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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Answer key: Proportion as equality of ratios worksheet — GCSE Foundation

Maths · GCSE Foundation · R7 · Proportion as equality of ratios · 15 questions

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(a) 0.6 litres — The ratio 1 : 9 means the solution has 1 + 9 = 10 equal parts in total. Each part is 6 ÷ 10 = 0.6 litres, and disinfectant is 1 part, so Priti needs 0.6 litres of disinfectant. Giving 0.667 litres divides by 9, the number of parts of water, instead of the total number of parts, 10 (6 ÷ 9 ≈ 0.667). Giving 6 litres is the total amount of solution, not just the disinfectant's share of it. Giving 5.4 litres works out the water's share (6 × 9 ÷ 10 = 5.4), not the disinfectant's.
(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.
(a) 375 g — Find the amount of flour needed per muffin: 150 ÷ 6 = 25 g. Multiply by the new number of muffins: 25 × 15 = 375 g. (60 g comes from using the scale factor the wrong way round, 6/15 × 150. 150 g comes from not scaling the recipe at all. 300 g comes from rounding the scale factor, 15 ÷ 6, down to 2 before multiplying.)
(d) 24 — Method: split 60 into 3 + 7 = 10 equal parts, find the value of one part, then use the difference in ratio parts. Working: 60 ÷ 10 = 6, so the numbers are 3 × 6 = 18 and 7 × 6 = 42, and their difference is 42 − 18 = 24. Answer: 24. 4 comes from finding the difference between the ratio numbers, 7 − 3, but forgetting to multiply by the value of one part. 60 comes from adding the two numbers back together instead of subtracting, which just repeats the given sum. 80 comes from dividing 60 by the first ratio number, 3, instead of by the total number of parts, 10, giving a part value of 20 and a difference of 7 × 20 − 3 × 20 = 80.
(d) £24 — Add the parts of the ratio: 2 + 3 + 7 = 12. Find the value of one part: £96 ÷ 12 = £8. Cara's share is 3 parts: 3 × £8 = £24. (£16 is Ben's share, 2 × £8, not Cara's. £56 is Dev's share, 7 × £8, not Cara's. £28.80 comes from wrongly adding the ratio parts as 10 instead of 12, giving one part = £9.60.)
(a) 160 g — Method: use the ratio 20:100 to find the mass of the whole solution from the mass of acid, then take the acid away to leave the water. Working: 20:100 = 40:m, and 40 ÷ 20 = 2, so m = 2 × 100 = 200 g of solution; the water is 200 − 40 = 160 g. Answer: 160 g. The distractors: 200 g is the mass of the whole solution, which is the middle step and includes the acid the question asks you to leave out; 8 g comes from working out 20% of 40 g, which treats the 40 g as the whole solution rather than as the 20% inside it; 10 g comes from reading the 40 g as the 80% that is water, giving a solution of 50 g and a difference of 50 − 40.
(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.)
(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) 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.
(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.
(a) 18 — Method: write both numbers with the same multiplier, turn the second ratio into an equation by cross-multiplying, solve for the multiplier and then build A from it. Working: let A = 3k and B = 5k, so 3k : (5k + 6) = 1 : 2; cross-multiplying gives 2 × 3k = 5k + 6, so 6k = 5k + 6 and k = 6; A = 3 × 6 = 18. Answer: 18, and the check works, because B = 30, B + 6 = 36 and 18:36 = 1:2. The distractors: 9 comes from reading the 6 as the difference between the two numbers — 5 − 3 = 2 parts, so one part is 3 and A is 3 × 3 — but the 6 is added to B, it is not the gap between A and B; 6 comes from solving A : (A + 6) = 1 : 2, adding the 6 to A instead of to B; 30 is the value of B, found from the correct multiplier but given in place of A.
(c) 250 miles — Find the distance travelled in 1 hour: 150 ÷ 3 = 50 miles. Multiply by 5 hours: 50 × 5 = 250 miles. Giving 300 miles doubles the original distance (150 × 2 = 300) using a scale factor of 2 instead of the correct 5 ÷ 3. Giving 200 miles adds only one extra hour's distance, 50, instead of the two extra hours actually needed (150 + 50 = 200, rather than 150 + 100). Giving 90 miles divides by the scale factor instead of multiplying (150 × 3 ÷ 5 = 90).
(d) 132 — Method: find the value of one part of the ratio, use it to find Leo's pages, then add both amounts together. Working: 84 ÷ 7 = 12 (value of one part). Leo's pages = 12 × 4 = 48. Total = 84 + 48 = 132. Wrong options: 48 gives only Leo's pages and forgets to add Mia's; 147 comes from reversing the ratio parts (84 ÷ 4 × 7 = 147) and stopping there; 231 comes from reversing the ratio parts and then adding Mia's pages (84 + 147).
(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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