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 salt solution has a concentration of 10%. The solution contains 50 g of salt. Work out the total mass of the solution.
- 2.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.
- 3.Two numbers are in the ratio 3 : 7. Their sum is 60. Work out the positive difference between the two numbers.
- 4.Ollie says that the ratio 6 : 15 is equivalent to the ratio 2 : 5. Is he correct? Give a reason for your answer.
- 5.£96 is shared between Ben, Cara and Dev in the ratio 2:3:7. Work out how much Cara receives.
- 6.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.
- 7.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.
- 8.A car travels 90 km using 6 litres of petrol. Work out how far the car can travel, at the same rate, using 9 litres of petrol.
- 9.y is directly proportional to x. When x = 8, y = 20. Work out the value of y when x = 12.
- 10.Two numbers A and B are in the ratio 3:5. When 6 is added to B, the ratio of A to the new value of B is 1:2. Work out the value of A.
- 11.A rectangle has its height and its width in the ratio 3:2. The height of the rectangle is 18 cm. Work out the width of the rectangle.
- 12.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.
- 13.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.
- 14.300 g of a salt solution of concentration 10% is poured into 200 g of a salt solution of concentration 25%. Work out the concentration of salt in the mixture.
- 15.An acid solution has a concentration of 20%. The solution contains 40 g of acid, and the rest of the solution is water. Work out the mass of water in the solution.
Answer key
- (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) 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.
- (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.
- (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.
- (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.)
- (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.)
- (a) 135 km — Method: find the distance travelled on one litre, then scale up to 9 litres. Working: 90 ÷ 6 = 15 km per litre, so 15 × 9 = 135 km. Answer: 135 km. 45 km comes from working out the extra distance for the extra 3 litres (15 × 3) but forgetting to add the original 90 km. 60 km comes from using the ratio the wrong way round, 90 × 6 ÷ 9, instead of finding the rate per litre first. 99 km comes from simply adding the number of litres, 9, onto the original distance, 90, instead of scaling the whole journey.
- (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) 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.
- (b) 12 cm — Method: height:width = 3:2 means both lengths are built from parts of the same size, so write 3:2 = 18:x, find one part and multiply by the number of parts in the width. Working: the height is 3 parts and measures 18 cm, so one part is 18 ÷ 3 = 6 cm; the width is 2 parts, so 2 × 6 = 12 cm. Answer: 12 cm. The distractors: 27 cm comes from using the ratio the wrong way round, 18 ÷ 2 × 3, which makes the width longer than the height even though 2 is the smaller part; 6 cm is the value of one part, given as the width instead of being doubled; 17 cm comes from treating the ratio as a difference — 3 − 2 = 1, so 1 cm is taken off the height — but a ratio scales the lengths, it does not subtract from them.
- (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.
- (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) 16% — Method: scale each concentration to its own mass to find the salt it contains, add the two masses of salt, then write the ratio of salt to mixture per 100 g. Working: 10:100 = x:300 gives 30 g of salt, and 25:100 = y:200 gives 50 g of salt; the mixture holds 30 + 50 = 80 g of salt in 300 + 200 = 500 g of solution; 80:500 = 16:100. Answer: 16%. The distractors: 17.5% is the mean of 10% and 25%, which ignores that there is more of the weaker solution than of the stronger one; 19% comes from swapping the two concentrations over, working out (300 × 25% + 200 × 10%) ÷ 500; 26.7% comes from dividing the 80 g of salt by the 300 g of the first solution rather than by the 500 g of mixture.
- (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.
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