11.8 Proportion Problems

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: inverse proportion, telling direct and inverse proportion apart, and multi-step problems. 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

• Recognise when two quantities are in inverse proportion
• Solve inverse proportion problems by finding the total amount of work first
• Recognise the graphs and tables of direct and inverse proportion
• Solve mixed proportion problems in real-life contexts

Key words

• Inverse proportion: When one quantity is multiplied by k, the other is divided by k. Double one and the other halves.
• Product: The answer when two numbers are multiplied. In inverse proportion x × y is always the same.
• Direct proportion: When one quantity is multiplied by k, so is the other. y ÷ x is always the same.
• Rate: How fast something happens, such as km per hour or bottles per hour.

Inverse proportion

More workers finish a job faster. If the number of workers doubles, the time halves. Find the total work first (workers × days), then share it out.

Worked example: painters

4 painters take 6 days to paint a school. How long would 3 painters take, working at the same rate?
Total work = 4 × 6 = 24 painter-days. (1 painter would take 24 days.)
3 painters: 24 ÷ 3 = 8 days

Answer: 8 days

Key fact: Inverse proportion: multiply to find the total, then divide. Ask yourself, “Should the answer be bigger or smaller?”

Graphs and tables

For a 120 km journey, speed × time = 120. As the speed goes up, the time goes down. The graph is a curve that gets closer to the axes but never touches them.

• Time (hours); 20 6; 30 4; 40 3; 60 2; 120 1

Worked example: using a table

y is inversely proportional to x. When x = 5, y = 12. Find y when x = 15.
x × y = 5 × 12 = 60. y = 60 ÷ 15 = 4

Answer: y = 4

Direct or inverse?

Ask: “If I double the first amount, does the second double (direct) or halve (inverse)?”

• Number of identical tins and their total mass; Type direct
• Number of people sharing a bill and the amount each pays; Type inverse
• Hours worked at a fixed rate and the pay; Type direct
• Speed and time for the same journey; Type inverse

Worked example: deciding

A field can be cut by 2 mowers in 9 hours. Is the time in direct or inverse proportion to the number of mowers?
Doubling the mowers to 4 halves the time to 4.5 hours.

Answer: Inverse proportion

Multi-step problems

Some problems change part-way through, or mix direct and inverse proportion. Write down what stays the same.

Worked example: camp food

A camp has enough food for 120 campers for 15 days. 30 more campers arrive at the start. How long will the food last now?
Total food = 120 × 15 = 1800 camper-days. Campers now = 150.
1800 ÷ 150 = 12 days

Answer: 12 days

Worked example: two steps

6 machines fill 900 bottles in 2 hours. How long do 4 machines take to fill 1200 bottles?
1 machine fills 900 ÷ 6 = 150 bottles in 2 hours, so 75 bottles an hour.
4 machines fill 4 × 75 = 300 bottles an hour. 1200 ÷ 300 = 4 hours

Answer: 4 hours

Common mistakes

• Using direct proportion when more workers should mean less time.
• Dividing first in an inverse problem. Multiply to find the total work, then divide.
• Calling any line that slopes downwards “inverse proportion”. The inverse graph is a curve.
• Forgetting to change units, such as 2 hours 30 minutes = 2.5 hours, not 2.3 hours.

Revision questions

Try these, then check your answers below each one.

Question 1. 3 workers take 10 days to fit a kitchen. How many days would 5 workers take, working at the same rate?

Answer: 6 days

Working: 3 × 10 = 30 worker-days. 30 ÷ 5 = 6 days.

Question 2. A journey takes 2.5 hours at an average speed of 48 mph. How many hours does it take at 60 mph?

Answer: 2 hours

Working: Distance = 48 × 2.5 = 120 miles. 120 ÷ 60 = 2 hours.

Question 3. Which of these pairs are in <b>inverse</b> proportion? (Choose from: Number of identical books and their total mass, Number of people sharing a prize equally and the amount each gets, Hours worked at £11 an hour and the pay.)

Answer: Number of people sharing a prize equally and the amount each gets

Working: Twice as many people means each gets half as much, so that pair is inverse. The others are direct.

Related lessons

• Previous lesson: 11.7 Proportion and Graphs
• Next lesson: 12.1 Pythagoras' Theorem 1
• In this chapter: 11.1 Writing Ratios
• In this chapter: 11.2 Using Ratios 1
• In this chapter: 11.3 Ratios and Measures

Continue in the full course

This lesson is section 11.8 (EGCMF1108) 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.