Ratio and proportion
Proportion questions give a price or a time and ask about a different number. Ratio questions compare two amounts, and ask you to simplify a ratio or share an amount in one. Here’s how to answer each kind, and the traps the wrong answers set.
Chapter 1
What are ratio and proportion questions?
Proportion questions give you two amounts that belong together, like a number of pens and their price, and ask about a different number. Ratio questions compare two amounts, like 3 red sweets for every 4 white.
What you know. 3 pens cost £2.10.
What it asks. The price of a different number of pens: 4.
Rush asks for a price or a time for a different number, like this one, and asks you to simplify or share a ratio. The exam adds a few longer kinds, in chapter 4. Most of them come down to one idea: find what one is worth, then scale.
3 pens cost £2.10. What do 4 pens cost?
The options
- Option 1: £2.80
- Option 2: 70p
- Option 3: £2.70
- Option 4: £3.50
A Maths question from Rush, level 3.
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Reading a ratio
Red to white is 3 : 4: for every 3 red sweets, there are 4 white. Here there are 6 red and 8 white, which is still 3 : 4. The order follows the words, so white to red is 4 : 3.
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A ratio isn’t a fraction
Each group of 3 red and 4 white has 7 sweets, so 3/7 of them are red, not 3/4. To turn a ratio into a fraction, the bottom is all the parts added up.
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Apples 2 → 4 both
doubleCost 40p → 80p Painters 2 → 4 one up,
one downDays 8 → 4 Which way does it go?
In proportion, two amounts change together. Twice the apples cost twice as much. But some go opposite ways: twice the painters finish in half the days.
Chapter 2
Solve it in 4 steps.
Here’s the question, one step at a time.
The question
3 pens cost £2.10. What do 4 pens cost?
The options
- Option 1: £2.80
- Option 2: 70p
- Option 3: £2.70
- Option 4: £3.50
A Maths question from Rush, level 3.
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Step 1
Up or down?
Before any sums, decide which way the answer goes. 4 pens is more than 3, so they cost more than £2.10. That already rules out 70p.
- 3 pens cost £2.10
- 4 pens cost more than £2.10
More pens, more money.
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Step 2
Find what one is worth.
Work in pence, so £2.10 is 210p. Share it between the 3 pens: 210p ÷ 3 = 70p for one pen.
210p ÷ 3 = 70p each.
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Step 3
Scale to the new number.
One pen is 70p, so 4 pens cost 4 × 70p = 280p. Write it back in pounds: £2.80.
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Step 4
Check it another way.
4 pens is the 3 you know and one more: £2.10 + 70p = £2.80. Both ways give the same answer, so it’s right.
Three mistakes this question sets
| Pens | Cost |
|---|---|
| 1 | 70p |
| 2 | £1.40 |
| 3 | £2.10 |
| 4 | £2.80 |
| 5 | £3.50 |
| The mistake | Do this instead |
|---|---|
| Stopping at one pen. 70p is the price of one pen. That’s step 2, and there are two more. | Look again at the number the question asks about, then scale to it. |
| A slip in the times. £2.70 is 10p short: 4 × 70p is 280p, not 270p. | Check it another way: the 3 pens you know, plus one more. |
| One pen too many. £3.50 is 5 pens at 70p, not 4. | Count up in 70p and stop at 4 pens: 70p, £1.40, £2.10, £2.80. |
So it’s £2.80. Four pens at 70p each. And it’s more than £2.10, just as step 1 said.
Remember: Up or down? Find one. Then scale.
Chapter 3
Everything else Rush asks.
Each card is one kind of question: the rule, one example, and the trap. Most use the same idea: find what one is worth.
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The price of one
Share the total by how many there are.
5 lollies cost £4.50. In pence, 450p ÷ 5 = 90p. Trap: £4.50 is usually one of the options too. One lolly can’t cost as much as five.
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- 4 buns cost 75p
- 12 buns is 3 lots of 4
- 3 × 75p = £2.25
Spot an easy jump
If one number goes into the other, skip finding one.
4 buns cost 75p, and 12 buns is 3 lots of 4, so 3 × 75p = £2.25. It works going down too: if 6 pens cost £1.20, 3 pens cost half, 60p. Trap: multiplying by 12, not by the jump, 3.
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24 : 40=3 : 5
÷ 8 on both sides
Simplify a ratio
Divide both sides by the biggest number that goes into both.
8 goes into 24 and 40, so 24 : 40 is 3 : 5. Traps: 12 : 20 halves once and stops, but 4 still goes into both. And 5 : 3 has the right numbers the wrong way round. Keep the order.
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Share in a ratio
Add the parts, find one part, then multiply.
Share 40 in the ratio 3 : 5. That’s 3 + 5 = 8 parts, and 40 ÷ 8 = 5 in each. The smaller share is 3 × 5 = 15. Trap: the other share, 25, is always an option. Read “larger” or “smaller” again.
More workers, fewer days
Find how long one would take alone, then share it out.
4 painters paint a hall in 6 days. How many days for 8 painters? One painter alone would take 4 × 6 = 24 days. 8 painters share those 24 days of work: 24 ÷ 8 = 3 days. Trap: 12, from doubling the days as if it were a price. More painters can’t take longer.
6 days
3 days
Chapter 4: In the exam
Ratios with three parts.
In the exam, a ratio can have three parts, like the one in GL’s free Maths Familiarisation 1 paper. Rush only shares in two, but the method is the same with one more part.
The questionZoe, Raj and Mia share 48 stickers in the ratio 5 : 2 : 1. How many stickers does Zoe get?
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Step 1
Add all the parts.
Every number in the ratio is a part, so count all three: 5 + 2 + 1 = 8 parts.
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Step 2
Find one part.
Share the stickers between the 8 parts: 48 ÷ 8 = 6 in each.
48 ÷ 8 = 6 in each part.
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Step 3
Multiply by Zoe’s parts.
Zoe has 5 parts: 5 × 6 = 30. Check: Raj has 2 × 6 = 12 and Mia 6, and 30 + 12 + 6 = 48.
Raj gets 12 and Mia 6.
Three mistakes this question sets
| The mistake | Do this instead |
|---|---|
| Giving the ratio number. 5 is how many parts Zoe has, not how many stickers. | Find one part first, then multiply by 5. |
| Giving someone else’s share. 12 is Raj’s and 6 is Mia’s. Exam options often include them. | Read the name in the question again before you choose. |
| Forgetting a part. Adding only 5 + 1 makes 6 parts, so 8 in each and Zoe 40. That leaves Raj with nothing. | Add every number in the ratio. Check the shares make 48. |
So it’s 30. Five parts of 6. And the three shares add up to 48.
Remember: Add every part. Find one. Times.
Other ways the exam asks it
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You know one part, not the total
Boys to girls is 2 : 3, and there are 12 boys. The 2 boys’ parts are 12, so one part is 6. Girls: 3 × 6 = 18. The whole class: 5 × 6 = 30.
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Twice as many
Amy has twice as many sweets as Ben, and together they have 36. Twice as many is the ratio 2 : 1, which is 3 parts: 36 ÷ 3 = 12. Ben has 12 and Amy 24. Trap: halving 36 to 18 makes them equal.
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“For every” or “in every”?
1 red for every 4 white is the ratio 1 : 4, so 1 in every 5 counters is red. “1 in every 4 is red” is a fraction, 1/4. In a bag of 40, the first gives 8 red and the second 10.
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- For 4: 300 g flour, 2 eggs
- For 2: 150 g flour, 1 egg
- For 6: 450 g flour, 3 eggs
Scaling a recipe
A recipe for 4 people uses 300 g of flour and 2 eggs. For 6 people, halve it for 2, then multiply by 3: 450 g and 3 eggs. Trap: scaling the flour and forgetting the eggs. Every ingredient changes.
Chapter 5
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- Up or down?
- Find what one is worth.
- Scale to the new number.
- Check it another way.
Remember: Up or down? Find one. Then scale.
For parents.
What parents ask us about ratio and proportion.
Questions parents ask
What is the difference between ratio and proportion?
A ratio compares parts with each other: 3 boys to 4 girls. Proportion is about amounts that change together: twice as many pens cost twice as much. Both come down to finding what one part, or one item, is worth.
How do you share an amount in a ratio?
Add the parts, divide the total by that number, then multiply by each part. For 35 shared in the ratio 3 : 4: 3 + 4 = 7 parts, 35 ÷ 7 = 5, so the shares are 15 and 20. Check that they add back to 35.
When is ratio taught at school?
Ratio first appears in the national curriculum in Year 6, so a child sitting the test in September may not have met it in class yet. It rests on times tables and division, so it doesn't take long to learn once those are quick.
What is inverse proportion?
It's when one amount goes up as the other goes down, like workers and time. Find how long one worker would take, then divide by the new number: if 3 taps fill a tank in 10 minutes, one tap would take 30 minutes, so 5 taps take 6 minutes.
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