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SATs practice · Year 6Chapter 3 · Fractions
Year 6 · Chapter 3

Fractions

By the end of this chapter you will be able to

  • Simplify a fraction by dividing by a common factor
  • Find equivalent fractions and compare fractions with different denominators
  • Convert between mixed numbers and improper fractions
  • Add and subtract fractions, including mixed numbers
  • Multiply and divide fractions, and write a fraction as a decimal

You should already know

  • Equivalent fractions and simple fractions of amounts (Year 4 and 5 maths)
  • Common multiples and factors (Year 5 maths)

Where this leads

  • Decimals and percentages, including fractions as percentages (Chapter 4 of this book)

KS2 SATs practice for arithmetic (Year 6). Original questions in the style of the national tests. Content: Compare and order fractions, including fractions greater than 1 (Year 5-6), Identify, name and write equivalent fractions, and use common factors to simplify (Year 5-6), Recognise mixed numbers and improper fractions, and convert between them (Year 5), Add and subtract fractions with the same denominator or with denominators that are multiples of the same number (Year 5), Add and subtract fractions with different denominators and mixed numbers, using equivalent fractions and common multiples (Year 6), Multiply proper fractions and mixed numbers by whole numbers (Year 5); multiply simple pairs of proper fractions in simplest form (Year 6), Divide proper fractions by whole numbers; associate a fraction with division and find decimal equivalents (Year 6).

Exercise P  Do you remember?

Non-calculator
1
FWork out 1/3 of 27.
2
FWork out 3/4 of 12.
3
FWhich fraction is equal to 1/2? Choose one: 2/5, 3/4 or 4/8.
4
FWork out 2/5 + 1/5.
5
FWork out 5 × 4.
3.1

Equivalent fractions and simplifying

Year 6
Identify, name and write equivalent fractions, and use common factors to simplify (Year 5-6)
Video lesson: Equivalent fractions and simplifyingComing soon
Practice material. Non-calculator practice. This book is practice in the style of the Year 6 arithmetic test. It is not the official test, and no official test item is used. Do every question without a calculator. Marks and timings are a practice guide only.
Story. Maya makes a fruit salad. The recipe says 36/48 of the bowl should be apples. She wants the simplest way to write that fraction, so that she can check it on the recipe card.

Rule

Equivalent fractions have the same value but different numbers. Multiply the top and the bottom by the same number to make an equivalent fraction.

To simplify, divide the top and bottom by a common factor. A fraction is in its simplest form when no number divides into both except 1.

For the full explanation, read Year 6 maths, chapter 2.

Key ideas

  • Multiply or divide the top and bottom by the same number to get an equivalent fraction.
  • The highest common factor (HCF) gives the simplest form in one step.
  • Check a simplified fraction by seeing whether the top and bottom still share a factor.
Worked example 3.1.1
Simplify 36/48.

Method. Find the highest common factor of the top and bottom, then divide both by it.

  1. Find a number that divides into both the top and the bottom. The highest common factor of 36 and 48 is 12.
    HCF of 36 and 48 is 12
  2. Divide the top and the bottom by it. Dividing both by the same number keeps the fraction the same size.
    36 ÷ 12 = 3, 48 ÷ 12 = 4
  3. Write the answer.
    3/4

Check. 3 × 12 = 36 and 4 × 12 = 48, so 3/4 is the same fraction. ✓

Watch out. Dividing by 2 only gives 18/24, which can still be simplified.

Worked example 3.1.2
Write 3/5 as an equivalent fraction with denominator 20.

Method. Find what 5 is multiplied by to make 20, then do the same to the top.

  1. Multiply the top and the bottom by the same number. Both are multiplied by 4, so the value of the fraction does not change.
    3 × 4 = 12 and 5 × 4 = 20
  2. Check the bottom. 5 × 4 = 20, which is the new bottom number.
    5 × 4 = 20
  3. Write the answer.
    3/5

Check. 20 ÷ 5 = 4, so the top is 3 × 4 = 12. ✓

Worked example 3.1.3
Fill in the missing number: 5/8 = ☐/40.

Method. Multiply the bottom by 5, and do the same to the top.

  1. Multiply the top and the bottom by the same number. Both are multiplied by 5, so the value of the fraction does not change.
    5 × 5 = 25 and 8 × 5 = 40
  2. Check the bottom. 8 × 5 = 40, which is the new bottom number.
    8 × 5 = 40
  3. Write the answer.
    5/8

Check. 40 ÷ 8 = 5, so the top is 5 × 5 = 25. ✓

Exercise 3A  Practice: equivalent fractions and simplifying

Non-calculator
1
FSimplify 12/18.
2
FSimplify 45/60.
3
FSimplify 28/42.
4
FSimplify 16/40.
5
FSimplify 36/54.
6
FSimplify 24/30.
7
FSimplify 56/72.
8
FSimplify 9/27.
9
FWrite 3/4 with a denominator of 20.
10
FWrite 2/7 with a denominator of 35.
11
FFill in the box: 4/9 = ☐/36.
12
PSAli eats 6 of the 12 slices of a pizza. Write the fraction he ate in its simplest form.
13
PSA class has 35 pupils. 49 pupils took part in the trip. Write 35/49 in its simplest form.
14
RSam says 5/10 and 1/2 are not equal because the numbers look different. Is he right? Explain.
3.2

Comparing and ordering fractions

Year 6
Compare and order fractions, including fractions greater than 1 (Year 5-6)
Video lesson: Comparing and ordering fractionsComing soon
Story. In a race, Tom has run 3/4 of a lap and Ruby has run 5/8 of a lap. The coach wants to know who is further round the track.

Rule

To compare fractions, write them with the same denominator. The fraction with the bigger top number is bigger, when the bottoms are the same.

Fractions greater than 1 work the same way: 7/4 is bigger than 5/3 once both are written with 12 as the denominator.

Key ideas

  • Write the fractions with a common denominator, then compare the tops.
  • The bigger the top for the same bottom, the bigger the fraction.
  • Fractions greater than 1 can be compared the same way.
Worked example 3.2.1
Which is greater: 3/4 or 5/8? Write the sign that goes in the box: 3/4 ☐ 5/8.

Method. Write both fractions with the same denominator, then compare.

  1. Find a common denominator. 4 and 8 both divide into 8.
    8
  2. Write both fractions with that denominator. Multiply each top and bottom by the same number.
    3/4 = 6/8 and 5/8 = 5/8
  3. Compare the top numbers. 6 and 5 tell us which fraction is bigger.
    6 > 5, so 3/4 > 5/8
  4. Write the answer.
    >

Check. 3/4 = 6/8, which is bigger than 5/8. ✓

Watch out. Compare the tops only after the bottoms are equal.

Worked example 3.2.2
Which is greater: 7/4 or 5/3? Write the sign that goes in the box: 7/4 ☐ 5/3.

Method. Use the common denominator 12 for both.

  1. Find a common denominator. 4 and 3 both divide into 12.
    12
  2. Write both fractions with that denominator. Multiply each top and bottom by the same number.
    7/4 = 21/12 and 5/3 = 20/12
  3. Compare the top numbers. 21 and 20 tell us which fraction is bigger.
    21 > 20, so 7/4 > 5/3
  4. Write the answer.
    >

Check. 7/4 = 1 3/4 and 5/3 = 1 2/3, and 1 3/4 is bigger. ✓

Worked example 3.2.3
Put 5/6, 2/3 and 7/12 in order, smallest first.

Method. Write each fraction with the denominator 12, then order the tops.

  1. Write with denominator 12. 5/6 = 10/12, 2/3 = 8/12 and 7/12 stays the same.
    10/12, 8/12, 7/12
  2. Order the tops from smallest. 7 is smallest, then 8, then 10.
    7/12 < 8/12 < 10/12
  3. Write the original fractions. Put back the original fractions in the same order.
    7/12, 2/3, 5/6
  4. Write the answer.
    7/12, 2/3, 5/6

Check. The order is 7/12 (0.58), 2/3 (0.67), 5/6 (0.83). ✓

Exercise 3B  Practice: comparing and ordering

Non-calculator
1
FFill in the box with <, > or =: 3/4 ☐ 5/8
2
FFill in the box with <, > or =: 2/3 ☐ 3/5
3
FFill in the box with <, > or =: 7/10 ☐ 3/4
4
FFill in the box with <, > or =: 5/6 ☐ 7/9
5
FFill in the box with <, > or =: 4/5 ☐ 9/11
6
FFill in the box with <, > or =: 1/2 ☐ 4/9
7
FFill in the box with <, > or =: 7/4 ☐ 5/3
8
PSPut these in order, largest first: 5/8, 3/4, 1/2.
9
PSWhich is greater: 7/8 or 11/12? Give the greater fraction.
10
RFill in the box with <, > or =: 9/4 ☐ 7/3
11
RAli says 1/3 is greater than 1/2 because 3 is greater than 2. Is he right? Explain.
3.3

Mixed numbers and improper fractions

Year 6
Recognise mixed numbers and improper fractions, and convert between them (Year 5)
Video lesson: Mixed numbers and improper fractionsComing soon
Story. A baker has 4 3/8 kilograms of flour in a sack and wants to put it into bags of 1/8 kilogram. He writes the amount as an improper fraction to count the eighths.

Rule

A mixed number has a whole number and a fraction, such as 2 3/4. An improper fraction has a top number that is bigger than its bottom number, such as 11/4.

To change a mixed number to an improper fraction: multiply the whole number by the bottom, add the top, and keep the bottom. To change back, divide the top by the bottom.

Key ideas

  • 2 3/4 = 11/4, because 2 × 4 + 3 = 11.
  • 17/5 = 3 2/5, because 17 ÷ 5 = 3 remainder 2.
  • An improper fraction has a top number bigger than its bottom number.
Worked example 3.3.1
Write 2 3/4 as an improper fraction.

Method. Multiply the whole number by the bottom, then add the top.

  1. Multiply the whole number by the bottom number. Each whole has 4 parts, so 2 wholes have 2 × 4 parts.
    2 × 4 = 8
  2. Add the top number. Add the 3 parts that are left over.
    8 + 3 = 11
  3. Keep the same bottom number. The size of each part does not change.
    11/4
  4. Write the answer.
    11/4

Check. 11/4 = 2 3/4, because 4 × 2 = 8 and 8 + 3 = 11. ✓

Worked example 3.3.2
Write 17/5 as a mixed number.

Method. Divide the top by the bottom: the whole number and the remainder make the mixed number.

  1. Divide the top number by the bottom number. 17 ÷ 5 = 3 with 2 left over.
    17 ÷ 5 = 3 r 2
  2. Write the answer as a mixed number. The whole number is 3, and the 2 left over goes over 5.
    3 2/5
  3. Write the answer.
    3 2/5

Check. 3 × 5 + 2 = 17. ✓

Watch out. The remainder goes on top of the same bottom: 2/5, not 2/3.

Worked example 3.3.3
Fill in the box with <, > or =: 2 1/3 ☐ 7/3.

Method. Change the mixed number to an improper fraction, then compare.

  1. The bottom numbers are the same. So compare the top numbers.
    7/3 and 7/3
  2. Compare the top numbers. 7 and 7 tell us which fraction is bigger.
    7 = 7, so 7/3 = 7/3
  3. Write the answer.
    =

Check. 2 1/3 = 7/3. ✓

Exercise 3C  Practice: mixed numbers and improper fractions

Non-calculator
1
FWrite 1 2/3 as an improper fraction.
2
FWrite 3 1/2 as an improper fraction.
3
FWrite 2 3/8 as an improper fraction.
4
FWrite 5 1/4 as an improper fraction.
5
FWrite 4 3/5 as an improper fraction.
6
FWrite 2 1/6 as an improper fraction.
7
FWrite 11/4 as a mixed number.
8
FWrite 17/6 as a mixed number.
9
FWrite 23/5 as a mixed number.
10
FWrite 14/3 as a mixed number.
11
FWrite 19/8 as a mixed number.
12
PSA baker has 4 3/8 kg of flour. Write this as an improper fraction, counting eighths.
13
PSA rope is 50/7 metres long. Write its length as a mixed number of metres.
14
RExplain why 9/4 is the same as 2 1/4.
15
PSWrite 13/4 as a mixed number.
16
FWrite 3 2/5 as an improper fraction.
3.4

Adding and subtracting fractions

Year 6
Add and subtract fractions with the same denominator or with denominators that are multiples of the same number (Year 5), Add and subtract fractions with different denominators and mixed numbers, using equivalent fractions and common multiples (Year 6)
Video lesson: Adding and subtracting fractionsComing soon
Story. Sam walks 3/5 of a kilometre to the library and 1/4 of a kilometre to the shop. He wants the total distance, so he writes the two fractions with a common denominator.

Rule

If the bottoms are the same, add or subtract the tops and keep the bottom.

If the bottoms are different, first write both fractions with a common denominator, which is a common multiple of the bottoms. Then add or subtract the tops.

Key ideas

  • Same bottom: add or subtract the tops only.
  • Different bottoms: find a common multiple, make equivalent fractions, then add or subtract.
  • Give the answer in its simplest form, and as a mixed number if the answer is bigger than 1.
Worked example 3.4.1
Work out 2/7 + 3/7.

Method. The bottoms are the same, so add the tops.

  1. The bottom numbers are the same. So add the top numbers and keep the bottom number.
    2 + 3 = 5, so 5/7
  2. Write the answer. The fraction cannot be simplified.
    5/7
  3. Write the answer.
    5/7

Check. 5/7 is less than one whole, so the answer is sensible. ✓

Worked example 3.4.2
Work out 3/4 + 2/5. Give the answer as a mixed number.

Method. Find a common denominator of 4 and 5, which is 20.

  1. Find a common denominator. 4 and 5 both divide into 20.
    20
  2. Make equivalent fractions. Multiply each top and bottom to get the common denominator.
    3/4 = 15/20 and 2/5 = 8/20
  3. Add the top numbers. The bottom number stays the same.
    15 + 8 = 23, so 23/20
  4. Write the answer. The fraction cannot be simplified.
    23/20
  5. Write the answer as a mixed number. 23 ÷ 20 = 1 remainder 3.
    23/20 = 1 3/20
  6. Write the answer.
    1 3/20

Check. 15/20 + 8/20 = 23/20 = 1 3/20. ✓

Worked example 3.4.3
Work out 5/6 - 1/4.

Method. Find a common denominator of 6 and 4, which is 12.

  1. Find a common denominator. 6 and 4 both divide into 12.
    12
  2. Make equivalent fractions. Multiply each top and bottom to get the common denominator.
    5/6 = 10/12 and 1/4 = 3/12
  3. Subtract the top numbers. The bottom number stays the same.
    10 - 3 = 7, so 7/12
  4. Write the answer. The fraction cannot be simplified.
    7/12
  5. Write the answer.
    7/12

Check. 10/12 - 3/12 = 7/12. ✓

Watch out. Do not subtract the bottoms: 5/6 - 1/4 is not 4/2.

Exercise 3D  Practice: adding and subtracting fractions

Non-calculator
1
FWork out 2/9 + 4/9.
2
FWork out 3/10 + 1/2.
3
FWork out 4/7 + 2/7.
4
FWork out 1/2 + 1/3.
5
FWork out 3/4 + 1/8.
6
FWork out 2/5 + 1/10.
7
FWork out 5/6 - 1/3.
8
FWork out 7/8 - 1/4.
9
FWork out 2/3 - 2/9.
10
PSWork out 2 1/3 + 1 1/2. Give the answer as a mixed number.
11
PSWork out 3 3/4 - 1 1/4. Give the answer as a mixed number.
12
PSSam walks 3/5 km to the library and 1/4 km to the shop. How far does he walk altogether?
13
RAli says 1/2 + 1/3 = 2/5. Is he right? Explain, and give the correct answer.
14
RWork out 9/10 - 3/5.
3.5

Multiplying and dividing fractions, and decimals

Year 6
Multiply proper fractions and mixed numbers by whole numbers (Year 5), Multiply simple pairs of proper fractions, giving the answer in its simplest form (Year 6), Divide proper fractions by whole numbers; associate a fraction with division and find decimal equivalents (Year 6)
Video lesson: Multiplying and dividing fractions, and decimalsComing soon
Story. A jug holds 3/4 of a litre of juice. The class has 8 jugs of the same size and needs to know how much juice that is in total. Later, a teacher shares 3/4 of a litre equally between 6 cups.

Rule

To multiply a fraction by a whole number, multiply the top and keep the bottom. Then simplify.

To multiply two fractions, multiply the tops and multiply the bottoms, then simplify.

To divide a fraction by a whole number, keep the top and multiply the bottom by the whole number.

A fraction is a division: 4/5 means 4 ÷ 5 = 0.8.

Key ideas

  • Fraction × whole number: multiply the top.
  • Fraction × fraction: multiply the tops and the bottoms.
  • Fraction ÷ whole number: multiply the bottom.
  • A fraction can be written as a decimal by making the bottom 10, 100 or 1000.
Worked example 3.5.1
Work out 3/4 × 8.

Method. Multiply the top by 8, then simplify.

  1. Multiply the top number by the whole number. 3 × 8 = 24, so the top becomes 24.
    3 × 8 = 24
  2. Simplify the fraction. Divide the top and the bottom by 4.
    24/4 = 6
  3. Write the answer.
    6

Check. 8 ÷ 4 = 2 and 2 × 3 = 6. ✓

Watch out. Simplify 24/4 to get a whole number: 6.

Worked example 3.5.2
Work out 2/3 × 3/5. Give the answer in its simplest form.

Method. Multiply the tops, multiply the bottoms, then simplify.

  1. Multiply the top numbers. 2 × 3 = 6.
    2 × 3 = 6
  2. Multiply the bottom numbers. 3 × 5 = 15.
    3 × 5 = 15
  3. Simplify. Divide the top and bottom by 3.
    6/15 = 2/5
  4. Write the answer.
    2/5

Check. 6/15 = 2/5 when both are divided by 3. ✓

Worked example 3.5.3
Work out 3/4 ÷ 6.

Method. Keep the top, and multiply the bottom by 6.

  1. Keep the top number the same. The top stays at 3.
    3
  2. Multiply the bottom number by the whole number. Sharing into 6 equal parts makes each part 6 times smaller.
    4 × 6 = 24
  3. Simplify. Divide the top and bottom by 3.
    3/24 = 1/8
  4. Write the answer.
    1/8

Check. 1/8 × 6 = 6/8 = 3/4. ✓

Exercise 3E  Practice: multiplying and dividing fractions

Non-calculator
1
FWork out 3/8 × 4. Give the answer as a whole number or mixed number.
2
FWork out 5/6 × 9. Give the answer as a whole number or mixed number.
3
FWork out 2/5 × 15. Give the answer as a whole number or mixed number.
4
FWork out 5/12 × 8. Give the answer as a whole number or mixed number.
5
FWork out 2/3 × 3/5.
6
FWork out 3/4 × 2/9.
7
FWork out 5/6 × 3/10.
8
FWork out 3/4 ÷ 6.
9
FWork out 2/5 ÷ 4.
10
FWork out 5/6 ÷ 10.
11
FWrite 4/5 as a decimal.
12
FWrite 3/4 as a decimal.
13
RWrite 1/8 as a decimal.
14
PSA jug holds 3/4 of a litre. How much juice is in 8 jugs, in litres?
15
PSA teacher shares 3/4 of a litre equally between 6 cups. How much goes in each cup, as a fraction of a litre?
16
RWork out 4/5 × 5/6 and give the answer in its simplest form.
17
RExplain why 1/2 ÷ 2 is smaller than 1/2.

Chapter summary

SkillMethod
SimplifyDivide the top and bottom by their highest common factor: 36/48 = 3/4.
Equivalent fractionMultiply the top and bottom by the same number: 3/5 = 12/20.
CompareWrite with a common denominator, then compare the tops.
Mixed and improper2 3/4 = 11/4 (2 × 4 + 3); 17/5 = 3 2/5 (17 ÷ 5 = 3 remainder 2).
Add and subtractCommon denominator, equivalent fractions, then add or subtract the tops.
MultiplyFraction × whole: multiply the top. Fraction × fraction: multiply tops and bottoms.
Divide by wholeKeep the top, multiply the bottom: 3/4 ÷ 6 = 3/24 = 1/8.
DecimalMake the bottom 10, 100 or 1000: 4/5 = 8/10 = 0.8.

Can you…?

  • I can simplify a fraction to its simplest form
  • I can find equivalent fractions and compare fractions with different bottoms
  • I can change between mixed numbers and improper fractions
  • I can add and subtract fractions with different bottoms
  • I can multiply and divide fractions, and simplify the answer
  • I can write a fraction as a decimal

Key terms

equivalent fraction
a fraction with the same value written with different numbers
simplest form
a fraction that cannot be made smaller by dividing the top and bottom by the same number
highest common factor
the biggest number that divides exactly into two or more numbers
common denominator
the same bottom number for two or more fractions
mixed number
a whole number with a fraction, such as 2 3/4
improper fraction
a fraction with a top number bigger than the bottom, such as 11/4
proper fraction
a fraction with a top number smaller than the bottom, such as 3/4

Exercise 3Z  Check-up

Non-calculator
1
Simplify 42/56.
2
Write 5/8 with a denominator of 24.
3
Fill in the box with <, > or =: 5/6 ☐ 7/8
4
Write 19/6 as a mixed number.
5
Write 3 5/9 as an improper fraction.
6
Work out 3/7 + 2/7.
7
Work out 7/9 - 1/3.
8
Work out 2/5 × 15.
9
Work out 5/6 × 3/10.
10
Write 3/5 as a decimal.

Total 19 marks. Allow about 19 minutes.

Exercise 3X  Practice test questions

Non-calculator
1
FSimplify 48/64.[1 mark]
2
FWrite 2/3 as an equivalent fraction with denominator 12.[1 mark]
3
FFill in the box with <, > or =: 3/5 ☐ 4/7[2 marks]
4
FWork out 5/9 + 2/9.[1 mark]
5
FWork out 3/4 + 1/6. Give the answer in its simplest form.[2 marks]
6
PSWork out 4 1/2 - 2 3/4. Give the answer as a mixed number.[2 marks]
7
FWork out 5/8 × 16.[1 mark]
8
FWork out 3/5 × 4/9.[2 marks]
9
FWork out 4/9 ÷ 2.[2 marks]
10
FWrite 7/20 as a decimal.[2 marks]
11
RA recipe needs 2/3 of a cup of milk, and Ali makes it 3 times. Explain how much milk he needs, as a mixed number.[3 marks]
← Chapter 2: Multiplication and divisionChapter 4: Decimals and percentages →

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