Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Ratio, proportion and rates of change
Practise Direct and inverse proportion. unlimited generated questions on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Two quantities are in direct proportion when they increase at the same rate (double one and the other doubles), and in inverse proportion when one halves as the other doubles. Both tiers use recipes, best buys and 'workers and days' problems; at Higher tier you also form and use equations such as \(y = kx^2\) or \(y = \frac{k}{\sqrt{x}}\).
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
3
Scale a recipe up or downFind the amount for one (or a scale factor) and multiply every ingredient by the same number.
4
Compare value for money using unit pricesWork out the cost per item, or per 100 g, for each pack and compare.
4
Solve inverse proportion problems like workers and daysFind the total work (workers × days), then divide by the new number of workers.
5
Recognise direct and inverse proportion graphsDirect proportion is a straight line through the origin; inverse proportion is a curve that gets closer and closer to both axes.
Notes
Direct proportion
y is directly proportional to x, written \(y \propto x\), when \(y = kx\) for a constant k.
If x doubles, y doubles; if x is halved, y is halved. The ratio y : x stays the same.
The graph of \(y = kx\) is a straight line through the origin, with gradient k.
Unitary method: find the value for one, then multiply. 6 pens cost £4.50, so 1 pen costs 75p and 10 pens cost £7.50.
Best buys: compare the cost of the same amount (e.g. per 100 g), or the amount you get for £1.
Inverse proportion
y is inversely proportional to x when \(y = \frac{k}{x}\), so the product xy = k is always the same.
If x doubles, y halves. Being inversely proportional to x is the same as being proportional to \(\frac{1}{x}\).
The graph is a curve that gets closer and closer to both axes but never meets them.
Workers and days: 4 builders take 9 days, so the job is 36 'builder-days'. 6 builders take 36 ÷ 6 = 6 days, if they all work at the same rate.
Cheatsheet
Direct proportion: \(y = kx\); the graph is a straight line through the origin
Inverse proportion: \(y = \frac{k}{x}\), so xy = k
\(\propto\) means 'is proportional to'
Unitary method: find the value for one, then multiply
Inverse problems: workers × time stays the same
How to answer each type of question
Scale a recipe (direct proportion)
2 marks3
Find the amount for one (divide), or the scale factor (new number ÷ old number).
Multiply the quantity you need by the new number, or by the scale factor.
Check the answer is sensible: more flapjacks need more of each ingredient.
Example. Here are the ingredients needed to make 12 flapjacks. 150 g oats, 90 g butter, 60 g sugar, 45 g golden syrup Kate wants to make 20 flapjacks. How much butter does she need?
Show the model answer
90 ÷ 12 = 7.5 g of butter for one flapjack (M1) 7.5 × 20 = 150 g (A1)
Decide which pack is the best value
3 marks4
Work out the cost of one item (or of 100 g, or of 1 litre) for each pack, in the same units.
Or scale the packs up to the same size and compare the prices.
Write a conclusion naming the better value pack, with your figures.
Example. Tea bags are sold in two sizes. A small box has 80 tea bags and costs £2.40. A large box has 240 tea bags and costs £6.60. Which box is the better value for money? You must show how you get your answer.
Show the model answer
Small box: 240p ÷ 80 = 3p per tea bag (M1) Large box: 660p ÷ 240 = 2.75p per tea bag (M1) The large box is better value, because 2.75p is less than 3p (C1)
Solve an inverse proportion problem in context
2 marks4
Check it is inverse: more machines (or workers) means less time.
Multiply to find the total amount of work, e.g. machines × hours.
Divide by the new number of machines or workers.
Example. It takes 5 machines 12 hours to complete an order. How long would it take 3 machines to complete the same order? Assume that all the machines work at the same rate.
Direct: both go up together at the same rate. Inverse: one goes up as the other goes down, and their product stays the same.
Graph check: direct proportion must pass through (0, 0). A straight line that misses the origin is not direct proportion.
Always write the equation with k first; it is often worth a mark on its own.
Best buys: compare like with like, e.g. pence per tea bag for every pack.
Where marks are lost
Treating an inverse problem as direct: more workers cannot take more days.
Forgetting the power when you substitute: in \(y = kx^2\) with x = 3, you need 9k, not 3k.
Leaving k in the final formula: 'find a formula' needs the value of k, e.g. \(y = 5x^2\).
Comparing best buys in mixed units (pounds and pence), or not saying which is better value.
Giving the value of k when the question asks for y (or x).
Exam technique
'Find a formula for y in terms of x' means an equation starting y = ... with a number in place of k.
When asked why a graph shows direct proportion, say it is a straight line through the origin.
Read the wording carefully: 'the square of x' is \(x^2\), 'the square root of x' is \(\sqrt{x}\) and 'the cube of x' is \(x^3\).
For best buys, show a value per unit for every option, then write a sentence naming the best value.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
It takes 2 painters 8 days to paint a house. How many days would it take 4 painters to paint the house? Assume that all the painters work at the same rate.[2]
Show the answer and mark scheme
Answer: 4 days
M1 for \(2 \times 8 = 16\) (painter-days) or \(2 \times 8 \div 4\)
A1 for 4 cao
Worked solution: Total work \(= 2 \times 8 = 16\) painter-days. \(16 \div 4 = 4\) days
Question 2Medium3 marks
6 decorators can decorate a hotel in 20 days. The job needs to be finished in 12 days. How many more decorators are needed? Assume that all the decorators work at the same rate.[3]
Show the answer and mark scheme
Answer: 4
P1 for \(6 \times 20 = 120\) (decorator-days)
P1 for \(120 \div 12 = 10\) decorators
A1 for 4 cao
Worked solution: \(6 \times 20 = 120\) decorator-days. \(120 \div 12 = 10\) decorators are needed, so \(10 - 6 = 4\) more.