Direct Proportion and the Unitary Method
Two quantities are in direct proportion when they increase or decrease at the same rate, so their ratio stays constant and doubling one quantity always doubles the other. The unitary method is the standard way to solve direct proportion problems: find the value of a single unit, such as one item or one hour, first, then multiply by however many units the question asks about, the same idea that underlies best-value comparisons.
Method
- Identify the two quantities that are in direct proportion - as one increases, the other increases by the same multiplier.
- Use the unitary method: divide to find the value of a single unit (one item, one person, one hour, and so on).
- Multiply this single-unit value by however many units the question asks for, to scale up or down to the answer.
- Alternatively, find the multiplier between the two given amounts of the same quantity, and apply that same multiplier to the other quantity.
- For best-value or best-buy problems, find the price (or amount) per common unit for each option, then compare like with like.
- Keep track of units throughout - convert to the same unit before dividing or multiplying if the question mixes, for example, grams and kilograms.
- Sense-check the answer: if a smaller amount was given and a larger one is being found, the answer should be bigger than the value started from, and vice versa.
Worked example
9 identical bricks weigh 20.7 kg in total. Work out the weight of 14 of the same bricks.
- Find the weight of a single brick (the unitary method): 20.7 / 9 = 2.3 kg.
- Multiply the weight of one brick by the number of bricks needed: 2.3 x 14.
- Calculate: 2.3 x 14 = 32.2 kg.
- Sense-check: 14 bricks is close to double 9, so the answer should be close to double 20.7 kg - 32.2 kg is a sensible size for that.
Practice questions
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Q112 identical calculators cost 54 pounds in total. Work out the cost of 5 of these calculators.Show answer
Answer: 22.50 pounds (54 / 12 = 4.50 pounds each, then 4.50 x 5)
Q2A printer produces 45 leaflets in 6 minutes, printing at a constant rate. How many leaflets does it produce in 14 minutes?Show answer
Answer: 105 leaflets (45 / 6 = 7.5 per minute, then 7.5 x 14)
Q3A stationery shop sells a pack of 8 notebooks for 9.60 pounds and a pack of 15 notebooks for 16.50 pounds. Which pack offers the better value per notebook?Show answer
Answer: The pack of 15 (1.10 pounds per notebook, compared with 1.20 pounds per notebook for the pack of 8)
Q4A minibus uses 15 litres of fuel to travel 84 miles at a constant rate of consumption. Work out how far the minibus can travel on 40 litres of fuel.Show answer
Answer: 224 miles (84 / 15 = 5.6 miles per litre, then 5.6 x 40)
Q5A 3D printer extrudes 54 g of plastic filament every 12 minutes it runs, at a constant rate. Work out how much filament it extrudes in 50 minutes.Show answer
Answer: 225 g (54 / 12 = 4.5 g per minute, then 4.5 x 50)
Q6A bank exchanges 65 pounds for 78.65 US dollars. Using the same exchange rate, work out how many US dollars 100 pounds would exchange for.Show answer
Answer: 121 US dollars (78.65 / 65 = 1.21 dollars per pound, then 1.21 x 100)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A charity fun run sells wristbands to raise money. 8 wristbands raise 27.60 pounds for the charity. The organisers want to raise at least 200 pounds from wristband sales alone. Work out the smallest number of wristbands they need to sell, showing your method.
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A hotel charges for its conference room in direct proportion to the number of hours booked. Booking the room for 3.5 hours costs 122.50 pounds. (a) Work out the hourly rate. (b) A company books the room for a full day event lasting 7.5 hours. Work out the total cost.
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Free printable worksheet
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This topic is chapter 38 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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