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KS3 Maths · Year 8Chapter 4 · Ratio and proportion
Year 8 · Chapter 4

Ratio and proportion

By the end of this chapter you will be able to

  • Write a ratio in its simplest form, with two or three parts
  • Share a total in a given ratio, and check the shares add up to the total
  • Find the fraction of a total that one part of a ratio makes up
  • Use the unitary method to find the value of one, and then the value of any number of things
  • Calculate speed, density and price per unit, and use a map scale to find real distances

You should already know

  • Equivalent fractions, and the highest common factor (Year 7, Chapters 3 and 4)
  • Multiplying and dividing whole numbers and decimals (Year 7)
  • Changing between units of length, mass and money (Year 7)

Where this leads

  • Ratio with algebra, and the rule of direct proportion y = kx (Year 9 and GCSE)
  • Compound measures such as speed and density in more detail (Year 9)
  • Percentages as ratios and multipliers (Year 8, Chapter 3)

National Curriculum for England, Key Stage 3 mathematics, Year 8. Content: Write, simplify and use ratios, including sharing a quantity in a given ratio, Link ratio to fractions, Use direct proportion and the unitary method, Use rates such as speed, density or price per unit, and map scales.

Exercise P  Do you remember?

Non-calculator
1
FFind the highest common factor of 12 and 18.
2
FWork out 5 × 6 + 4.
3
FWhat is 1/4 of 60?
4
FWrite 250 g as a fraction of 1 kg.
5
FA pen costs 50p. How much do 4 pens cost?
4.1

Writing and simplifying ratios

Year 8
Write, simplify and use ratios
Video lesson: Writing and simplifying ratiosComing soon
Priya makes orange squash. She uses 3 cups of juice for every 9 cups of water. She wants to write this ratio in its simplest form, so that she can scale it up for a party.

A ratio compares amounts of the same kind. The ratio of juice to water is written 3:9.

To simplify a ratio, divide every part by the highest common factor (HCF) of all the parts. The ratio 3:9 simplifies to 1:3.

Before you write a ratio, make sure both amounts have the same units. For example 40 cm : 1 m must become 40 cm : 100 cm.

Key ideas

  • A ratio is written with a colon: 2:3.
  • Divide every part by the same number to simplify a ratio.
  • Both parts of a ratio must be in the same units before you simplify.
  • A ratio with 1 as one part is written 1:n, for example 1:3.
Worked example 4.1.1
Write 12:18 in its simplest form.

Method. Divide every part of the ratio by the highest common factor of all the parts.

  1. Find the highest common factor of all the parts. The biggest number that divides into every part is 6.
    HCF of 12:18 = 6
  2. Divide every part by the same number. Dividing every part by the same number keeps the ratio the same.
    12:18 ÷ 6 = 2:3
  3. Write the answer. The ratio is now in its simplest form.
    2:3

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

Watch out. Divide every part, not just the first one. The parts must stay in the same order.

Worked example 4.1.2
Write 6:9:12 in its simplest form.

Method. Divide every part of the ratio by the highest common factor of all the parts.

  1. Find the highest common factor of all the parts. The biggest number that divides into every part is 3.
    HCF of 6:9:12 = 3
  2. Divide every part by the same number. Dividing every part by the same number keeps the ratio the same.
    6:9:12 ÷ 3 = 2:3:4
  3. Write the answer. The ratio is now in its simplest form.
    2:3:4

Check. Multiply back by 3: 2:3:4 × 3 = 6:9:12. ✓

Watch out. Divide every part, not just the first one. The parts must stay in the same order.

Worked example 4.1.3
Write 40 cm : 1 m as a ratio in its simplest form.

Method. Change the units first, so both amounts are in centimetres. Then simplify.

  1. Change 1 m to centimetres. There are 100 cm in 1 m.
    1 m = 100 cm
  2. Write the ratio in the same units. Now both parts are in cm.
    40 cm : 100 cm = 40:100
  3. Simplify. The HCF of 40 and 100 is 20.
    40 ÷ 20 = 2 and 100 ÷ 20 = 5
  4. Write the answer.
    2:5

Check. 2 × 20 = 40 and 5 × 20 = 100.

Watch out

Watch out: 40 cm : 1 m is not 40:1. Change the units first, so both parts are in the same units.

Exam tip

Exam tip. A ratio is simplest when no number divides into all the parts. Check your answer by multiplying back.

Exercise 4A  Writing and simplifying ratios

Non-calculator
1
FWrite 12:18 in its simplest form.
2
FWrite 15:25 in its simplest form.
3
FWrite 8:20 in its simplest form.
4
FWrite 24:36 in its simplest form.
5
FWrite 6:9:12 in its simplest form.
6
FWrite 10:15:25 in its simplest form.
7
FWrite 20:30 in its simplest form.
8
FWrite 7:21 in its simplest form.
9
FWrite 45:30 in its simplest form.
10
PSWrite 40 cm : 1 m as a ratio in its simplest form.
11
PSSimplify the ratio 3:9 for Priya’s juice and water.
12
PSA box has 14 red sweets and 21 green sweets. Write the ratio of red to green in its simplest form.
13
PSA bag has 12 red sweets and 20 blue sweets. Write the ratio of red to blue in its simplest form.
14
RAli says that 2:3 is the same ratio as 4:5. Is Ali right? Explain how you know.
15
PSWrite these three quantities as a ratio in simplest form: 2 kg, 500 g, 1 kg 500 g. Hint: change to grams.
16
PSWrite 6:4:2 in its simplest form.
17
PSSimplify 30 minutes : 2 hours, giving the ratio in its simplest form.
18
RExplain why the ratio 1:3 and the ratio 3:1 are different. Give an example.
4.2

Sharing in a ratio

Year 8
Share a quantity in a given ratio
Video lesson: Sharing in a ratioComing soon
Tom and Ben find £60 in a drawer. They agree to share it in the ratio 2:3, because Tom helped to clean the drawer first. Tom gets 2 parts and Ben gets 3 parts.

To share a quantity in a ratio, follow these steps:

  1. add the parts of the ratio to find the total number of parts;
  2. divide the total by the number of parts to find the value of one part;
  3. multiply each part of the ratio by the value of one part.

Always check that your shares add up to the total you started with.

Key ideas

  • Add the parts of the ratio to find the number of equal parts.
  • Divide the total by the number of parts to find one part.
  • Multiply each ratio part by one part to find each share.
  • Check the shares add up to the total.
Worked example 4.2.1
Share £60 in the ratio 2:3.

Method. Share the total into equal parts for each part of the ratio, then multiply each part by one part.

  1. Add the parts of the ratio. The ratio has 5 equal parts altogether.
    2 + 3 = 5
  2. Find the value of one part. Share the total into 5 equal parts.
    60 ÷ 5 = 12£
  3. Multiply each part of the ratio by the value of one part. Each share is its number of parts times the value of one part.
    2 × 12 = 24£ and 3 × 12 = 36£
  4. Check the shares add up to the total. The shares must add up to the total we started with.
    24 + 36 = 60£
  5. Write the answer.
    24£, 36£

Check. 24 + 36 = 60£, so the shares are correct.

Watch out. Add the parts first. Do not divide the total by the number in the ratio itself, such as 2 or 3.

Worked example 4.2.2
Share 120 minutes in the ratio 2:3:1.

Method. Share the total into equal parts for each part of the ratio, then multiply each part by one part.

  1. Add the parts of the ratio. The ratio has 6 equal parts altogether.
    2 + 3 + 1 = 6
  2. Find the value of one part. Share the total into 6 equal parts.
    120 ÷ 6 = 20 min
  3. Multiply each part of the ratio by the value of one part. Each share is its number of parts times the value of one part.
    2 × 20 = 40 min and 3 × 20 = 60 min and 1 × 20 = 20 min
  4. Check the shares add up to the total. The shares must add up to the total we started with.
    40 + 60 + 20 = 120 min
  5. Write the answer.
    40 min, 60 min, 20 min

Check. 40 + 60 + 20 = 120 min, so the shares are correct.

Watch out. Add the parts first. Do not divide the total by the number in the ratio itself, such as 2 or 3.

Worked example 4.2.3
Share 45 sweets between Ali and Ben in the ratio 4:5.

Method. Share the total into equal parts for each part of the ratio, then multiply each part by one part.

  1. Add the parts of the ratio. The ratio has 9 equal parts altogether.
    4 + 5 = 9
  2. Find the value of one part. Share the total into 9 equal parts.
    45 ÷ 9 = 5
  3. Multiply each part of the ratio by the value of one part. Each share is its number of parts times the value of one part.
    4 × 5 = 20 and 5 × 5 = 25
  4. Check the shares add up to the total. The shares must add up to the total we started with.
    20 + 25 = 45
  5. Write the answer.
    20, 25

Check. 20 + 25 = 45, so the shares are correct.

Watch out. Add the parts first. Do not divide the total by the number in the ratio itself, such as 2 or 3.

Watch out

Watch out: a ratio of 2:3 means 5 equal parts, not 2 parts and 3 parts of the total. Add the parts first.

Exam tip

Exam tip. Write a line that says how many parts there are altogether, and what one part is worth. This gets method marks even if you slip.

Exercise 4B  Sharing in a ratio

Non-calculator
1
FShare £60 in the ratio 2:3.
2
FShare 45 sweets in the ratio 4:5.
3
FShare £84 in the ratio 3:4.
4
FShare 120 minutes in the ratio 2:3:1.
5
FShare £240 in the ratio 5:3.
6
FShare 96 cm in the ratio 1:3.
7
FShare 150 g in the ratio 2:3:5.
8
PSPriya and Sam share £40 in the ratio 3:5. How much does Sam get?
9
PSGift money of £72 is shared between Ali, Ben and Cara in the ratio 1:2:3. How much does Cara get?
10
PSA drink is made with 500 ml in the ratio 2:3 of juice to water. How many millilitres of water are there?
11
PSSweets are shared between two children in the ratio 3:2. There are 35 sweets altogether. How many does each child get?
12
PSShare 90 minutes in the ratio 4:5. How many minutes is each part?
13
RJess says that 40 shared in the ratio 1:3 gives 10 and 30. Is Jess right? Show your working.
14
PSA cake is shared between Jo and Lee in the ratio 5:4. Jo gets 20 slices. How many slices are there altogether?
15
RSam shares £50 with Tom in the ratio 2:3. Sam says he gets £20. Is Sam right? Explain how you know.
16
PSA recipe is shared into 4 equal bags in the ratio 1:1:2. Write the amount in each part if the total is 400 g.
17
PSAli and Jo share 84 stickers in the ratio 3:4. How many stickers does Jo get? Explain your method.
4.3

Ratios and fractions

Year 8
Link ratio to fractions
Video lesson: Ratios and fractionsComing soon
A bag has red and blue counters in the ratio 3:2. Nina wants to know what fraction of the counters are red, so she can say it in a game without counting.

A ratio tells us how the parts compare. To find a fraction of the whole, use the total number of parts.

For the ratio 3:2, there are 3 + 2 = 5 parts altogether. The red part is 3 of the 5 parts, so the fraction red is 3/5.

The fractions for every part of the ratio add up to 1.

Key ideas

  • The fraction for one part is that part divided by the total number of parts.
  • The fractions for all the parts of a ratio add up to 1.
  • Use a ratio to find a fraction of a total, by multiplying the total by the fraction.
  • A fraction can be written as a ratio, for example 1/3 of the class is 1:2 with the rest.
Worked example 4.3.1
Write the fraction of red counters if the ratio of red to blue is 3:2.

Method. The fraction for one part is that part divided by the total number of parts.

  1. Add the parts to find the total. The total number of parts is 5.
    3 + 2 = 5
  2. Write the part as a fraction of the total. The part we want has 3 parts.
    3/5
  3. Simplify. Divide the top and bottom by any common factor.
    3/5

Check. The other parts add up to 2 parts, so the fractions add to 1.

Worked example 4.3.2
The ratio of blue to green is 1:4. What fraction is green?

Method. The fraction for one part is that part divided by the total number of parts.

  1. Add the parts to find the total. The total number of parts is 5.
    1 + 4 = 5
  2. Write the part as a fraction of the total. The part we want has 4 parts.
    4/5
  3. Simplify. Divide the top and bottom by any common factor.
    4/5

Check. The other parts add up to 1 parts, so the fractions add to 1.

Worked example 4.3.3
A bag of 40 counters is in the ratio red:blue 3:5. How many are red?

Method. Find the total number of parts, then find the fraction of the total that is red.

  1. Add the parts. 3 + 5 = 8 parts altogether.
    8 parts
  2. Find the fraction that is red. Red is 3 of the 8 parts.
    3/8
  3. Multiply the total by the fraction. 3/8 of 40 is 40 ÷ 8 = 5, then 5 × 3 = 15.
    3/8 × 40 = 15
  4. Write the answer.
    15 red counters

Check. 15 red and 25 blue make 40. 25 = 5 × 5, which is right.

Watch out

Watch out: a ratio of 3:2 does not mean that 3/2 are red. The fraction is out of the total parts, so it is 3/5.

Exam tip

Exam tip. Write the total number of parts first. Then the fraction is always the part over the total.

Exercise 4C  Ratio and fractions

Non-calculator
1
FThe ratio of the parts is 3:2. What fraction is part 1?
2
FThe ratio of the parts is 2:7. What fraction is part 1?
3
FThe ratio of the parts is 1:4. What fraction is part 2?
4
FThe ratio of the parts is 3:5:7. What fraction is part 3?
5
FThe ratio of the parts is 2:3:5. What fraction is part 2?
6
FThe ratio of the parts is 5:3. What fraction is part 2?
7
FThe ratio of red to blue counters is 3:5. What fraction of the counters are red?
8
PSA bag has 40 counters, in the ratio red:blue 3:5. How many are red?
9
PSA class has boys and girls in the ratio 3:2. What fraction of the class are boys?
10
PSIn a school, 2/5 of pupils walk. What is the ratio of walkers to non-walkers, in its simplest form?
11
PSOne third of a class are girls. Write the ratio of boys to girls.
12
RAli says that if the ratio of red to blue is 1:3, then 1/4 of the counters are red. Is Ali right? Explain how you know.
13
PSA bag of 60 sweets has a ratio of lemon to mint of 2:3. How many lemon sweets are there?
14
PSIn a survey, the ratio of people who like tea to people who like coffee is 3:1. What fraction of people like coffee?
15
RExplain why the fractions for the parts of the ratio 2:3:5 add up to 1. Accept any answer that shows 2/10 + 3/10 + 5/10 = 1.
4.4

Direct proportion and the unitary method

Year 8
Use direct proportion and the unitary method
Video lesson: Direct proportion and the unitary methodComing soon
Mia buys 5 pens for £3.50. Her friend asks how much 8 pens would cost. Mia uses the unitary method: she finds the cost of one pen first.

Two quantities are in direct proportion when they increase at the same rate. If 5 pens cost £3.50, then 10 pens cost twice as much.

The unitary method finds the value of one first, then multiplies to find the value you need:

  1. divide to find the value of one;
  2. multiply by the number you want.

Key ideas

  • Direct proportion: when one quantity doubles, the other doubles too.
  • The unitary method: divide to find one, then multiply to find the answer.
  • Find the value of one first. This is safer than scaling by a ratio.
  • Check your answer makes sense: more items should cost more.
Worked example 4.4.1
5 pens cost £3.50. How much do 8 pens cost?

Method. Use the unitary method: first find the value of one, then multiply by the number you want.

  1. Find the value of one. Divide the cost for 5 by 5.
    £3.50 ÷ 5 = £0.70
  2. Multiply to find the answer. Multiply the value of one by 8.
    £0.70 × 8 = £5.60
  3. Write the answer.
    £5.60

Check. Check: £5.60 ÷ 8 = £0.70, which is the value of one.

Watch out. Find the value of one first. Multiplying straight away by the ratio gives the wrong answer.

Worked example 4.4.2
A recipe for 4 people uses 300 g of flour. How much flour for 6 people?

Method. Use the unitary method: first find the value of one, then multiply by the number you want.

  1. Find the value of one. Divide the cost for 4 by 4.
    300 g ÷ 4 = 75 g
  2. Multiply to find the answer. Multiply the value of one by 6.
    75 g × 6 = 450 g
  3. Write the answer.
    450 g

Check. Check: 450 g ÷ 6 = 75 g, which is the value of one.

Watch out. Find the value of one first. Multiplying straight away by the ratio gives the wrong answer.

Worked example 4.4.3
A machine makes 60 bottles in 12 minutes. How many bottles in 30 minutes?

Method. Use the unitary method: first find the value of one, then multiply by the number you want.

  1. Find the value of one. Divide the cost for 12 by 12.
    60 bottles ÷ 12 = 5 bottles
  2. Multiply to find the answer. Multiply the value of one by 30.
    5 bottles × 30 = 150 bottles
  3. Write the answer.
    150 bottles

Check. Check: 150 bottles ÷ 30 = 5 bottles, which is the value of one.

Watch out. Find the value of one first. Multiplying straight away by the ratio gives the wrong answer.

Watch out

Watch out: if 5 pens cost £3.50, one pen does not cost £3.50 × 1/5 in your head. Divide: £3.50 ÷ 5 = £0.70.

Exam tip

Exam tip. Write down the value of one first. Examiners give method marks for it.

Exercise 4D  Direct proportion and the unitary method

Non-calculator
1
F5 pens cost £3.50. How much do 8 pens cost?
2
F4 people need 300 g of flour. How much flour do 6 people need?
3
F3 tickets cost £12. How much do 5 tickets cost?
4
FA machine makes 60 bottles in 12 minutes. How many bottles does it make in 30 minutes?
5
PSA car uses 6 litres of petrol to travel 90 km. How many litres does it use to travel 150 km? Assume the car uses petrol at the same rate.
6
PSBen pays £18 for 4 kg of apples. How much does he pay for 7 kg of apples?
7
PSA tap fills 15 litres in 5 minutes. How many litres does it fill in 12 minutes?
8
PSy is in direct proportion to x. When x = 4, y = 12. Find y when x = 7.
9
RSam says that if 5 pens cost £3.50, then 10 pens cost £7. Is Sam right? Explain how you know.
10
PSA car park charges £2.40 for 3 hours. What would 5 hours cost at the same rate?
11
RIs the total cost of pens directly proportional to the number of pens? Explain your answer.
12
PSA jug holds 2 litres. How many 250 ml cups can be filled from 5 jugs? Give your answer as a number of cups.
13
PS5 people share a bill of £35. How much is the bill for 8 people, at the same cost each?
14
RA photocopier makes 75 copies in 3 minutes. How long does it take to make 200 copies? Give your answer in minutes.
4.5

Rates: speed, density, price per unit, and map scales

Year 8
Use rates such as speed, density or price per unit, and map scales
Video lesson: Rates: speed, density, price per unit, and map scalesComing soon
A walking route is drawn on a map with a scale of 1:50 000. Ali measures a road as 4 cm long on the map. He wants to know the real length in kilometres, so he can plan the walk.

A rate compares two quantities with different units. Three common rates are:

A map scale such as 1:50 000 means 1 cm on the map is 50 000 cm in real life. Multiply the map distance by the scale, then change the units.

Key ideas

  • Speed = distance ÷ time. Density = mass ÷ volume. Price per unit = cost ÷ units.
  • Map scale 1:n: multiply the map distance by n to get the real distance.
  • 100 cm = 1 m, and 1000 m = 1 km. Change the units at the end.
  • Use the same units for distance and time before you divide.
Worked example 4.5.1
A car travels 150 km in 2 hours. Find its average speed.

Method. Speed = distance ÷ time.

  1. Write the formula. Speed is how far you go in one unit of time.
    speed = distance ÷ time
  2. Put in the numbers. The distance is 150 km and the time is 2 hours.
    150 ÷ 2
  3. Work out the speed. Divide the distance by the time.
    150 ÷ 2 = 75
  4. Write the answer with units. Speed is in distance units per time unit.
    75 km per hour

Check. 75 × 2 = 150, which is the distance.

Watch out. Use the same units for distance and time. Change minutes to hours first if you need to.

Worked example 4.5.2
A map has a scale of 1:50 000. A road is 4 cm long on the map. Find the real length in km.

Method. Multiply the map distance by the scale factor to get the real distance. Then change the units.

  1. Find the scale. A scale of 1:50,000 means 1 cm on the map is 50,000 cm in real life.
    1 cm : 50,000 cm
  2. Multiply the map distance by the scale. 4 cm on the map is 4 times as far.
    4 × 50,000 = 200,000 cm
  3. Change the units. There are 100 cm in 1 m and 1000 m in 1 km.
    200,000 cm = 2 km
  4. Write the answer.
    2 km

Check. Divide back: 2 km is 200,000 cm, and 200,000 ÷ 50,000 = 4 cm on the map.

Watch out. Change the units at the end. Do not multiply a scale in km by a map distance in cm without changing units first.

Worked example 4.5.3
A piece of metal has a mass of 540 g and a volume of 200 cm3. Find its density.

Method. Density = mass ÷ volume.

  1. Write the formula. Density tells us how much mass is in each cubic centimetre.
    density = mass ÷ volume
  2. Put in the numbers. The mass is 540 g and the volume is 200 cm3.
    540 ÷ 200
  3. Work out the density. Divide the mass by the volume.
    540 ÷ 200 = 2.7
  4. Write the answer with units.
    2.7 g/cm3

Check. 2.7 × 200 = 540, which is the mass.

Watch out. Mass divided by volume. Do not multiply them.

Watch out

Watch out: 1:50 000 does not mean 1 cm on the map is 50 km. It means 1 cm is 50 000 cm, which is 0.5 km.

Exam tip

Exam tip. Write the units in your answer. A speed without units, such as 75, is not a full answer.

Exercise 4E  Rates and map scales

Non-calculator
1
FA car travels 150 km in 2 hours. What is its average speed?
2
FA train travels 240 km in 3 hours. What is its average speed?
3
FA metal block has a mass of 540 g and a volume of 200 cm3. What is its density?
4
FA stone has a mass of 360 g and a volume of 120 cm3. What is its density?
5
FA bag of 3 kg of potatoes costs £4.50. What is the price per kilogram?
6
FA pack of 5 kg of rice costs £7.50. What is the price per kilogram?
7
PSWork out the real length of a path that is 4 cm long on a map with the scale 1:50 000. Give your answer in km.
8
PSA map has a scale of 1:20 000. A river is 3 cm long on the map. How long is the river in real life? Give your answer in m.
9
PSA road is 5 km long in real life. How long is it on a map with the scale 1:50 000? Give your answer in cm.
10
PSA road is 3 km long. How long is it on a map with a scale of 1:25 000? Give your answer in cm.
11
PSA cyclist rides 60 km in 3 hours. How far does she ride in 5 hours, at the same speed?
12
RAli says that 120 km in 2 hours is faster than 150 km in 3 hours. Is Ali right? Show your working.
13
PSA map has a scale of 1:25 000. A road is 8 cm long on the map. How long is the road in real life? Give your answer in km.
14
RAli says that a density of 2.7 g/cm3 means 2.7 g of the material has a volume of 1 cm3. Is he right? Explain how you know.

Chapter summary

SkillMethod
Simplify a ratioDivide every part by the highest common factor of all the parts
Share in a ratioAdd the parts, find one part, then multiply each part of the ratio by one part
Ratio to fractionPart ÷ total number of parts (add the parts to find the total)
Unitary methodFind the value of one, then multiply to find the value you need
Speed, density, priceSpeed = distance ÷ time; density = mass ÷ volume; price = cost ÷ units
Map scale 1:nReal distance = map distance × n, then change the units

Can you…?

  • I can write a ratio in its simplest form, with units the same
  • I can share an amount in a ratio, and check the shares add up
  • I can find the fraction that one part of a ratio makes up
  • I can use the unitary method to solve direct proportion problems
  • I can calculate speed, density and price per unit
  • I can use a map scale to find real distances, and a real distance to find a map distance

Key terms

ratio
a comparison of amounts of the same kind, written with a colon such as 2:3
simplest form
a ratio or fraction that cannot be made smaller by dividing every part by the same number
direct proportion
two quantities that change at the same rate, so doubling one doubles the other
unitary method
a method that finds the value of one first, then the value of any number
rate
a comparison of two quantities with different units, such as speed
density
mass divided by volume, for example in g/cm3
map scale
the ratio of a map distance to the real distance, such as 1:50 000

Exercise 4Z  Check-up

Non-calculator
1
Write 45:75 in its simplest form.
2
Share £100 in the ratio 3:2.
3
Write the fraction for the part that is 2 in the ratio 2:5.
4
6 cakes cost £9. How much do 10 cakes cost?
5
A car travels 240 km in 4 hours. Find its average speed.
6
A stone has a mass of 630 g and a volume of 300 cm3. Find its density.
7
A pack of 4 kg of flour costs £2.40. What is the price per kilogram?
8
Map scale 1:100 000. A road is 3 cm long on the map. Find the real length in km.
9
A map has a scale of 1:50 000. A road is 6 km long. How long is it on the map? Give your answer in cm.
10
Write 25 cm : 1 m in its simplest form.

Total 34 marks. Allow about 34 minutes.

Exercise 4X  Exam-style questions

Non-calculator
1
FWrite 30:45 in its simplest form.[2 marks]
2
FBen and Ann share 81 cards in the ratio 4:5. How many cards does Ann get?[3 marks]
3
FA bag has red and white counters in the ratio 2:7. What fraction of the counters are white?[2 marks]
4
F5 identical books cost £42. How much do 8 of the same books cost?[3 marks]
5
FA 900 g bag of flour is shared in the ratio 1:2 between two bags. How many grams are in each bag?[3 marks]
6
FMap scale 1:40 000. A lake is 5 cm long on the map. Find the real length of the lake in km.[4 marks]
7
FA 4 cm line on a map represents 2 km. What is the scale of the map? Give your answer as 1:n.[4 marks]
8
FA taxi charges £12 for 4 km. Mia pays £21. How far did she travel, if the charge is directly proportional to distance?[3 marks]
9
FAli shares £90 with his sister in the ratio 5:4. Ali says he gets £50. Is Ali right? Explain your answer.[3 marks]
10
FA cyclist rides 24 km in 1 hour 30 minutes. Find the average speed in km per hour.[4 marks]
11
FExplain why the ratio 1:4 and the fraction 1/4 are the same for a bag split into two parts. Accept any answer that gives the 1 part and 4 parts out of 5 total, and the fraction 1/5.[3 marks]
← Chapter 3: Fractions and percentagesChapter 5: Expanding and factorising →

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