Printable · GCSE Foundation · ages 14-16
Ratio, proportion and rates of change worksheet — GCSE Foundation
Fifteen questions across the ratio, proportion and rates of change statements at Foundation tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
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Answer key: Ratio, proportion and rates of change worksheet — GCSE Foundation
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- (a) 20% — Method: a percentage concentration compares the sugar with the whole solution, so add the two masses to get the mass of solution and then scale the ratio of sugar to solution to a denominator of 100. Working: the solution has a mass of 200 + 50 = 250 g; sugar:solution = 50:250, and scaling to per 100 gives 50 ÷ 250 × 100 = 20, so the ratio is 20:100. Answer: 20%. The distractors: 25% comes from comparing the sugar with the 200 g of water, 50:200, rather than with the whole solution; 80% is the percentage of the solution that is water, 200:250, which answers for the wrong part of the mixture; 0.2% comes from working out 50 ÷ 250 = 0.2 and writing that decimal down as a percentage without multiplying by 100.
- (b) 75 cm² — Areas of similar shapes are in the ratio of the squares of their lengths. Squaring both parts of 3 : 5 gives an area ratio of 9 : 25, so the larger area is 25/9 of the smaller one. Working with the smaller area: 27 ÷ 9 = 3, and 3 × 25 = 75. The area of the larger rectangle is 75 cm².
- (d) 2:5 — The ratio of the y-values equals the ratio of the coefficients of x, since x cancels: 2x : 5x = 2 : 5.
- (a) 180 g flour and 120 g butter — The ratio 3 : 2 means flour and butter must scale by the same factor. Scaling both parts by 60 gives 180 g flour and 120 g butter, since 3 × 60 = 180 and 2 × 60 = 120, which is the correct pair. Swapping the amounts gives 120 g flour and 180 g butter, which is in the ratio 2 : 3 — butter to flour, not flour to butter. Scaling the flour by 60 but the butter by only 45 gives 180 g flour and 90 g butter, using two different scale factors on the two parts of the ratio. Scaling the flour by 60 but the butter by 70 gives 180 g flour and 140 g butter, the opposite inconsistency.
- (c) 10 g — Method: mass = density × volume. Working: 2 × 5 = 10, in grams because the density is in grams per cubic centimetre. Answer: 10 g. The distractors: 2.5 g comes from dividing the volume by the density, 5 ÷ 2, instead of multiplying; 0.4 g comes from dividing the density by the volume, 2 ÷ 5; 7 g comes from adding the density and the volume.
- (a) 135 minutes — Method: to change hours into minutes, multiply by 60. Working: 2.25 × 60 = 135 minutes. So the film runs for 135 minutes. Distractor 145 minutes comes from reading the '.25' as 25 minutes instead of a quarter of an hour, giving 2 hours 25 minutes. Distractor 225 minutes comes from multiplying by 100 instead of 60. Distractor 150 minutes comes from rounding 2.25 hours to 2.5 hours before converting.
- (b) The ratio C : m is not constant because the formula includes a fixed charge of £3 as well as the charge per mile. — For the ratio C : m to stay constant, C must be directly proportional to m, i.e. C = km with no constant term. Because of the +3 fixed charge, C is not directly proportional to m: for example m = 1 gives C = 5.5 (ratio 5.5 : 1), while m = 10 gives C = 28 (ratio 2.8 : 1) — the ratio has changed.
- (d) 2:3:5 — The highest common factor of 12, 18 and 30 is 6. Divide every part by 6: 12 ÷ 6 = 2, 18 ÷ 6 = 3, 30 ÷ 6 = 5, giving 2 : 3 : 5. Dividing by 2 instead of 6 gives 6 : 9 : 15, which still shares a common factor of 3, so it is not fully simplified. Dividing by 3 instead of 6 gives 4 : 6 : 10, which still shares a common factor of 2, so it is not fully simplified either. Swapping the first two parts gives 3 : 2 : 5, the parts in the wrong order.
- (d) 5 — Method: rearrange y = kx to make the constant the subject, then substitute the pair of values given. Working: k = y ÷ x, so k = 10 ÷ 2 = 5. Answer: 5. The distractors: 20 comes from multiplying 10 by 2 instead of dividing, which is the rearrangement done the wrong way round; 12 comes from adding the pair, 10 + 2, treating the relationship as y = x + k; 8 comes from working out 10 − 2, the same additive reading with the operation reversed.
- (d) 60 — Method: 75% is three quarters, so divide by 4 to find one quarter and then multiply by 3. Working: 80 ÷ 4 = 20 for one quarter, and 3 × 20 = 60. Answer: 60 milk chocolates. The distractors: 20 is one quarter of 80, which is 25% and not 75%; 55 comes from subtracting 25 from 80 rather than taking 25% of 80 away from it; 75 comes from quoting the percentage itself as a number of chocolates.
- (c) L = d/5 — The scale 1 : 20 means each cm on the drawing represents 20 cm in real life, so the real length in cm is 20d. Converting to metres by dividing by 100: L = 20d/100 = d/5.
- (d) 2 : 3 — x is 2/3 of y means for every 3 parts of y, x is 2 parts, so x : y = 2 : 3. 3 : 2 comes from writing the ratio the wrong way round. 2 : 5 comes from comparing x with the total of x and y (2 parts out of 5), instead of with y alone. 3 : 5 comes from comparing y with the total of x and y (3 parts out of 5), instead of with x.
- (b) 2/5 — Method: a fraction taken 'of the total' has the whole batch as its denominator, so add the two masses first and then write the butter over that total. Working: the total mass is 450 g + 300 g = 750 g, so the fraction is 300/750; the highest common factor of 300 and 750 is 150, and 300 ÷ 150 = 2 while 750 ÷ 150 = 5. Answer: 2/5 of the batch. The distractors: 2/3 comes from comparing the butter with the flour, 300/450, a part-to-part fraction when the question asks for a part compared with the whole; 3/5 comes from writing the flour over the total, 450/750, which answers about the wrong ingredient; 3/2 comes from writing the flour over the butter, 450/300, which both uses the wrong denominator and reverses the order.
- (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.
- (c) 5/7 — First find the number of children: 84 − 35 = 49. The question compares the adults with the children, not with everyone on the bus, so the denominator is 49 and the numerator is 35, giving 35/49. Both parts divide by 7: 35 ÷ 7 = 5 and 49 ÷ 7 = 7. In its simplest form the fraction is 5/7.
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