Direct and inverse proportion, find k, then use it
Learn how to spot direct and inverse proportion using pizza slices and shared bills. Worked examples, exam-style questions with answers.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
Download the PDF (660 KB)If 4 friends share a pizza bill and 8 friends share it instead, why does each person pay half, not double?
Proportion questions are about how two amounts change together. Picture a pizza party: more guests need more pizza (direct), but more guests sharing one bill each pay less (inverse). Every question is the same three moves: find k, write the formula, use it.
The picture to keep in your head: A pizza party
Direct proportion: each guest gets 2 slices, so more guests means more slices, in step: double the guests, double the slices (slices = 2 × guests). Inverse proportion: one pizza bill is shared, so more guests means each pays less: double the guests, half each (each pays = bill ÷ guests). The fixed number (2 slices each, or the bill) is k, the constant of proportionality. Every question is: find k, write the formula, use it.
The method, every time
- Decide: direct (both grow together) or inverse (one grows as the other shrinks)?
- Write the formula with k: y = kx, y = k/x, y = kx2, and so on.
- Put in the pair of values you are given to find k.
- Write the formula again with the number in place of k.
- Use it to find the value the question asks for, x or y.
Words you'll see
- direct proportion
- both amounts grow together: double one, double the other
- inverse proportion
- one grows as the other shrinks: double one, halve the other
- k
- the constant of proportionality, the fixed number in the formula
- formula
- a rule linking the two amounts, e.g. y = 4.5x
Double one, double the other
At a pizza party, if each guest gets 2 slices, more guests mean more slices. This is direct proportion: y = kx. Find k with one pair, then use the formula for any other value.
Watch out
- Adding the difference instead of multiplying. Use multiplication, not addition, to find y
- Forgetting to write the formula with k. Always write y = kx before using it
Double one, halve the other
At a pizza party, if more guests share the same bill, each pays less. This is inverse proportion. When y is inversely proportional to x, their product is a fixed number k. So, to find k, multiply x and y. Then use k to find the new y when x changes.
Watch out
- Using y = kx for an inverse question. Use y = k/x for inverse proportion
- Writing k = y/x instead of k = xy. Multiply x and y to find k, not divide
Proportional to x2, or to the square root
Some amounts grow faster than in step. A pizza twice as wide has four times the area, because area depends on the width squared. So read the wording carefully: 'proportional to the square of x' means y = kx2, and 'inversely proportional to the square root of x' means y = k/√(x). The method never changes: find k from the pair you're given, write the formula, then use it.
Watch out
- Forgetting to square x when finding k. Always square x before dividing y by x² to find k
- Squaring y instead of x. Use the correct variable in the formula: y = kx², not y² = kx
Tables and graphs
At a pizza party, if more guests come, more slices are needed for direct proportion. If more guests share the same pizza, each pays less for inverse proportion. In a table, direct proportion has y ÷ x constant; inverse has x × y constant. Graphs show direct as a straight line through (0, 0), and inverse as a curve that never touches the axes.
Watch out
- Saying any straight line is direct proportion. Check if the line passes through (0, 0)
- Checking only one pair of values. Check all pairs to confirm the constant
21 exam-style direct and inverse proportion questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · Direct, real life
- 1A recipe for 4 cakes needs 200g of flour. How much flour is needed for 10 cakes?[2 marks]
Show the answer
500g
4 cakes = 200g flour, so 1 cake = 50g. 10 cakes = 50 × 10 = 500g. Formula: flour = 50 × cakes. - 25 pens cost £3.40. How much do 8 pens cost?[2 marks]
Show the answer
£5.44
5 pens = £3.40, so 1 pen = £0.68. 8 pens = 0.68 × 8 = £5.44. Formula: cost = 0.68 × pens. - 3An exchange rate is £1 = €1.25. How many euros do you get for £20?[2 marks]
Show the answer
€25
£1 = €1.25, so £20 = 1.25 × 20 = €25. Formula: euros = 1.25 × pounds. - 4A car uses 6 litres of fuel for 120 km. How much fuel is needed for 200 km?[2 marks]
Show the answer
10 litres
120 km = 6 litres, so 1 km = 0.05 litres. 200 km = 0.05 × 200 = 10 litres. Formula: fuel = 0.05 × distance.
B · y = kx and y = k/x
- 5Each guest at a party gets 4 slices of pizza. Write a formula for the total number of slices, s, for g guests, and find s when there are 10 guests.[3 marks]
Show the answer
s = 4g, 40 slices
Direct proportion with k = 4: s = 4g, so s = 4 × 10 = 40. - 6y is directly proportional to x. When x = 6, y = 15. Find y when x = 10.[3 marks]
Show the answer
25
k = 15 ÷ 6 = 2.5, so y = 2.5x and y = 2.5 × 10 = 25. - 7A £20 pizza bill is shared equally. Write a formula for the cost per person, c, when there are n people, and find c for 8 people.[3 marks]
Show the answer
c = 20/n, £2.50
Inverse proportion with k = 20: c = 20/n = 20 ÷ 8 = 2.50. - 8y is inversely proportional to x. When x = 4, y = 9. Find x when y = 6.[3 marks]
Show the answer
x = 6
k = 4 × 9 = 36, so y = 36/x; 6 = 36/x gives x = 6. - 9A £100 bill is shared equally by 8 people. How much would each person pay if only 5 people shared it?[3 marks]
Show the answer
£20
k = 100: 100 ÷ 5 = £20 each (it was £12.50 each for 8).
C · Squares, cubes and roots (Higher)
- 10y is directly proportional to x squared. When x = 3, y = 27. Write the formula for y in terms of x and find y when x = 5.[4 marks]
Show the answer
y = 3x², y = 75
y = kx². 27 = k×3² → k = 27 ÷ 9 = 3. y = 3x². y = 3×5² = 3×25 = 75. - 11y is inversely proportional to x squared. When x = 2, y = 5. Write the formula for y in terms of x and find y when x = 4.[4 marks]
Show the answer
y = 20/x², y = 1.25
y = k/x². 5 = k/2² → k = 5×4 = 20. y = 20/x². y = 20/4² = 20/16 = 1.25. - 12y is directly proportional to the square root of x. When x = 16, y = 8. Write the formula for y in terms of x and find y when x = 25.[4 marks]
Show the answer
y = 2√x, y = 10
y = k√x. 8 = k×√16 → k = 8 ÷ 4 = 2. y = 2√x. y = 2×√25 = 2×5 = 10. - 13y is inversely proportional to x cubed. When x = 2, y = 1. Write the formula for y in terms of x and find y when x = 1.[4 marks]
Show the answer
y = 8/x³, y = 8
y = k/x³. 1 = k/2³ → k = 1×8 = 8. y = 8/x³. y = 8/1³ = 8. - 14y is directly proportional to x cubed. When x = 2, y = 32. Write the formula for y in terms of x and find x when y = 128.[4 marks]
Show the answer
y = 4x³, x ≈ 3.17
y = kx³. 32 = k×2³ → k = 32 ÷ 8 = 4. y = 4x³. 128 = 4x³ → x³ = 32 → x = ∛32 ≈ 3.1748.
D · Workers and time
- 15A group of 6 painters take 10 days to paint a house. How long would it take 4 painters to paint the same house?[3 marks]
Show the answer
15 days
More painters mean less time. So time = k ÷ painters. 10 = k ÷ 6 → k = 60. Time = 60 ÷ 4 = 15 days. - 16A journey takes 3 hours at 60 km/h. How long does the same journey take at 90 km/h?[3 marks]
Show the answer
2 hours
Time is inversely proportional to speed: k = 3 × 60 = 180, time = 180 ÷ 90 = 2. - 178 taps fill a tank in 45 minutes. How long would 6 taps take?[3 marks]
Show the answer
60 minutes
More taps, less time: k = 8 × 45 = 360, time = 360 ÷ 6 = 60.
E · Spot and explain (Higher)
- 18x: 1, 2, 5 and y: 7, 14, 35. Is y directly proportional, inversely proportional, or neither? Give a reason.[2 marks]
Show the answer
directly proportional: y ÷ x = 7 every time
7 ÷ 1 = 14 ÷ 2 = 35 ÷ 5 = 7, so y = 7x. - 19x: 1, 2, 4 and y: 12, 6, 2. Is y inversely proportional to x? Give a reason.[2 marks]
Show the answer
No: x × y is 12, 12 and 8, not the same each time
For inverse proportion x × y must stay the same; 4 × 2 = 8 breaks it. - 20y is inversely proportional to x². What happens to y when x is doubled?[2 marks]
Show the answer
y is divided by 4
y = k/x²; doubling x multiplies x² by 4, so y becomes a quarter of what it was. - 21Which of these shows direct proportion: y = 3x, y = 3x + 2 or y = 3/x? Explain.[2 marks]
Show the answer
y = 3x: a straight line through (0, 0)
y = 3x + 2 is straight but misses (0, 0); y = 3/x is inverse.
Direct and inverse proportion exam checklist
- Write the formula with k first, then substitute values.
- Find k from the given pair of numbers.
- Direct: divide y by x to find k. Inverse: multiply x and y to find k.
- Read 'square' and 'square root' carefully in the question.
- Answer the question asked—make sure you find x or y, not just k.
Got stuck on a question?
Find the part of this guide it belongs to and go through the worked example again, line by line. Then put your answer back into the question: if it doesn't work, look for a sign or a step you skipped.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
Download the PDF (660 KB)Want maths to click like this?
Every Bloom lesson works the same way: one idea at a time, shown with a picture before it's explained, then practised until it sticks. Book a free assessment and see it applied to your child's own work.
