Solve ratio problems where the ratio changes, two ratios must be combined, or you need to set up an equation. It is on Foundation and Higher, aimed at grade 6. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Higher ratio questions mix algebra with parts. If A:B=2:5 and B:C=3:4, write A:B:C by making the Bs match: 6:15 and 15:20, so 6:15:20. If one amount is an expression, the parts become expressions and the story (total, difference, fraction of) becomes an equation.
A ratio of areas or a ratio after a percentage change is still parts, with an extra line first. Define a letter as 'one part' only when every quantity is a multiple of that part; otherwise let x be a named amount and write the others in terms of x.
Chain two ratios by matching the shared letter (LCM of the two middle numbers). Combined with a total, one part is total ÷ sum of parts. a:b=4:1 means a=4b.
Objectives
Find a cheaper price when one item costs a fixed amount more.
Share an amount in a ratio and take the larger part.
Use “twice as old” or a priced child to finish an equation.
Key points
Combine two ratios that share a quantity by scaling them to a common value.
Write the parts of a ratio as 3x and 5x and form an equation from the story.
Handle 'some is added / taken away and the ratio becomes…' problems.
Convert between a ratio and a fraction or equation, e.g. a:b=2:3 means 3a=2b.
Worked examples
Example 1
3 pens and 2 pencils cost £13. A pen costs £1 more than a pencil. Find the cost of a pencil.
2 marks
Let a pencil cost x. Then a pen costs x + 1
Form equation: 3(x + 1) + 2x = 13, so 5x + 3 = 13
Subtract 3: 5x = 10, so x = 2. A pencil costs £2
Answer: 2
Common slips
Slip: Assume pens and pencils cost the same Pens cost £1 more, so their price is x + 1.
Slip: Tom wrote 13 13 is the total bill, not the pencil price.
Slip: Stop at 5x = 10 Divide by 5 to find the cost of one pencil.
Example 2
Share £40 in the ratio 3:5. Find the larger share.
2 marks
Find total ratio parts: 3 + 5 = 8 parts
Value of one part: 40 / 8 = £5
Multiply by 5 for the larger share: 5 × 5 = £25
Answer: 25
Common slips
Slip: Divide 40 by 3 You must divide by the total number of parts, not one ratio number.
Slip: Uma wrote 15 15 is 3 parts, which is the smaller share.
Slip: State the smaller share £15 The question asks for the larger share.
Example 3
Val is twice as old as Wes and together they are 36. Find the younger age.
3 marks
Let Wes (the younger) be x. Then Val is 2x
Form equation: x + 2x = 36, giving 3x = 36
Divide by 3: x = 12. The younger age is 12
Answer: 12
Common slips
Slip: Assume they are both 18 One is twice as old, so their ages are in the ratio 1:2.
Slip: Val wrote 18 18 splits the age evenly in half instead of 1:2.
Slip: State the older age 24 The question specifically asks for the younger age.
Questions and answers
GCSE Maths problem-solving with ratio & algebra questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
4 pens and 4 pencils cost £24. A pen costs £2 more than a pencil. Find the cost of a pencil.
Show answer
Answer: 2
Four pens add 4 × £2 = £8 extra.
Subtract extra: £24 - £8 = £16 for 8 items, so 16 / 8 = 2.
M1 — Four pens add 4 × £2 = £8 extra.
A1 — Subtract extra: £24 - £8 = £16 for 8 items, so 16 / 8 = 2.
Question 2 Easy2 marks
2 pens and 3 pencils cost £13. A pen costs £2 more than a pencil. Find the cost of a pencil.
Show answer
Answer: 1.80
Two pens add 2 × £2 = £4 extra.
Subtract extra: £13 - £4 = £9 for 5 items, so 9 / 5 = 1.8, which is £1.80.
M1 — Two pens add 2 × £2 = £4 extra.
A1 — Subtract extra: £13 - £4 = £9 for 5 items, so 9 / 5 = 1.8, which is £1.80.
Question 3 Medium2 marks
Share £48 in the ratio 1:2. Find the larger share.
Show answer
Answer: 32
Total parts: 1 + 2 = 3 parts. One part is 48 / 3 = 16.
Larger share is 2 parts: 2 × 16 = 32.
M1 — Total parts: 1 + 2 = 3 parts. One part is 48 / 3 = 16.
A1 — Larger share is 2 parts: 2 × 16 = 32.
Question 4 Medium2 marks
Share £36 in the ratio 2:7. Find the larger share.
Show answer
Answer: 28
Total parts: 2 + 7 = 9 parts. One part is 36 / 9 = 4.
Larger share is 7 parts: 7 × 4 = 28.
M1 — Total parts: 2 + 7 = 9 parts. One part is 36 / 9 = 4.
A1 — Larger share is 7 parts: 7 × 4 = 28.
Question 5 Hard2 marks
Flo is twice as old as Gil and together they are 45. Find the younger age.
Show answer
Answer: 15
Total ratio parts: 1 + 2 = 3 parts.
Divide total: 45 / 3 = 15 for the younger age.
M1 — Total ratio parts: 1 + 2 = 3 parts.
A1 — Divide total: 45 / 3 = 15 for the younger age.
Question 6 Hard2 marks
Hap is twice as old as Ida and together they are 24. Find the younger age.
Show answer
Answer: 8
Total ratio parts: 1 + 2 = 3 parts.
Divide total: 24 / 3 = 8 for the younger age.
M1 — Total ratio parts: 1 + 2 = 3 parts.
A1 — Divide total: 24 / 3 = 8 for the younger age.
Question 7 Exam-style4 marks
5 adults and 3 children cost £55 in total. An adult ticket costs £3 more than a child ticket. Work out the cost of an adult ticket.
Show answer
Answer: 8
Let a child ticket cost c. An adult ticket costs c + 3.
Form the equation: 5(c + 3) + 3c = 55.
Expand and collect: 8c + 15 = 55, so 8c = 40 and c = 5.
An adult ticket costs 5 + 3 = £8.
Check: 5 × 8 + 3 × 5 = 40 + 15 = 55.
M1 — Adult price written as c + 3
M1 — 5(c + 3) + 3c = 55
M1 — 8c = 40 or c = 5
A1 — Adult ticket £8
Common mistakes
Most lost marks on this topic are one slip.
Giving the whole bill as the price of one pencil
Remove the extra pounds, then share the rest
Giving the smaller part when the question asks for the larger share
Use the larger number in the ratio
Halving an age total when one person is twice the other
Split the total into 3 parts
Spot the slip
3 pens and 2 pencils cost £13. A pen costs £1 more than a pencil. Sue says a pencil is £13. What does a pencil cost?
2
13
3
Show the slip
2 Three extra pounds for the pens leave £10 for five items, so a pencil is £2.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
State your variable clearly ('Let the red counters be 3x') — this line is often the first method mark.
Write the new ratio as a fraction equation and cross-multiply; it is the cleanest way to show the algebra.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.