a mixed number and simplify if you can.

  1. Check the denominators. To add fractions, the parts must be the same size.
    Both denominators are 5, so we can add straight away.
  2. Add the whole numbers. Add the wholes first, then deal with the fractions.
    2 + 1 = 3
  3. Add the fractions. Add the numerators. The denominator stays the same because the parts are the same size.
    15 + 25 = 1 + 25 = 35
  4. Write the answer.
    215 + 125 = 335

Check. Take away: 335 − 125 = 215. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Worked example 6.6.2
Work out 234 + 134.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Add the whole numbers, then add the numerators. Change the answer to a mixed number and simplify if you can.

  1. Check the denominators. To add fractions, the parts must be the same size.
    Both denominators are 4, so we can add straight away.
  2. Add the whole numbers. Add the wholes first, then deal with the fractions.
    2 + 1 = 3
  3. Add the fractions. Add the numerators. The denominator stays the same because the parts are the same size.
    34 + 34 = 3 + 34 = 64
  4. The fraction part is 1 or more, so change it to a whole number and a fraction. 64 means 6 ÷ 4.
    64 = 124
    3 + 124 = 424
  5. Simplify the fraction part. 2 and 4 can both be divided by 2.
    24 = 2 ÷ 24 ÷ 2 = 12
    So the answer is 412.
  6. Write the answer.
    234 + 134 = 412

Check. Take away: 412 − 134 = 234. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Worked example 6.6.3
Work out 312 + 214.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Add the whole numbers, then add the numerators. Change the answer to a mixed number and simplify if you can.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 4, so 4 will do.
    4 ÷ 2 = 2, so 2 goes into 4
  2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
    312 → 12 = 1 × 22 × 2 = 24
  3. Add the whole numbers. Add the wholes first, then deal with the fractions.
    3 + 2 = 5
  4. Add the fractions. Add the numerators. The denominator stays the same because the parts are the same size.
    24 + 14 = 2 + 14 = 34
  5. Write the answer.
    312 + 214 = 534

Check. Take away: 534 − 214 = 312. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Worked example 6.6.4
Work out 156 + 213.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into.

Method. Make sure the denominators are the same. Add the whole numbers, then add the numerators. Change the answer to a mixed number and simplify if you can.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 6, so 6 will do.
    6 ÷ 3 = 2, so 3 goes into 6
  2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
    213 → 13 = 1 × 23 × 2 = 26
  3. Add the whole numbers. Add the wholes first, then deal with the fractions.
    1 + 2 = 3
  4. Add the fractions. Add the numerators. The denominator stays the same because the parts are the same size.
    56 + 26 = 5 + 26 = 76
  5. The fraction part is 1 or more, so change it to a whole number and a fraction. 76 means 7 ÷ 6.
    76 = 116
    3 + 116 = 416
  6. Write the answer.
    156 + 213 = 416

Check. Take away: 416 − 213 = 156. ✓

Watch out. Add the numerators only. The denominator does not get added: 1/4 + 1/4 is 2/4, not 2/8.

Worked example 6.6.5
Work out 456 − 116.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into. Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Make the denominators the same, then subtract. If the fraction you are taking away is bigger, change both numbers to improper fractions first.

  1. Check the denominators. To subtract fractions, the parts must be the same size.
    Both denominators are 6, so we can subtract straight away.
  2. Subtract the whole numbers. The fraction part of the first number is big enough, so we can subtract the wholes and the fractions separately.
    4 − 1 = 3
  3. Subtract the fractions. Subtract the numerators. The denominator stays the same.
    56 − 16 = 46
  4. Simplify the fraction part. 4 and 6 can both be divided by 2.
    46 = 4 ÷ 26 ÷ 2 = 23
    So the answer is 323.
  5. Write the answer.
    456 − 116 = 323

Check. Add back: 323 + 116 = 456. ✓

Watch out. Never take the smaller number from the bigger one in each column (3⅕ − 1⅗ is not 2⅖). Change to improper fractions when you need to.

Worked example 6.6.6
Work out 578 − 234.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into. Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Make the denominators the same, then subtract. If the fraction you are taking away is bigger, change both numbers to improper fractions first.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 8, so 8 will do.
    8 ÷ 4 = 2, so 4 goes into 8
  2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
    234 → 34 = 3 × 24 × 2 = 68
  3. Subtract the whole numbers. The fraction part of the first number is big enough, so we can subtract the wholes and the fractions separately.
    5 − 2 = 3
  4. Subtract the fractions. Subtract the numerators. The denominator stays the same.
    78 − 68 = 18
  5. Write the answer.
    578 − 234 = 318

Check. Add back: 318 + 234 = 578. ✓

Watch out. Never take the smaller number from the bigger one in each column (3⅕ − 1⅗ is not 2⅖). Change to improper fractions when you need to.

Worked example 6.6.7
Work out 315 − 135.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into. Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Make the denominators the same, then subtract. If the fraction you are taking away is bigger, change both numbers to improper fractions first.

  1. Check the denominators. To subtract fractions, the parts must be the same size.
    Both denominators are 5, so we can subtract straight away.
  2. The first fraction part (1) is smaller than the second (3), so we cannot take them away. Change both numbers to improper fractions. Each whole is worth 5 fifths, so the numbers become fractions over 5.
    315 = 3 × 5 + 15 = 165
    135 = 1 × 5 + 35 = 85
  3. Subtract the numerators. Now both numbers are fractions with the same denominator.
    165 − 85 = 16 − 85 = 85
  4. The answer is 1 or more, so change it to a mixed number or a whole number. 85 means 8 ÷ 5.
    85 = 135
  5. Write the answer.
    315 − 135 = 135

Check. Add back: 135 + 135 = 315. ✓

Watch out. Never take the smaller number from the bigger one in each column (3⅕ − 1⅗ is not 2⅖). Change to improper fractions when you need to.

Worked example 6.6.8
Work out 4 − 138.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into. Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Make the denominators the same, then subtract. If the fraction you are taking away is bigger, change both numbers to improper fractions first.

  1. The first fraction part (0) is smaller than the second (3), so we cannot take them away. Change both numbers to improper fractions. Each whole is worth 8 eighths, so the numbers become fractions over 8.
    408 = 4 × 8 + 08 = 328
    138 = 1 × 8 + 38 = 118
  2. Subtract the numerators. Now both numbers are fractions with the same denominator.
    328 − 118 = 32 − 118 = 218
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 218 means 21 ÷ 8.
    218 = 258
  4. Write the answer.
    4 − 138 = 258

Check. Add back: 258 + 138 = 4. ✓

Watch out. Never take the smaller number from the bigger one in each column (3⅕ − 1⅗ is not 2⅖). Change to improper fractions when you need to.

Worked example 6.6.9
Work out 214 − 12.

Key words. To add or subtract fractions, the denominators (bottom numbers) must be the same. A common denominator is a bottom number that both denominators go into. Key words. An improper fraction has a top number that is bigger than (or equal to) the bottom number, like 74. A mixed number is a whole number and a fraction together, like 134.

Method. Make the denominators the same, then subtract. If the fraction you are taking away is bigger, change both numbers to improper fractions first.

  1. Find a common denominator. The denominators must be the same before we compare or add. Every smaller denominator goes into 4, so 4 will do.
    4 ÷ 2 = 2, so 2 goes into 4
  2. Change the fractions to the common denominator. Multiply the top and the bottom by the same number, so each fraction keeps its value.
    12 = 1 × 22 × 2 = 24
  3. The first fraction part (1) is smaller than the second (2), so we cannot take them away. Change both numbers to improper fractions. Each whole is worth 4 quarters, so the numbers become fractions over 4.
    214 = 2 × 4 + 14 = 94
    24 = 0 × 4 + 24 = 24
  4. Subtract the numerators. Now both numbers are fractions with the same denominator.
    94 − 24 = 9 − 24 = 74
  5. The answer is 1 or more, so change it to a mixed number or a whole number. 74 means 7 ÷ 4.
    74 = 134
  6. Write the answer.
    214 − 12 = 134

Check. Add back: 134 + 12 = 214. ✓

Watch out. Never take the smaller number from the bigger one in each column (3⅕ − 1⅗ is not 2⅖). Change to improper fractions when you need to.

Watch out

Do not take the smaller fraction from the bigger one in each column. 315 − 135 is not 225 (that takes 1 fifth from 3 fifths the wrong way round). When the fraction you are taking away is bigger, change to improper fractions. Also do not leave an answer like 364. The fraction part must be less than 1.

Exercise 6F  Adding and subtracting mixed numbers

Non-calculator
1
FWork out each addition.
(a)115 + 225
(b)327 + 137
(c)218 + 438
(d)5310 + 2410
2
FWork out each addition. Each answer is a mixed number with the fraction part less than 1.
(a)234 + 134
(b)356 + 236
(c)145 + 335
(d)278 + 158
3
FWork out each addition. Change to a common denominator first.
(a)112 + 214
(b)3310 + 115
(c)223 + 156
(d)434 + 158
4
FWork out each subtraction.
(a)456 − 116
(b)678 − 238
(c)3910 − 1410
(d)545 − 215
5
FWork out each subtraction. Change to a common denominator first.
(a)578 − 234
(b)356 − 113
(c)6910 − 225
(d)434 − 112
6
FWork out each subtraction. Change to improper fractions when you need to.
(a)315 − 135
(b)414 − 234
(c)218 − 158
(d)6110 − 3710
7
FWork out each subtraction from a whole number.
(a)4 − 138
(b)5 − 214
(c)3 − 125
(d)6 − 2710
8
FUse the number line to work out 212 + 134.
2214224234331432433444144244345S+1+ 3/4
9
PSA cake recipe uses 112 cups of flour. A second recipe uses 234 cups. How much flour do they use altogether?
10
PSZainab runs 234 km. Liam runs 112 km. How much further does Zainab run?
11
PSA plank is 5 m long. Jack cuts off a piece 134 m long and then a piece 212 m long. How much of the plank is left?
12
FFind each missing number.
(a)234 + ? = 414
(b)? − 115 = 235
13
RRavi says, “315 − 135 = 225.” What mistake has Ravi made? What is the correct answer?
14
RMia says, “234 + 134 = 364.” Is her answer finished? Explain and give the correct answer.
15
RChallenge! A rectangle is 2½ cm long and 1¾ cm wide. Work out its perimeter.
2½ cm1¾ cm
6.7

Multiplying fractions by whole numbers and fractions of amounts

Year 5
Multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams, Fractions of amounts, Solve problems with bar models

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Multiplying fractions by whole numbers and fractions of amountsComing soon

Multiplying by a whole number is repeated addition. 3 × 25 means three lots of two fifths: 25 + 25 + 25 = 65 = 115. Look at the picture: 3 groups of 2 fifths make 6 fifths, which is one whole bar and 1 fifth. Multiply the numerator by the whole number and keep the denominator.

3 × 25 = 65 = 115. Each colour is one lot of 25.

For a mixed number, multiply the whole part and the fraction part separately, then add. 4 × 112: 4 × 1 = 4 and 4 × 12 = 42 = 2. Together 4 + 2 = 6. You can check by changing to an improper fraction: 112 = 32 and 4 × 32 = 122 = 6.

Fractions of amounts (a reminder). To find 34 of 36, divide by the denominator to find one quarter, then multiply by the numerator: 36 ÷ 4 = 9, and 3 × 9 = 27.

369999
34 of 36: each quarter is 9, and three quarters is 27.

Key ideas

  • Fraction × whole number: multiply the numerator, keep the denominator. 3 × 25 = 65. Then write it as a mixed number and simplify.
  • Mixed number × whole number: multiply the whole part and the fraction part separately, then add (or change to an improper fraction first).
  • Fraction of an amount: ÷ by the denominator, × by the numerator.
  • Finding the whole: if 35 of a number is 18, one fifth is 18 ÷ 3 = 6, so the whole is 5 × 6 = 30.
Worked example 6.7.1
Work out 3 × 25.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the numerator by the whole number and keep the denominator. If the answer is an improper fraction, change it to a mixed number or a whole number.

  1. Think of 3 × 25 as 3 lots of 25. Multiplying by a whole number is repeated addition.
    25 + 25 + 25
  2. Multiply the numerator by 3. Keep the denominator. We are counting fifths. 3 lots of 2 fifths is 6 fifths.
    3 × 25 = 3 × 25 = 65
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 65 means 6 ÷ 5.
    65 = 115
  4. Write the answer.
    3 × 25 = 115

Check. Divide back: 115 ÷ 3 = 25. ✓

Watch out. Multiply the top only. The bottom stays the same: 3 × 2/5 is 6/5, not 6/15.

Worked example 6.7.2
Work out 4 × 38.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the numerator by the whole number and keep the denominator. If the answer is an improper fraction, change it to a mixed number or a whole number.

  1. Think of 4 × 38 as 4 lots of 38. Multiplying by a whole number is repeated addition.
    38 + 38 + 38 + 38
  2. Multiply the numerator by 4. Keep the denominator. We are counting eighths. 4 lots of 3 eighths is 12 eighths.
    4 × 38 = 4 × 38 = 128
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 128 means 12 ÷ 8.
    128 = 148
  4. Simplify the fraction part. 4 and 8 can both be divided by 4.
    48 = 4 ÷ 48 ÷ 4 = 12
    So the answer is 112.
  5. Write the answer.
    4 × 38 = 112

Check. Divide back: 112 ÷ 4 = 38. ✓

Watch out. Multiply the top only. The bottom stays the same: 3 × 2/5 is 6/5, not 6/15.

Worked example 6.7.3
Work out 6 × 13.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the numerator by the whole number and keep the denominator. If the answer is an improper fraction, change it to a mixed number or a whole number.

  1. Think of 6 × 13 as 6 lots of 13. Multiplying by a whole number is repeated addition.
    13 + 13 + 13 + 13 + 13 + 13
  2. Multiply the numerator by 6. Keep the denominator. We are counting thirds. 6 lots of 1 thirds is 6 thirds.
    6 × 13 = 6 × 13 = 63
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 63 means 6 ÷ 3.
    63 = 2
  4. Write the answer.
    6 × 13 = 2

Check. Divide back: 2 ÷ 6 = 13. ✓

Watch out. Multiply the top only. The bottom stays the same: 3 × 2/5 is 6/5, not 6/15.

Worked example 6.7.4
Five friends each drink 34 of a litre of water. How much water do they drink altogether?

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the numerator by the whole number and keep the denominator. If the answer is an improper fraction, change it to a mixed number or a whole number.

  1. Think of 5 × 34 as 5 lots of 34. Multiplying by a whole number is repeated addition.
    34 + 34 + 34 + 34 + 34
  2. Multiply the numerator by 5. Keep the denominator. We are counting quarters. 5 lots of 3 quarters is 15 quarters.
    5 × 34 = 5 × 34 = 154
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 154 means 15 ÷ 4.
    154 = 334
  4. Write the answer.
    5 × 34 = 334

Check. Divide back: 334 ÷ 5 = 34. ✓

Watch out. Multiply the top only. The bottom stays the same: 3 × 2/5 is 6/5, not 6/15.

Worked example 6.7.5
Work out 4 × 112.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the whole part and the fraction part separately, then add the answers. (Or change to an improper fraction first.)

  1. Split the mixed number into its whole part and its fraction part. We multiply each part by the whole number, then add the answers (the distributive law).
    112 = 1 + 12
  2. Multiply the whole part. The whole numbers multiply as usual.
    4 × 1 = 4
  3. Multiply the fraction part. Multiply the numerator by 4 and keep the denominator.
    4 × 12 = 42 = 2
  4. Add the two answers. Put the whole parts and the fraction parts back together.
    4 + 2 = 6
  5. Write the answer.
    4 × 112 = 6

Check. Other way: 112 = 32, and 4 × 32 = 122 = 6. ✓

Watch out. Do not forget to multiply the whole part as well as the fraction part.

Worked example 6.7.6
Work out 3 × 214.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the whole part and the fraction part separately, then add the answers. (Or change to an improper fraction first.)

  1. Split the mixed number into its whole part and its fraction part. We multiply each part by the whole number, then add the answers (the distributive law).
    214 = 2 + 14
  2. Multiply the whole part. The whole numbers multiply as usual.
    3 × 2 = 6
  3. Multiply the fraction part. Multiply the numerator by 3 and keep the denominator.
    3 × 14 = 34
  4. Add the two answers. Put the whole parts and the fraction parts back together.
    6 + 34 = 634
  5. Write the answer.
    3 × 214 = 634

Check. Other way: 214 = 94, and 3 × 94 = 274 = 634. ✓

Watch out. Do not forget to multiply the whole part as well as the fraction part.

Worked example 6.7.7
Work out 34 of 36.

Key words. “Of” means multiply. To find a fraction of an amount, divide by the denominator, then multiply by the numerator.

Method. To find a fraction of an amount, divide by the denominator to find one part. Then multiply by the numerator.

  1. Divide the amount by the denominator. The denominator 4 tells us to cut the amount into 4 equal parts. One part is 14 of the amount.
    36 ÷ 4 = 9
  2. Multiply by the numerator. The numerator 3 tells us how many of those parts we want.
    3 × 9 = 27
  3. Write the answer.
    34 of 36 = 27
    369999

Check. Other way: 3 × 36 = 108, then ÷ 4 = 27. ✓

Watch out. Divide by the bottom number first. Then multiply by the top number.

Worked example 6.7.8
Sofia has £48. She spends 58 of it. How much does she spend?

Key words. “Of” means multiply. To find a fraction of an amount, divide by the denominator, then multiply by the numerator.

Method. To find a fraction of an amount, divide by the denominator to find one part. Then multiply by the numerator.

  1. Divide the amount by the denominator. The denominator 8 tells us to cut the amount into 8 equal parts. One part is 18 of the amount.
    £48 ÷ 8 = £6
  2. Multiply by the numerator. The numerator 5 tells us how many of those parts we want.
    5 × £6 = £30
  3. Write the answer.
    58 of £48 = £30
    £48£6£6£6£6£6£6£6£6

Check. Other way: 5 × £48 = £240, then ÷ 8 = £30. ✓

Watch out. Divide by the bottom number first. Then multiply by the top number.

Worked example 6.7.9
35 of a number is 18. What is the number?

Key words. “Of” means multiply. To find a fraction of an amount, divide by the denominator, then multiply by the numerator.

Method. Use a bar model. The known part tells us the value of one small part. Then multiply to find the whole.

  1. 3 parts are worth 18. Find the value of one part. The top number 3 is how many parts make 18, so divide by 3.
    18 ÷ 3 = 6
  2. Find the whole. The whole is 5 parts. The bottom number 5 is how many equal parts there are altogether.
    5 × 6 = 30
  3. Check with the fraction. Take the fraction of the whole you found.
    35 of 30 = 6 × 3 = 18 ✓
?66666

Check. 35 of 30 = 18. ✓

Watch out. Divide by the numerator first to find one part, then multiply by the denominator.

Watch out

Multiply the numerator only. 3 × 25 = 65, not 615. The denominator tells the size of the parts and it does not change. And in a mixed number, do not forget to multiply the whole part: 4 × 112 is not 1 + 2.

Exercise 6G  Multiplying by whole numbers and fractions of amounts

Non-calculator
1
FThe picture shows 3 lots of two fifths. Write the multiplication and work out the answer.
2
FWork out each multiplication.
(a)3 × 14
(b)4 × 16
(c)5 × 13
(d)6 × 18
3
FWork out each multiplication. Write each answer as a mixed number.
(a)4 × 37
(b)5 × 23
(c)3 × 58
(d)6 × 310
4
FWork out each multiplication. Give each answer in its simplest form.
(a)4 × 38
(b)6 × 512
(c)6 × 710
(d)8 × 34
5
FWork out each multiplication.
(a)2 × 114
(b)3 × 123
(c)4 × 212
(d)5 × 1310
6
FFind each missing fraction.
(a)3 × ? = 910
(b)3 × ? = 65
(c)5 × ? = 158
7
FWork out each fraction of an amount.
(a)34 of 36
(b)25 of 45
(c)58 of 48
(d)710 of 90
(e)23 of 27
(f)56 of 42
8
FWork out each answer.
(a)25 of £35
(b)38 of 64 kg
(c)712 of 60 minutes
(d)310 of 250 ml
9
PSSolve each problem.
(a)Five friends each drink 34 of a litre of water. How much do they drink altogether?
(b)Four bottles each hold 310 of a litre. How much do they hold altogether?
10
PSSolve each problem.
(a)A cake needs 212 cups of flour. How much flour is needed for 4 cakes?
(b)Each piece of rope is 113 m long. What is the total length of 3 pieces?
11
RTom says, “3 × 25 = 615.” What is wrong? What is the correct answer?
12
RAlways, sometimes or never?
(a)When you multiply a proper fraction by a whole number bigger than 1, the answer is bigger than the fraction.
(b)When you multiply a proper fraction by 3, the answer is greater than 1.
13
RWhich is the odd one out? Explain how you know.    3 × 23   6 × 13   4 × 12   2 × 113
14
RChallenge! I think of a fraction. I multiply it by 6 and the answer is 412. What fraction did I think of?

Exercise 6H  Fraction problems

Non-calculator
1
PSUse a bar model to solve each problem.
(a)35 of the stickers in a pack are shiny. There are 18 shiny stickers. How many stickers are in the pack?
(b)Leo spends 23 of his money on a football. The football costs £24. How much money did Leo have?
2
PSThere are 40 children in a club. 38 walk to school and 14 cycle. The rest come by bus. How many children come by bus?
walk 3/8cycle 1/4bus ?40
3
PSA chocolate bar has 12 squares. Ravi eats 14 of the bar and Priya eats 13 of the bar. How many squares are left?
4
RIs 34 of 20 more than 23 of 24? Show how you know.
5
PSDev reads three tenths of a book on Monday and two fifths of it on Tuesday.
(a)What fraction of the book has he read?
(b)What fraction of the book is left to read?
6
FWork out 12 + 14 + 18.
7
PSIn a class of 28 children, 37 are boys. How many girls are in the class?
8
PSAisha has £60. She spends 14 of it on a book and 25 of it on trainers. How much money does she have left?
9
PSA tank is 34 full. It holds 36 litres. How many litres does the tank hold when it is full?
10
RMei says, “3 × 25 and 2 × 35 have the same answer.” Is she right? Explain.
11
RChallenge! One third of a number is 7 less than half of the number. What is the number?

Chapter summary

SkillMethod
Equivalent fractionsMultiply or divide the top and the bottom by the SAME number: 23 = 812. 1 tenth = 10 hundredths.
SimplifyDivide the top and the bottom by a common factor (the HCF does it in one go). Simplest form: the only common factor is 1.
Compare and orderSame denominator: bigger numerator = bigger fraction. Otherwise change to a common denominator first. Write the original fractions in the answer.
Improper to mixedDivide the top by the bottom: whole number = answer, new top = remainder. 114 = 234.
Mixed to improperWhole × bottom + top, over the same bottom: 234 = 114.
Fractions as divisionThe fraction line means divide: 3 ÷ 4 = 34 and 7 ÷ 4 = 134.
Add and subtract fractionsSame denominator: add or subtract the numerators and keep the denominator. Different: common denominator first. Simplify, and write answers over 1 as mixed numbers.
Mixed numbersAdd (or subtract) the wholes and the fractions. If the fraction part is 1 or more, carry a whole. If you cannot take the fraction away, change to improper fractions.
Fraction × whole numberMultiply the numerator and keep the denominator: 3 × 25 = 65 = 115. Mixed numbers: multiply the whole part and the fraction part, then add.
Fraction of an amountDivide by the denominator, multiply by the numerator: 34 of 36 = 36 ÷ 4 × 3 = 27. To find the whole, divide by the numerator then multiply by the denominator.
ReasoningSpot the mistake by doing it yourself. Never add the denominators. “Prove it” needs working and a sentence. One counter-example breaks “always”.

Can you…?

  • I can find equivalent fractions by multiplying or dividing the top and the bottom by the same number, including tenths and hundredths.
  • I can simplify a fraction by dividing by a common factor.
  • I can compare and order fractions by changing them to a common denominator.
  • I can change improper fractions to mixed numbers and mixed numbers to improper fractions.
  • I can write a division as a fraction, such as 3 ÷ 4 = 34.
  • I can add and subtract fractions with the same denominator and with denominators that are multiples of one another.
  • I can add and subtract mixed numbers, including when I cross a whole number.
  • I can multiply fractions and mixed numbers by whole numbers.
  • I can find a fraction of an amount, and find the whole when I know a fraction of it.
  • I can solve word problems with bar models and explain my reasoning.

Key terms

Fraction
A number that shows equal parts of a whole, such as 38.
Numerator
The top number of a fraction. It says how many parts we have.
Denominator
The bottom number of a fraction. It says how many equal parts the whole is cut into.
Equivalent fractions
Fractions that are worth the same amount, like 12 and 48.
Simplify
To write a fraction with smaller numbers by dividing the top and the bottom by a common factor.
Common factor
A number that divides exactly into two or more numbers.
Highest common factor (HCF)
The biggest number that divides exactly into two numbers.
Simplest form
A fraction whose top and bottom have no common factor except 1.
Common denominator
A bottom number that the denominators of two or more fractions all go into.
Unit fraction
A fraction with a numerator of 1, such as 15.
Improper fraction
A fraction with a top number bigger than or equal to the bottom number, such as 74.
Mixed number
A whole number and a fraction together, such as 134.
Fraction wall
A picture of bricks that shows fractions of one whole, row by row.
Ascending
In order from the smallest to the largest.
Descending
In order from the largest to the smallest.
Tenths and hundredths
A tenth is one of 10 equal parts and a hundredth is one of 100 equal parts. 1 tenth = 10 hundredths.
Counter-example
One example that shows a statement is false.
Bar model
A drawing of bars that shows the parts and the whole of a problem.

Exercise 6Z  Check-up (non-calculator)

Non-calculator
1
Write the missing number.   34 = ?20
2
Write 1830 in its simplest form.
3
Write < or > in the box.   710 [ ? ] 35
4
Write these fractions in order, smallest first.    23   712   56
5
Write 236 as a mixed number.
6
Write 438 as an improper fraction.
7
Work out 710 + 45. Write your answer as a mixed number.
8
Work out 314 − 134.
9
Work out 5 × 23. Write your answer as a mixed number.
10
A film lasts 90 minutes. 23 of the film has been shown. How many minutes of the film have been shown?

Total 19 marks. Allow about 19 minutes.

Exercise 6X  SATs-style questions

Non-calculator
1
FWhich fraction is equivalent to 35?   Tick one box.    A 610   B 58   C 310   D 65[1 mark]
2
FWrite the missing number.   56 = ?18[1 mark]
3
FWrite 175 as a mixed number.[1 mark]
4
FWrite these fractions in order, starting with the smallest.    58   34   12   916[2 marks]
5
FWork out 356 + 113.   Show your working.[2 marks]
6
FWork out 416 − 256.   Show your working.[2 marks]
7
RNoah says, “712 is greater than 35.” Is Noah correct? Prove it.[2 marks]
8
PSA loaf of bread needs 38 kg of flour. How much flour is needed for 4 loaves?   Give your answer as a mixed number in its simplest form.[2 marks]
9
PSMr Khan has 72 stickers. He gives 14 of them to Leo and 13 of them to Mei. How many stickers does he have left?   Show your working.[3 marks]
10
PSAisha reads 14 of her book on Monday and 38 of it on Tuesday. She has 21 pages left to read. How many pages are in the whole book?   Show your working.[3 marks]
← Chapter 5: DivisionChapter 7: Decimals and percentages →

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