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Pearson Edexcel MathsRatio, proportion and rates of change

Direct and inverse proportion

Recognise and solve relationships between changing quantities.

Pearson EdexcelGCSE MathsRatio, proportion and rates of changeFoundation and Higher
Visual model

Direct goes up together, inverse goes down

directinverse
Gold-standard guide
20 mins

What you will learn

Recognise and solve relationships between changing quantities.
Use a clear step-by-step method for direct and inverse proportion.
Check your answer and avoid the most common exam mistake.
Useful before you start
Core number skillsEarlier ratio, proportion and rates of change skillsShowing clear working
Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Method

Direct proportion: y = kxn — as x increases, y increases

Step 1

Write the proportionality equation

y ∝ x2, so y = kx2

Step 2

Substitute the known pair to find k

45 = k(3)2 = 9k → k = 5

Step 3

Find y when x = 5

y = 5(5)2 = 5 × 25 = 125

Watch out

Watch out

Students write the formula as y = kx rather than y = kx2 because they misread 'proportional to x2'

f
Direct

y=kx.y = kx.

f
Inverse

y=kx.y = \frac{k}{x}.

Worked example

y is directly proportional to x2. When x = 3, y = 45. Find y when x = 5 and find x when y = 80.

1

Write the proportionality equation: y ∝ x2, so y = kx2.

2

Substitute the known pair to find k: 45 = k(3)2 = 9k → k = 5.

3

Find y when x = 5: y = 5(5)2 = 5 × 25 = 125.

4

Find x when y = 80: 80 = 5x2 → x2 = 16 → x = 4.

Final answer

y = 125 when x = 5; x = 4 when y = 80

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

y is directly proportional to x2. When x = 3, y = 45. Find y when x = 5 and find x when y = 80.

4 marks4 minsdirect-and-inverse-proportion-worked
Show solution
Worked solution
  1. 1.Write the proportionality equation: y ∝ x2, so y = kx2.
  2. 2.Substitute the known pair to find k: 45 = k(3)2 = 9k → k = 5.
  3. 3.Find y when x = 5: y = 5(5)2 = 5 × 25 = 125.
  4. 4.Find x when y = 80: 80 = 5x2 → x2 = 16 → x = 4.
Final answer

y = 125 when x = 5; x = 4 when y = 80

Mark points
  • M1: write the proportionality equation
  • M1: substitute the known pair to find k
  • M1: find y when x = 5
  • M1: find x when y = 80
  • A1: y = 125 when x = 5; x = 4 when y = 80
Watch out

Students write the formula as y = kx rather than y = kx2 because they misread 'proportional to x2'.Always underline the power in the question and include it in the equation before finding k.

Diagnosticrecall

y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.

1 mark2 minsdirect-and-inverse-proportion-q1
Show solution
Worked solution
  1. 1.Spot the skill: Direct proportion: y = kxn — as x increases, y increases.
  2. 2.Use the write the proportionality equation stage first, then substitute the known pair to find k.
  3. 3.Keep the final answer visible: 35.
Final answer

35

Mark points
  • M1: use the correct direct proportion: y = kxn — as x increases, y increases.inverse proportion: y = k/xn — as x increases, y decreases.always find k first using the given pair, then use k to answer the question.
  • A1: 35
Watch out

Students write the formula as y = kx rather than y = kx2 because they misread 'proportional to x2'.Always underline the power in the question and include it in the equation before finding k.

Easyprocedure

P is inversely proportional to Q. When Q = 5, P = 12. Find P when Q = 3.

2 marks3 minsdirect-and-inverse-proportion-q2
Show solution
Worked solution
  1. 1.Spot the skill: Direct proportion: y = kxn — as x increases, y increases.
  2. 2.Use the substitute the known pair to find k stage first, then find y when x = 5.
  3. 3.Keep the final answer visible: 20.
Final answer

20

Mark points
  • M1: use the correct direct proportion: y = kxn — as x increases, y increases.inverse proportion: y = k/xn — as x increases, y decreases.always find k first using the given pair, then use k to answer the question.
  • A1: 20
Watch out

Students write the formula as y = kx rather than y = kx2 because they misread 'proportional to x2'.Always underline the power in the question and include it in the equation before finding k.

Mediumreasoning

F is directly proportional to m × a. When m = 2 and a = 3, F = 24. Find F when m = 5 and a = 4.

3 marks4 minsdirect-and-inverse-proportion-q3
Show solution
Worked solution
  1. 1.Spot the skill: Direct proportion: y = kxn — as x increases, y increases.
  2. 2.Use the find y when x = 5 stage first, then find x when y = 80.
  3. 3.Keep the final answer visible: 80.
Final answer

80

Mark points
  • M1: use the correct direct proportion: y = kxn — as x increases, y increases.inverse proportion: y = k/xn — as x increases, y decreases.always find k first using the given pair, then use k to answer the question.
  • A1: 80
Watch out

Students write the formula as y = kx rather than y = kx2 because they misread 'proportional to x2'.Always underline the power in the question and include it in the equation before finding k.

Hardproblem solving

y ∝ 1/x2. When x = 2, y = 9. Find y when x = 6.

3 marks5 minsdirect-and-inverse-proportion-q4
Show solution
Worked solution
  1. 1.Spot the skill: Direct proportion: y = kxn — as x increases, y increases.
  2. 2.Use the find x when y = 80 stage first, then write the proportionality equation.
  3. 3.Keep the final answer visible: 1.
Final answer

1

Mark points
  • M1: use the correct direct proportion: y = kxn — as x increases, y increases.inverse proportion: y = k/xn — as x increases, y decreases.always find k first using the given pair, then use k to answer the question.
  • A1: 1
Watch out

Students write the formula as y = kx rather than y = kx2 because they misread 'proportional to x2'.Always underline the power in the question and include it in the equation before finding k.

Exam-stylemulti-step

It takes 6 workers 8 days to complete a job. How long for 4 workers?

4 marks6 minsdirect-and-inverse-proportion-q5
Show solution
Worked solution
  1. 1.Spot the skill: Direct proportion: y = kxn — as x increases, y increases.
  2. 2.Use the write the proportionality equation stage first, then substitute the known pair to find k.
  3. 3.Keep the final answer visible: 12 days.
Final answer

12 days

Mark points
  • M1: use the correct direct proportion: y = kxn — as x increases, y increases.inverse proportion: y = k/xn — as x increases, y decreases.always find k first using the given pair, then use k to answer the question.
  • A1: 12 days
Watch out

Students write the formula as y = kx rather than y = kx2 because they misread 'proportional to x2'.Always underline the power in the question and include it in the equation before finding k.

Grade 9 stretchproblem solving

y is inversely proportional to the square of x. When x = 3, y = 8. Find y when x = 12.

4 marks7 minsproportion-g9
Show solution
Worked solution
  1. 1.Write y = k/x2.
  2. 2.Use the first pair to find k.
  3. 3.Substitute x = 12.
Final answer

y = 12\frac{1}{2}

Mark points
  • M1: k = 72
  • M1: use 721447\frac{2}{1}44
  • A1: 12\frac{1}{2}
Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint
12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Direct and inverse proportion - 2 marksy is directly proportional to x. When x = 4, y = 20. Find y when x = 7.Mark answer
Answer

35

2Ratio and sharing - 2 marksTwo people share profit in ratio 5:3. Total profit £640. Find each share.Mark answer
Answer

£400 and £240

3Fractions and ratios - 2 marksA map scale is 1:25,000. Express as a fraction.Mark answer
Answer

125\frac{1}{2}5,000

4Percentage change - 3 marksA car depreciates by 20% per year. It was worth £12,000 new. Find its value after 2 years.Mark answer
Answer

£7,680

Mastery check
  • I can explain the method for direct and inverse proportion.
  • I can show clear working without skipping key steps.
  • I can avoid this mistake: Students write the formula as y = kx rather than y = kx2 because they misread 'proportional to x2'.Always underline the power in the question and include it in the equation before finding k.
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Related topics
Ratio and sharingFractions and ratiosPercentage change
Official exam-board sources

This guide follows the Pearson Edexcel GCSE Mathematics 1MA1 specification. Practice questions are original Learnova questions shaped around official content and exam skills.

Official specification Official past papers
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