14.5 Direct and Inverse Proportion
GCSE Maths Foundation · Edexcel · Chapter 14: Multiplicative reasoning
A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: the unitary method, best buys, proportion graphs and y = kx. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.
What you will learn
- Solve direct proportion problems using the unitary method, including recipes
- Compare value for money (best buys)
- Recognise direct proportion graphs and use y = kx
- Solve inverse proportion problems, such as workers and time
Key words
- Direct proportion: Two quantities increase at the same rate: double one and the other doubles. Their ratio stays the same.
- Inverse proportion: As one quantity increases the other decreases: double one and the other halves. Their product stays the same.
- Unitary method: Find the value of one item (one unit) first, then multiply.
- Constant, k: The constant of proportionality: the fixed number in y = kx. It is the value of y when x = 1.
Direct proportion and the unitary method
If quantities are in direct proportion, find the value for one, then multiply up.
Worked example: pens
6 pens cost £4.20. How much do 15 pens cost?
One pen: 4.20 ÷ 6 = £0.70. 15 pens: 15 × 0.70 = £10.50.
Answer: £10.50
Worked example: a recipe
A soup recipe for 4 people uses 300 g of lentils and 900 ml of stock. How much is needed for 10 people?
For 1 person: 75 g lentils, 225 ml stock. For 10: 750 g lentils, 2250 ml stock.
Answer: 750 g lentils, 2250 ml (2.25 litres) stock
Best buys
To compare value, work out the cost of the same amount of each (for example, per 100 g), or how much you get per £1.
Worked example: two jars of coffee
A 200 g jar costs £5.40. A 350 g jar costs £9.10. Which is better value?
Small: 540p ÷ 2 = 270p per 100 g. Large: 910p ÷ 3.5 = 260p per 100 g.
Answer: The 350 g jar is better value.
Direct proportion graphs and y = kx
The graph of two quantities in direct proportion is a straight line through the origin. Its equation is y = kx, where k is the gradient.
Worked example: finding k
40 litres of fuel cost £58. Write a formula for the cost, £C, of L litres.
C = kL. 58 = k × 40, so k = 58 ÷ 40 = 1.45. C = 1.45L.
Cost of 25 litres = 1.45 × 25 = £36.25.
Answer: C = 1.45L; £36.25
Key fact: A straight line that does not go through (0, 0) is not direct proportion. A taxi fare with a fixed charge is an example.
Inverse proportion
With inverse proportion, multiply the two quantities to find the fixed total, then divide.
Worked example: workers and days
4 builders take 9 days to build a wall. How long would 6 builders take, working at the same rate?
Total work = 4 × 9 = 36 builder-days. 36 ÷ 6 = 6 days.
Answer: 6 days
The formula has the form y = k/x. Here days = 36 ÷ workers.
Common mistakes
- Treating an inverse proportion as direct: more workers do not take more days.
- Comparing best buys using different amounts, such as price per 100 g against price per kg.
- Thinking any straight-line graph shows direct proportion. It must pass through the origin.
- Rounding the cost of one item too early, which spoils the final answer.
Revision questions
Try these, then check your answers below each one.
Question 1. 4 cinema tickets cost £26. Work out the cost of 7 tickets.
Answer: 45.50 £
Working: 26 ÷ 4 = 6.50. 7 × 6.50 = £45.50.
Question 2. 5 workers take 12 days to finish a job. How many days would 10 workers take, working at the same rate?
Answer: 6 days
Working: 5 × 12 = 60 worker-days. 60 ÷ 10 = 6 days.
Question 3. Which graph shows that y is directly proportional to x? (Choose from: A straight line through the origin, A straight line that crosses the y-axis at 5, A curve that gets closer to the x-axis, A horizontal line.)
Answer: A straight line through the origin
Working: y = kx is a straight line through (0, 0).
Related lessons
- Previous lesson: 14.4 Distance, Speed and Time
- Next lesson: 15.1 3D Solids
- In this chapter: 14.1 Percentages
- In this chapter: 14.2 Growth and Decay
- In this chapter: 14.3 Compound Measures
Continue in the full course
This lesson is section 14.5 (EGCMF1405) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.