Printable · GCSE Higher · ages 14-16
Ratio, proportion and rates of change worksheet — GCSE Higher
Fifteen questions across the ratio, proportion and rates of change statements at Higher tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
Answer key: Ratio, proportion and rates of change worksheet — GCSE Higher
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- (c) 2.5 — x : y = 2 : 5 means that for every matching pair of values, y ÷ x = 5 ÷ 2 = 2.5. So y = 2.5x, and comparing with y = kx gives k = 2.5. Dividing the other way round, 2 ÷ 5 = 0.4, gives x in terms of y — that is the constant for x = 0.4y, not for y = kx. Taking the y-part of the ratio on its own, 5, reads one number off the ratio instead of dividing the y-part by the x-part; 5 would only be right if the x-part were 1. Subtracting the two parts, 5 − 2 = 3, treats the ratio as a difference, but a ratio compares two quantities by multiplication, not by subtraction. The constant is k = 2.5.
- (a) 6 — Method: for inverse proportion the product xy is the same for every pair, so find that product and use it to work back to the missing value. Working: xy = 2 × 15 = 30, so when x = 5 the equation 5y = 30 gives y = 30 ÷ 5 = 6. Answer: 6. The distractors: 37.5 comes from treating the pair as direct proportion and scaling y up with x, 15 × 5 ÷ 2, although in inverse proportion y falls as x rises; 30 is the constant product itself, given as a value of y rather than used to find one; 12 comes from additive thinking — x rises by 3, so 3 is taken off y — which would make the two quantities differ by a constant instead of multiplying to one.
- (a) 1:8 — Convert 2 kg to grams: 2 kg = 2000 g. The ratio is 250 : 2000. The highest common factor of 250 and 2000 is 250. Divide both parts by 250: 250 ÷ 250 = 1 and 2000 ÷ 250 = 8, so the ratio is 1 : 8. Leaving the kilograms unconverted gives 250 : 2, which simplifies to 125 : 1 — the units on each side are different, so this does not compare like with like. Dividing by 50 instead of 250 gives 5 : 40, which still shares a common factor of 5, so it is not fully simplified. Swapping the order gives 8 : 1, grams to kilograms the wrong way round.
- (b) £60 — £48 represents 100% − 20% = 80% of the original price. 1% = £48 ÷ 80 = £0.60, so 100% = £0.60 × 100 = £60.
- (c) m = 6c — Method: the whole is the sum of the parts in the ratio, and the cement is 1 part, so one part weighs c kg. Working: the mix has 5 + 1 = 6 parts, each of mass c kg, so the total mass is 6 × c, giving m = 6c. Answer: m = 6c. The distractors: m = 5c uses the 5 gravel parts as the multiplier and forgets that the cement is in the mix too, so it gives the mass of the gravel and not the total; m = c + 5 comes from reading the ratio as '5 more than' and adding, which treats a number of parts as a mass in kilograms; m = c/6 turns the relationship upside down, as though the total were shared into the cement rather than the cement multiplied up to the total.
- (c) Average rate, t = 2 to 6, is −2°C/min — First find the temperature at each end of the interval. At t = 2, T = 80 − 6 × 2 + 0.5 × 2² = 80 − 12 + 2 = 70. At t = 6, T = 80 − 6 × 6 + 0.5 × 6² = 80 − 36 + 18 = 62. The average rate of change over the interval is the change in T divided by the change in t: 62 − 70 = −8, then −8 ÷ 4 = −2°C per minute, so the statement about the average rate is correct. The instantaneous rate at t = 6 is not −2: completing the square gives T = 0.5(t − 6)² + 62, so t = 6 is the turning point of the curve, where the tangent is horizontal and the rate is 0°C per minute — the reaction has stopped cooling by then. The instantaneous rate at t = 2 is not −2 either: a short chord centred on t = 2, from t = 1.9 (T = 70.405) to t = 2.1 (T = 69.605), gives −0.8 ÷ 0.2 = −4°C per minute, so the reaction is cooling twice as fast at the start of the interval as the average over it. Saying the temperature falls 2°C in total confuses the RATE, −2°C per minute, with a TOTAL drop, which is 70 − 62 = 8°C over the four minutes. Always check whether a figure is a rate, per minute, or a total change.
- (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) 2748 — A 7% increase each year means the value becomes 100% + 7% = 107% of the previous year's value, and 107% = 1.07, so the multiplier is 1.07. Multiply by 1.07 for each of the 2 years: 2400 × 1.07 × 1.07 = 2747.76, which rounds to 2748. (2736 comes from using simple growth instead of compound: 2400 + 2 × (2400 × 0.07) = 2736. 2568 is the population after only 1 year, 2400 × 1.07, forgetting the second year's growth. 2747 comes from rounding 2747.76 down instead of up to the nearest whole number.)
- (d) 3 : 5 — Simplify the area ratio: 18 : 50 divides by 2 to give 9 : 25. Areas scale with the square of the length ratio, so take the square root of each part: the square root of 9 is 3, and the square root of 25 is 5, giving a side length ratio of 3 : 5. Giving 5 : 3 has the ratio the right way round for larger to smaller, not smaller to larger. Giving 9 : 25 is the simplified area ratio, without square-rooting it. Giving 18 : 50 is the area ratio before it has even been simplified.
- (a) The gradient is 3; the candidate's method is right. — The gradient of a tangent, like any straight line, is the change in y divided by the change in x between two points on it. Here the tangent passes through (1, 2) and (5, 14), so the change in y is 14 − 2 = 12 and the change in x is 5 − 1 = 4. The gradient is 12 ÷ 4 = 3, so the candidate's calculation is correct. Subtracting in the wrong order, (2 − 14) ÷ (5 − 1), gives −12 ÷ 4 = −3, the wrong sign. Adding the two changes instead of dividing them, 12 + 4 = 16, does not find a gradient at all. Dividing the change in x by the change in y instead of the other way round, 4 ÷ 12 = 1/3, inverts the calculation completely. Before accepting or rejecting a claimed gradient, always redo the calculation yourself in the same order — change in y over change in x — rather than trusting the arithmetic as given.
- (c) 135 g — Find the ratio of butter to sugar in the first batch: 240:160, which simplifies to 3:2. For the second batch, sugar = 90 g, so butter = 90 × 3/2 = 135 g. (60 g comes from using the ratio the wrong way round, 90 × 2/3. 170 g comes from subtracting the drop in sugar, 160 − 90 = 70 g, from the original butter amount, 240 − 70, instead of scaling. 240 g comes from not scaling the butter amount at all.)
- (c) £136 — 5% interest each year means the value becomes 100% + 5% = 105% of the previous year's value, and 105% = 1.05, so the multiplier is 1.05. Account A: £3200 × 1.05 × 1.05 = £3528. Account B (simple interest): £3200 + 2 × (£3200 × 0.03) = £3392. The difference is £3528 − £3392 = £136. (£128 comes from working out Account A with simple interest too, instead of compound: £3200 + 2 × (£3200 × 0.05) = £3520, then £3520 − £3392 = £128. £3528 is the value of Account A on its own, not the difference between the two accounts. £3392 is the value of Account B on its own, not the difference.)
- (a) S_{n+1} = 1.04S_n − 30 — Adding 4% interest multiplies the balance by 1 + 0.04 = 1.04. Withdrawing £30 afterwards subtracts a fixed 30, giving S_{n+1} = 1.04S_n − 30. Writing +30 instead of −30 mistakes a withdrawal for a deposit — the £30 leaves the account, so it must be subtracted. Writing 0.96 instead of 1.04 treats the 4% as a decrease rather than an increase, as if the interest were shrinking the balance instead of growing it. Writing 1.4 instead of 1.04 turns 4% into 40%, a common slip when converting a percentage to a multiplier — 4% as a decimal is 0.04, so the multiplier is 1.04, not 1.4. Always convert the percentage to a decimal first, then add 1 for growth or subtract from 1 for decay, before applying any fixed amount that is added or removed.
- (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) 3.375 — Method: the volume scale factor between similar shapes is the length scale factor cubed, not the length scale factor itself. Working: the length scale factor is 12 ÷ 8 = 1.5, and 1.5³ = 3.375. Answer: 3.375. Jayden's answer, 1.5, is only the LENGTH scale factor — he never cubed it to get the volume scale factor. 2.25 comes from squaring the length scale factor instead of cubing it, which would give the area scale factor. 4.5 comes from multiplying the length scale factor by 3 (the number of dimensions) instead of cubing it.
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