Printable · GCSE Foundation · ages 14-16
Proportion as equality of ratios worksheet — GCSE Foundation
Fifteen questions on "proportion as equality of ratios" — DfE statement R7. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Proportion as equality of ratios worksheet — GCSE Foundation
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- 1.A printer prints pages at a constant rate. It prints 45 pages in 6 minutes. Work out how many pages it prints in 10 minutes.
- 2.The ratio of cats to dogs in a shelter is 3:4. There are 18 cats. Work out the total number of animals in the shelter.
- 3.A recipe for 4 people needs 300 g of rice. Work out how much rice is needed for 6 people, assuming the amount is in direct proportion to the number of people.
- 4.A science technician mixes 400 g of a salt solution of concentration 5% with 100 g of a salt solution of concentration 25%. Work out the concentration of the mixture.
- 5.y is directly proportional to x. When x = 8, y = 20. Work out the value of y when x = 12.
- 6.A shop sells fruit juice concentrate that must be diluted with water in the ratio 2 : 5 (concentrate : water) to make squash. Priya has 8 litres of concentrate. Work out how much water she needs to keep the same ratio.
- 7.A cordial drink is made by mixing cordial concentrate and water in the ratio 1 : 6. Freya says that a 700 ml jug of this drink contains 100 ml of concentrate. Is she correct? Give a reason for your answer.
- 8.A salt solution has a concentration of 10%. The solution contains 50 g of salt. Work out the total mass of the solution.
- 9.A road sign 3 m tall casts a shadow 5 m long. At the same time, a nearby lamppost casts a shadow 10 m long. Work out the height of the lamppost.
- 10.5 decorators paint 3 rooms in 6 hours. All the decorators work at the same rate. Work out how long it takes 10 decorators to paint 5 rooms.
- 11.A map has a scale of 1:25000. A road on the map is 6 cm long. Work out the real-life length of the road, giving your answer in kilometres.
- 12.The ratio of the amount of money Harry has to the amount of money Isla has is 5:3. Harry has £35. Work out how much money Isla has.
- 13.Two numbers are in the ratio 5 : 8. The larger number is 40. Work out the smaller number.
- 14.Ollie says that the ratio 6 : 15 is equivalent to the ratio 2 : 5. Is he correct? Give a reason for your answer.
- 15.The ratio of the number of red beads to blue beads in a bracelet is 3 : 7. There are 21 blue beads. Work out the number of red beads.
Answer key
- (d) 75 pages — Method: find the number of pages printed in one minute, then scale up to 10 minutes. Working: 45 ÷ 6 = 7.5 pages per minute, so 7.5 × 10 = 75 pages. Answer: 75 pages. 27 pages comes from using the ratio upside down, 45 × 6 ÷ 10, instead of finding the rate per minute first. 55 pages comes from simply adding the extra minutes, 10, onto the original number of pages, 45. 70 pages comes from rounding the rate down to 7 pages per minute before multiplying by 10, instead of using the exact rate of 7.5.
- (a) 42 — Find the value of one part: 18 ÷ 3 = 6. Find the number of dogs: 4 × 6 = 24. Add the cats and the dogs to find the total: 18 + 24 = 42. (24 is the number of dogs only, not the total number of animals. 126 comes from multiplying the number of cats by the total number of parts, 18 × 7, instead of finding one part first. 25 comes from adding the ratio numbers 3 and 4 directly to the number of cats.)
- (d) 450 g — Find the amount of rice per person: 300 ÷ 4 = 75 g. Multiply by the new number of people: 75 × 6 = 450 g. Giving 200 g swaps which number the rice is divided and multiplied by (300 ÷ 6 × 4 = 200), scaling the wrong way. Giving 180 g uses 4 + 6 = 10 as the base instead of the original 4 people (300 × 6 ÷ 10 = 180). Giving 400 g assumes each of the 2 extra people needs 300 ÷ 6 = 50 g on top of the original 300 g (300 + 2 × 50 = 400), rather than scaling the whole amount in proportion.
- (c) 9% — Method: a percentage concentration is the ratio of salt to solution written per 100 g, so scale each concentration to the mass it belongs to, add the two masses of salt, then scale the ratio of salt to mixture back to a denominator of 100. Working: 5:100 = x:400 gives 5 ÷ 100 × 400 = 20 g of salt, and 25:100 = y:100 gives 25 g of salt; the mixture holds 20 + 25 = 45 g of salt in 400 + 100 = 500 g of solution; 45:500 = 9:100. Answer: 9%. The distractors: 15% is the mean of 5% and 25%, which would only be right if the two masses were equal, and here one is four times the other; 21% comes from attaching the concentrations to the wrong masses, working out (400 × 25% + 100 × 5%) ÷ 500; 0.9% comes from working out 45 ÷ 500 = 0.09 and then moving the decimal point one place instead of two when writing the decimal as a percentage.
- (c) 30 — Method: use y = kx and find k from the given pair of values, then substitute x = 12. Working: k = 20 ÷ 8 = 2.5, so y = 2.5 × 12 = 30. Answer: 30. 24 comes from treating the relationship as additive, adding the increase in x (12 − 8 = 4) straight onto y (20 + 4 = 24), instead of multiplying by k. 14.5 comes from finding k correctly (2.5) but then adding it to x instead of multiplying (12 + 2.5 = 14.5). 4.8 comes from finding k upside down, 8 ÷ 20 = 0.4, and multiplying by x: 12 × 0.4 = 4.8.
- (d) 20 litres — The ratio of concentrate to water is 2 : 5, so water = concentrate × 5 ÷ 2. 8 × 5 ÷ 2 = 20, so Priya needs 20 litres of water. Giving 40 litres multiplies by 5 but forgets to divide by 2 (8 × 5 = 40). Giving 3.2 litres uses the ratio inverted, multiplying by 2 ÷ 5 instead of 5 ÷ 2 (8 × 2 ÷ 5 = 3.2). Giving 11 litres uses additive reasoning instead of multiplicative: it adds the difference between the ratio parts, 5 − 2 = 3, onto the amount of concentrate (8 + 3 = 11), but ratios scale by multiplying, not by adding a fixed amount.
- (c) Yes — 700 ÷ 7 = 100 ml for the 1 part of concentrate — Method: add the ratio parts to find the total number of parts, divide the total volume by this, then use the ratio to find concentrate's share. Working: 1 + 6 = 7 parts. 700 ÷ 7 = 100 ml per part. Concentrate = 1 part = 100 ml, so Freya is correct. Wrong options: 'divide 700 by 6' uses only one of the ratio numbers instead of the total of 7 parts, giving about 117 ml; '600 ml is concentrate' swaps which ratio number belongs to the concentrate and which belongs to the water; 'half of 700 ml should be concentrate' ignores the ratio altogether and assumes an equal split.
- (c) 500 g — Method: a concentration of 10% is the ratio 10:100, and the salt and the solution in the beaker must be in that same ratio, so write 10:100 = 50:m and scale. Working: 50 ÷ 10 = 5, so the salt is 5 times the 10 of the ratio; the solution must be 5 times the 100 of the ratio, giving 5 × 100 = 500 g. Answer: 500 g. The distractors: 5 g comes from working out 10% of 50 g, which treats the 50 g as the whole solution when it is the salt inside it; 450 g comes from scaling correctly and then taking the 50 g of salt away, which gives the mass of water rather than the mass of the whole solution; 5000 g comes from dividing by 0.01 instead of 0.1, that is from writing 10% as 0.01.
- (c) 6.00 m — Method: the ratio of height to shadow length is the same for both objects. Working: road sign height ÷ shadow = 3 ÷ 5 = 0.6. Lamppost height = 0.6 × 10 = 6.00 m. Wrong options: 16.67 m comes from inverting the ratio, using shadow ÷ height instead of height ÷ shadow (10 × 5 ÷ 3); 8.00 m comes from adding the difference between the two shadow lengths to the road sign's height instead of scaling (3 + (10 − 5)); 1.50 m comes from multiplying by the ratio of the two shadow lengths the wrong way round (3 × 5 ÷ 10).
- (c) 5 hours — Method: measure the job in decorator-hours, which is in the same ratio as the number of rooms, then share the decorator-hours between the decorators available. Working: 5 decorators × 6 hours = 30 decorator-hours for 3 rooms, so one room takes 30 ÷ 3 = 10 decorator-hours; 5 rooms take 5 × 10 = 50 decorator-hours; shared between 10 decorators that is 50 ÷ 10 = 5 hours. Answer: 5 hours. The distractors: 3 hours comes from halving the 6 hours because the number of decorators doubles, while forgetting that there are also more rooms to paint; 10 hours comes from scaling the 6 hours up for the rooms only, 6 × 5 ÷ 3, and leaving the workforce at 5 decorators; 6 hours comes from assuming that doubling the decorators and increasing the rooms cancel each other out, which they do not, because the rooms rise by a factor of 5/3 and the workforce by a factor of 2.
- (b) 1.5 km — Multiply the map length by the scale factor: 6 × 25000 = 150000 cm. Convert to kilometres, using 100 cm = 1 m and 1000 m = 1 km, so 100000 cm = 1 km: 150000 ÷ 100000 = 1.5 km. (1500 km comes from converting only as far as metres, 150000 ÷ 100 = 1500 m, and then writing kilometres on the end. 15 km comes from dividing by 10000 instead of 100000. 0.15 km comes from dividing by 1000000 instead of 100000.)
- (c) £21 — Method: the two amounts are in the same ratio as 5:3, so write 5:3 = 35:x, find the value of one part and then take the number of parts asked for. Working: Harry's £35 is 5 parts, so one part is £35 ÷ 5 = £7; Isla has 3 parts, so 3 × £7 = £21. Answer: £21. The distractors: £7 is the value of a single part, given as Isla's share instead of being multiplied by the 3 parts she has; £28 comes from finding one part correctly and then working out £35 − £7, subtracting one part from Harry's money instead of taking three parts; £33 comes from treating the ratio additively — the parts differ by 5 − 3 = 2, so £2 is taken off Harry's £35, but a ratio compares by multiplying, not by subtracting.
- (c) 25 — Method: divide the larger number by its ratio part to find the value of one part, then multiply by the smaller number's ratio part. Working: 40 ÷ 8 = 5 (value of one part). Smaller number = 5 × 5 = 25. Wrong options: 64 comes from dividing by the smaller ratio part instead of the larger (40 ÷ 5 × 8); 45 comes from adding the value of one part onto 40 instead of scaling down (40 + 5); 35 comes from subtracting the value of one part from 40 (40 − 5) instead of multiplying it by the smaller ratio part.
- (a) Yes — dividing both parts of 6 : 15 by 3 gives 2 : 5 — Method: divide both parts of the ratio by their highest common factor and compare. Working: the highest common factor of 6 and 15 is 3. 6 ÷ 3 = 2 and 15 ÷ 3 = 5, giving 2 : 5, so Ollie is correct. Wrong options: '6 : 15 simplifies to 3 : 5' divides incorrectly, giving the wrong simplified ratio; 'cannot simplify a ratio that does not start with an even number' states a false rule — any ratio can be simplified if its parts share a common factor; '15 ÷ 6 is not a whole number, so it cannot be simplified' wrongly tries to divide one part by the other instead of finding a common factor.
- (b) 9 — Method: divide the known quantity by its ratio number to find the value of one part, then multiply by the other ratio number. Working: 21 ÷ 7 = 3 (value of one part). Red beads = 3 × 3 = 9. Wrong options: 49 comes from dividing by the wrong ratio number (21 ÷ 3 × 7); 24 comes from adding the ratio number for red (3) to 21 instead of scaling; 14 comes from subtracting the ratio number for blue (7) from 21 instead of scaling.
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