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SATs practice · Year 6Chapter 5 · Ratio and proportion
Year 6 · Chapter 5

Ratio and proportion

By the end of this chapter you will be able to

  • Simplify a ratio and write it as a fraction of the whole
  • Find a missing value in an equivalent ratio
  • Share an amount in a given ratio, including three parts
  • Find a scale factor and use it on a similar shape
  • Use the unit method, and write one amount as a percentage of another

You should already know

  • Equivalent fractions and simplifying (Chapter 3 of this book)
  • Percentages of amounts (Chapter 4 of this book)

Where this leads

  • Algebra and formulae, including simple proportion in a formula (Chapter 6 of this book)

KS2 SATs practice for arithmetic (Year 6). Original questions in the style of the national tests. Content: Solve problems involving the relative sizes of two quantities, finding missing values by integer multiplication or division (Year 5-6), Use the ratio language a:b and the link between ratio and fractions (Year 6), Solve problems of unequal sharing and grouping using knowledge of fractions and multiples (Year 6), Solve problems involving similar shapes where the scale factor is known or can be found (Year 6), Use the unit method for proportion, including percentages as a proportion of a whole (Year 5-6).

Exercise P  Do you remember?

Non-calculator
1
FWrite 6/8 in its simplest form.
2
FWork out 1/5 of 40.
3
FShare 12 sweets equally between 4 children. How many does each child get?
4
FWork out 3 × 7.
5
FWhat is 50% of 90?
5.1

Ratio language and fractions

Year 6
Use the ratio language a:b and the link between ratio and fractions (Year 6)
Video lesson: Ratio language and fractionsComing 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. In Year 6 a class has 3 boys for every 5 girls. The teacher writes this as the ratio 3:5 and asks the class what fraction of the pupils are boys.

Rule

A ratio compares two amounts and is written a:b. Simplify it by dividing both parts by their highest common factor: 12:20 = 3:5.

The total parts are a + b. The first part is a/(a + b) of the whole, and the second part is b/(a + b).

For the full work on ratio, read Year 6 maths, chapter 5.

Key ideas

  • A ratio is written a:b, and it compares two quantities.
  • Simplify a ratio by dividing both parts by the same number.
  • Add the parts to find the total number of parts.
  • The fraction for the first part is a/(a + b).
Worked example 5.1.1
Simplify the ratio 12:20.

Method. Divide both parts by the highest common factor.

  1. Find the highest common factor of the two numbers. The biggest number that divides into 12 and 20 is 4.
    HCF = 4
  2. Divide both parts of the ratio by it. Dividing both parts by the same number keeps the ratio the same.
    12 ÷ 4 = 3 and 20 ÷ 4 = 5
  3. Write the answer.
    3:5

Check. 3 × 4 = 12 and 5 × 4 = 20. ✓

Watch out. Simplify fully: 6:10 is not the simplest form, 3:5 is.

Worked example 5.1.2
In a class, boys : girls = 3 : 5. What fraction of the class are boys?

Method. Find the total number of parts, then write the boys' parts as a fraction of the total.

  1. Add the parts. 3 + 5 = 8 parts altogether.
    3 + 5 = 8
  2. Write the boys as a fraction. Boys are 3 of the 8 parts.
    3/8
  3. Write the answer.
    3/8

Check. The girls are 5/8, and 3/8 + 5/8 = 1. ✓

Worked example 5.1.3
A class of 24 pupils has boys : girls = 3 : 5. How many boys are there?

Method. Share the 24 pupils into the parts of the ratio.

  1. Add the parts of the ratio. 3 + 5 = 8, so the total is shared into 8 equal parts.
    3 + 5 = 8 parts
  2. Find the value of one part. Share the total into 8 equal parts.
    24 ÷ 8 = 3
  3. Find each share. Multiply the value of one part by each number in the ratio.
    3 × 3 = 9 and 5 × 3 = 15
  4. Write the answer.
    9

Check. 9 boys and 15 girls make 24 pupils. ✓

Exercise 5A  Practice: ratio language and fractions

Non-calculator
1
FWrite 12:20 in its simplest form.
2
FWrite 9:6 in its simplest form.
3
FWrite 14:21 in its simplest form.
4
FWrite 15:45 in its simplest form.
5
FWrite 8:24 in its simplest form.
6
FWrite 16:40 in its simplest form.
7
FWrite 30:45 in its simplest form.
8
FA ratio is 3 : 4. What fraction of the total is the first part?
9
FA ratio is 3 : 4. What fraction of the total is the second part?
10
PSA school has 45 pupils. The ratio of boys to girls is 4 : 5. How many boys are there?
11
PSA juice drink is made with orange and water in the ratio 1 : 4. There are 500 ml altogether. How much orange is there?
12
RAli says 2 : 6 is the same as 1 : 3. Is he right? Write Yes or No.
13
RA ratio of red to blue counters is 2 : 3. Explain why the blue counters are 3/5 of all the counters.
5.2

Missing values in equivalent ratios

Year 6
Solve problems involving the relative sizes of two quantities, finding missing values by integer multiplication or division (Year 5-6)
Video lesson: Missing values in equivalent ratiosComing soon
Story. Paint is mixed by using 1 part blue to 4 parts white. Ruby knows she has 5 litres of blue paint, and she needs to find how much white she must use to keep the same mix.

Rule

In an equivalent ratio, both parts are multiplied or divided by the same whole number.

To find a missing value, work out what one unit (or one part) is worth, then multiply by the number of units you need.

Key ideas

  • Multiply or divide both parts by the same whole number.
  • Find the value of one part, then multiply by the number of parts.
  • Check that the new ratio simplifies to the original one.
Worked example 5.2.1
The ratio of red to blue sweets is 3 : 5. There are 15 blue sweets. How many red sweets are there?

Method. Find the value of one part from the blue sweets, then multiply by the red parts.

  1. Find the value of one unit. The blue sweets is 5 units, so one unit is 15 ÷ 5.
    15 ÷ 5 = 3
  2. Multiply by the number of units you need. The other amount has 3 units.
    3 × 3 = 9
  3. Write the answer.
    9

Check. 9 red and 15 blue: 9 : 15 = 3 : 5. ✓

Worked example 5.2.2
Fill in the missing number: 4 : 7 = ☐ : 21.

Method. Find what 7 is multiplied by to make 21, then do the same to 4.

  1. Find the scale factor. 7 × 3 = 21, so multiply by 3.
    7 × 3 = 21
  2. Multiply the first part. 4 × 3 = 12.
    4 × 3 = 12
  3. Check the ratio. 12 : 21 simplifies to 4 : 7.
    12 : 21 = 4 : 7
  4. Write the answer.
    12

Check. 12 : 21 = 4 : 7. ✓

Worked example 5.2.3
A cake recipe uses flour : sugar = 2 : 1. There are 300 g of flour. How much sugar is needed?

Method. Find one part from the flour, then take one part for sugar.

  1. Find the value of one unit. The flour is 2 units, so one unit is 300 ÷ 2.
    300 ÷ 2 = 150
  2. Multiply by the number of units you need. The other amount has 1 units.
    150 × 1 = 150
  3. Write the answer.
    150 g

Check. 300 g of flour is 2 parts, so 1 part is 150 g. ✓

Exercise 5B  Practice: missing values in equivalent ratios

Non-calculator
1
FFill in the missing number: 3 : 4 = ☐ : 20.
2
FFill in the missing number: 5 : 8 = ☐ : 40.
3
FFill in the missing number: 2 : 9 = ☐ : 36.
4
FFill in the missing number: 4 : 9 = ☐ : 45.
5
FThe ratio of apples to bananas is 4 : 3. There are 28 apples. How many bananas are there?
6
FWrite 5 : 2 with the first number 25.
7
PSPaint is mixed in the ratio 1 blue : 4 white. Ruby uses 5 litres of blue. How many litres of white paint does she need?
8
PSIn a club, boys : girls = 3 : 2. There are 30 girls. How many boys are there?
9
PSA recipe uses sugar : flour = 1 : 8. There is 1200 g of flour. How many grams of sugar are needed?
10
PSThe ratio of red to green beads is 6 : 5. There are 360 red beads. How many green beads are there?
11
RExplain how you know that 6 : 9 is the same as 2 : 3.
12
RWhich ratio is equivalent to 2 : 3? Choose one: 4 : 9, 6 : 9, 3 : 2 or 5 : 6.
5.3

Sharing and grouping in a ratio

Year 6
Solve problems of unequal sharing and grouping using knowledge of fractions and multiples (Year 6)
Video lesson: Sharing and grouping in a ratioComing soon
Story. Ali and Ben earn a prize of £60 for the school quiz. They agree to share it in the ratio 2 : 3, because Ali answered fewer questions. Ruby then wants to share the same method among three friends.

Rule

To share in a ratio, add the parts, find the value of one part, then multiply each number in the ratio by that value.

Check the shares add up to the total. With three friends, add all three parts of the ratio.

Key ideas

  • Share in a ratio by finding the value of one part.
  • The total must be shared into all the parts of the ratio.
  • The shares must add up to the total.
Worked example 5.3.1
Share £60 between Ali and Ben in the ratio 2 : 3. How much does each get?

Method. Find the total parts, the value of one part, then each share.

  1. Add the parts of the ratio. 2 + 3 = 5, so the total is shared into 5 equal parts.
    2 + 3 = 5 parts
  2. Find the value of one part. Share the total into 5 equal parts.
    60 ÷ 5 = 12
  3. Find each share. Multiply the value of one part by each number in the ratio.
    2 × 12 = 24 and 3 × 12 = 36
  4. Write the answer.
    Ali £24, Ben £36

Check. £24 + £36 = £60. ✓

Worked example 5.3.2
Share 84 sweets in the ratio 3 : 4. How many sweets does each part get?

Method. Find one part by dividing by 7.

  1. Add the parts of the ratio. 3 + 4 = 7, so the total is shared into 7 equal parts.
    3 + 4 = 7 parts
  2. Find the value of one part. Share the total into 7 equal parts.
    84 ÷ 7 = 12
  3. Find each share. Multiply the value of one part by each number in the ratio.
    3 × 12 = 36 and 4 × 12 = 48
  4. Write the answer.
    36 and 48

Check. 36 + 48 = 84. ✓

Worked example 5.3.3
Share 120 in the ratio 2 : 3 : 5 between three children. How much does the child with 5 parts get?

Method. Add all three parts, then find the value of one part.

  1. Add the parts. 2 + 3 + 5 = 10 parts.
    10 parts
  2. Find one part. 120 ÷ 10 = 12.
    12
  3. Find the 5-part share. 5 × 12 = 60.
    60
  4. Write the answer.
    60

Check. 24 + 36 + 60 = 120. ✓

Exercise 5C  Practice: sharing and grouping in a ratio

Non-calculator
1
FShare 48 in the ratio 1 : 3. Write both shares.
2
FShare 75 in the ratio 2 : 3. Write both shares.
3
FShare 90 in the ratio 4 : 5. Write both shares.
4
FShare 120 in the ratio 3 : 5. Write both shares.
5
FShare 63 in the ratio 4 : 5. Write both shares.
6
FShare 54 in the ratio 1 : 8. Write both shares.
7
FShare 25 in the ratio 2 : 3. Write both shares.
8
PSA prize of £250 is shared between Ali and Ben in the ratio 3 : 2. How much does Ali get?
9
PS96 counters are sorted into red and blue in the ratio 5 : 7. How many are red?
10
PSThree children share 120 in the ratio 2 : 3 : 5. How much does the child with 2 parts get?
11
PSA bag has 36 marbles in the ratio 1 : 2 : 3 of three colours. How many marbles are the most common colour?
12
RBen says that if 40 is shared in the ratio 3 : 2, each person gets 20. Is he right? Explain.
13
R72 pupils are put into two groups in the ratio 1 : 2. How many pupils are in the bigger group?
5.4

Similar shapes and scale factors

Year 6
Solve problems involving similar shapes where the scale factor is known or can be found (Year 6)
Video lesson: Similar shapes and scale factorsComing soon
Story. A photo of a poster is 4 cm wide and 6 cm high. The printer enlarges it so that the new width is 14 cm. The class wants to know the new height.

Rule

In similar shapes, every length is multiplied by the same scale factor. Scale factor = new length ÷ old length.

A scale factor bigger than 1 makes the shape bigger; less than 1 makes it smaller.

Key ideas

  • Scale factor = new length ÷ old length.
  • Multiply every length by the same scale factor.
  • Check the scale factor using a second pair of lengths.
Worked example 5.4.1
A rectangle is 4 cm wide and 6 cm high. The enlargement is 14 cm wide. How high is the enlargement?

Method. Find the scale factor, then multiply the height by it.

  1. Find the scale factor. The scale factor is the new side divided by the old side.
    14 ÷ 4 = 7/2
  2. Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.
    6 × 7/2 = 21
  3. Write the answer.
    21 cm

Check. 21 ÷ 6 = 3.5, and 14 ÷ 4 = 3.5. ✓

Worked example 5.4.2
A line of 5 cm is enlarged to 35 cm. What is the scale factor, and what does a 2 cm line become?

Method. The scale factor is 35 ÷ 5, and then multiply the other length.

  1. Find the scale factor. The scale factor is the new side divided by the old side.
    35 ÷ 5 = 7
  2. Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.
    2 × 7 = 14
  3. Write the answer.
    7 and 14 cm

Check. The scale factor is 7, and 2 × 7 = 14. ✓

Worked example 5.4.3
A triangle side of 3 cm becomes 12 cm in an enlargement. A side of 5 cm becomes how long?

Method. Find the scale factor, then multiply.

  1. Find the scale factor. The scale factor is the new side divided by the old side.
    12 ÷ 3 = 4
  2. Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.
    5 × 4 = 20
  3. Write the answer.
    20 cm

Check. 12 ÷ 3 = 4, and 5 × 4 = 20. ✓

Exercise 5D  Practice: similar shapes and scale factors

Non-calculator
1
FA shape is enlarged so 5 cm becomes 35 cm. A side of 4 cm becomes how long?
2
FA shape is enlarged so 3 cm becomes 18 cm. A side of 7 cm becomes how long?
3
FA shape is enlarged so 2 cm becomes 8 cm. A side of 5 cm becomes how long?
4
FA shape is enlarged so 6 cm becomes 15 cm. A side of 4 cm becomes how long?
5
FA shape is enlarged so 9 cm becomes 12 cm. A side of 6 cm becomes how long?
6
FA shape is enlarged so 4 cm becomes 10 cm. A side of 6 cm becomes how long?
7
FA line of 4 cm is enlarged to 10 cm. What is the scale factor?
8
PSA rectangle 4 cm by 9 cm is enlarged so that the 4 cm side becomes 12 cm. How long is the other side?
9
PSA picture is 6 cm wide and 10 cm high. It is enlarged so that the width becomes 9 cm. What is the new height?
10
PSA model car is 12 cm long. The real car is 300 cm long. How many times longer is the real car than the model?
11
RAli says a rectangle 2 cm by 3 cm has the same shape as one 4 cm by 5 cm. Is he right? Explain.
12
RA rectangle 2 cm by 5 cm is enlarged to 8 cm by 20 cm. What scale factor is used for both lengths?
5.5

The unit method and percentages of a whole

Year 6
Use the unit method for proportion, including percentages as a proportion of a whole (Year 5-6)
Video lesson: The unit method and percentages of a wholeComing soon
Story. A bakery sells 3 packs of biscuits for £7.50. A café wants 5 of the same packs. The café also wants to know what percentage of the pupils at a school are girls.

Rule

The unit method works out the value of one unit first, then multiplies to find the amount you want.

A percentage of a whole is the part divided by the whole, multiplied by 100: 90 out of 240 is 90/240 × 100 = 37.5%.

Key ideas

  • Unit method: find one unit first, then multiply.
  • Percentage of a whole = part ÷ whole × 100.
  • Percentages are out of 100, so 50% is one half.
Worked example 5.5.1
Three packs of biscuits cost £7.50. How much do 5 packs cost?

Method. Find the cost of one pack, then the cost of 5 packs.

  1. Find the cost of one pack. £7.50 ÷ 3 = £2.50.
    £2.50
  2. Find the cost of 5 packs. £2.50 × 5 = £12.50.
    £12.50
  3. Write the answer.
    £12.50

Check. £12.50 ÷ 5 = £2.50. ✓

Worked example 5.5.2
A school has 240 pupils. 90 of them are girls. What percentage of the pupils are girls?

Method. Write the part as a fraction of the whole, then as a percentage.

  1. Write the part as a fraction of the whole. 90 out of 240.
    90/240
  2. Multiply by 100 to get a percentage. Percent means out of 100.
    90/240 × 100 = 37.5%
  3. Write the answer.
    37.5%

Check. 90 ÷ 240 = 0.375, and 0.375 × 100 = 37.5%. ✓

Worked example 5.5.3
Ali scores 36 out of 48 in a test. What percentage did he score?

Method. Write the score as a fraction of the total, then multiply by 100.

  1. Write the part as a fraction of the whole. 36 out of 48.
    36/48
  2. Multiply by 100 to get a percentage. Percent means out of 100.
    36/48 × 100 = 75%
  3. Write the answer.
    75%

Check. 36 ÷ 48 = 0.75, which is 75%. ✓

Exercise 5E  Practice: the unit method and percentages of a whole

Non-calculator
1
F4 cakes cost £12. How much do 6 cakes cost?
2
F5 cakes cost £15. How much do 9 cakes cost?
3
F3 cakes cost £9. How much do 8 cakes cost?
4
F6 cakes cost £24. How much do 10 cakes cost?
5
FWrite 18 out of 45 as a percentage.
6
FWrite 12 out of 80 as a percentage.
7
FWrite 45 out of 60 as a percentage.
8
PSA bag has 40 marbles and 14 are red. What percentage are red?
9
PSThree pens cost £15. How much would seven pens cost?
10
PSA town has 600 households. 150 households have a garden pond. What percentage is this?
11
PSA bag of 9 apples costs 360 pence. How many pence do 4 apples cost?
12
RAli says 25% of 80 and 80% of 25 are different. Is he right? Write Yes or No.
13
RIn a survey, 37 out of 50 pupils chose fruit. What percentage chose fruit?
14
RA shop reduces 80 items to 60 items by selling some. What percentage is left? Explain.

Chapter summary

SkillMethod
Simplify a ratioDivide both parts by the highest common factor: 12 : 20 = 3 : 5.
Ratio and fractionAdd the parts; the first part is a/(a + b) of the whole.
Missing valueFind the value of one part, then multiply by the number of parts.
SharingAdd the parts, find one part, then multiply each part by it: 60 in 2 : 3 is 24 and 36.
Scale factorNew length ÷ old length; multiply every length by it.
Unit methodFind one unit first: £7.50 for 3 packs is £2.50 each, so 5 packs cost £12.50.
Percentage of a wholePart ÷ whole × 100: 90 out of 240 is 37.5%.

Can you…?

  • I can simplify a ratio and write it as a fraction of the total
  • I can find a missing value in an equivalent ratio
  • I can share an amount in a ratio, including three parts
  • I can find a scale factor and use it on a similar shape
  • I can use the unit method and write one amount as a percentage of another

Key terms

ratio
a comparison of two or more amounts, written a : b
equivalent ratio
a ratio with the same relative sizes, such as 2 : 3 and 4 : 6
scale factor
the number every length is multiplied by in an enlargement or reduction
unit method
a way of working out one unit first, then the amount needed
similar shapes
shapes with the same shape but different sizes, with lengths in the same scale factor

Exercise 5Z  Check-up

Non-calculator
1
Write 18 : 30 in its simplest form.
2
A ratio of red : blue is 2 : 5. What fraction of the counters are red?
3
Boys : girls = 4 : 9. There are 20 boys. How many girls are there?
4
Share 72 in the ratio 1 : 5. Write the bigger share.
5
Share £400 between two people in the ratio 3 : 5. How much does the person with 3 parts get?
6
A rectangle 4 cm by 6 cm is enlarged so the 4 cm side becomes 10 cm. How long is the other side?
7
4 bags of rice weigh 600 g in total. How many grams do 3 bags weigh?
8
Write 42 out of 60 as a percentage.
9
Write 9 out of 36 as a percentage.
10
Explain why 2 : 5 and 4 : 10 are equivalent ratios.

Total 21 marks. Allow about 21 minutes.

Exercise 5X  Practice test questions

Non-calculator
1
FSimplify the ratio 15 : 25.[1 mark]
2
PSThe ratio of red to blue counters is 3 : 5. There are 24 red counters. How many blue counters are there?[2 marks]
3
PSShare 90 in the ratio 2 : 7. Write the smaller share.[2 marks]
4
PSA rectangle 3 cm by 8 cm is enlarged so the 3 cm side becomes 12 cm. How long is the other side?[2 marks]
5
PSSeven notebooks cost 525 pence in total. How many pence do 4 of the notebooks cost?[2 marks]
6
PSWrite 63 out of 90 as a percentage.[2 marks]
7
RWhich ratio is equivalent to 4 : 6? Choose one: 2 : 4, 6 : 4, 2 : 3 or 8 : 10.[1 mark]
8
RRuby says that if she shares 100 sweets in the ratio 1 : 4, the bigger share is 80. Is she right? Explain.[3 marks]
9
PSA tank holds 40 litres when it is 80% full. How many litres does it hold when it is full?[3 marks]
10
PSShare 45 in the ratio 4 : 5. Write the bigger share.[2 marks]
11
PSIn a pack, 25% of the 80 sweets are red. How many sweets are red?[1 mark]
← Chapter 4: Decimals and percentagesChapter 6: Algebra and formulae →

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