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) 3/5 — The quantity being described goes on the top of the fraction and the quantity it is compared with goes on the bottom. Here the potatoes are written as a fraction of the carrots, so the mass of the potatoes is the numerator and the mass of the carrots is the denominator. Both masses are already in kilograms, so no conversion is needed. This gives 3/5, and since 3 and 5 share no common factor it is already in its simplest form.
- (b) 1:3 — Let the number who cycle be 1 part. Since 3 times as many pupils walk, the number who walk is 3 parts. The ratio of cycle to walk is 1:3. (3:1 comes from writing the ratio the wrong way round, walk to cycle. 1:4 comes from treating “3 times as many” as “3 more”, so walk = cycle + 3, giving parts 1 and 4. 3:4 comes from comparing the number who walk with the total number of pupils, 3 parts out of 4 parts altogether, instead of comparing cycle with walk.)
- (d) 1:12 — Write the ratio model : real bus using the numbers in the question: 5 : 60. Divide both parts by their highest common factor, 5: 5÷5 = 1, 60÷5 = 12, giving 1 : 12. (12:1 comes from writing the ratio the wrong way round, real bus : model. 1:55 comes from dividing the first part correctly, 5÷5 = 1, but subtracting on the second part, 60 − 5 = 55, instead of dividing it by 5 as well. 5:12 comes from dividing only the real length by 5 and leaving the model length unchanged — both parts of a ratio must be divided by the same number.)
- (c) 9 g/cm³ — Density = mass ÷ volume, so 270 ÷ 30 = 9 g/cm³. Working out 270 − 30 = 240 subtracts the volume from the mass instead of dividing. Writing 270 gives the mass on its own, without dividing by the volume at all. Working out 270 × 30 = 8100 multiplies mass and volume together instead of dividing. The density of the block is 9 g/cm³.
- (d) 5:3:2 — German = 100% − 50% − 30% = 20%. The ratio 50 : 30 : 20 simplifies by dividing every part by 10 to give 5 : 3 : 2.
- (a) 4 — Method: find the constant of proportionality from the pair given, write the equation, then substitute the new value of y and solve. Working: k = 21 ÷ 7 = 3, so y = 3x; putting y = 12 gives 12 = 3x, and x = 12 ÷ 3 = 4. Answer: 4. The distractors: 36 comes from multiplying by the constant instead of dividing by it, 12 × 3, which is the proportion set up upside down; 84 comes from multiplying 12 by the 7 from the first pair, using a value of x as though it were the constant; 9 comes from working out 12 − 3, treating the equation as y = x + 3 rather than y = 3x.
- (c) £90 — Method: split the total amount into the number of parts shown by the ratio, then find Chloe's share. Working: the ratio 2:3 has 2 + 3 = 5 parts, so one part is £150 ÷ 5 = £30, and Chloe's share is 3 × £30 = £90. So Chloe receives £90. Distractor £60 is Ben's share, not Chloe's. Distractor £75 comes from splitting the money into two equal halves, ignoring the ratio. Distractor £30 is the value of one part, found correctly but never multiplied by 3.
- (b) 9 m/s — The gradient of line P is 21 ÷ 3 = 7, so P has a rate of 7 m/s. The gradient of line Q is 45 ÷ 5 = 9, so Q has a rate of 9 m/s. Because 9 is greater than 7, line Q is the steeper line, with gradient 9 m/s. Taking line P's gradient instead of Q's gives 7 m/s, the less steep line. Subtracting the two lines' coordinates directly, (45 − 21) ÷ (5 − 3) = 24 ÷ 2 = 12 m/s, mixes points from different lines rather than using one line's own two points. Adding the two gradients, 7 + 9 = 16 m/s, treats 'steeper' as a total rather than a comparison.
- (b) −1% — Method: write each change as a multiplier and multiply them. A 10% fall is × 0.9 and a 10% rise is × 1.1. Working: 0.9 × 1.1 = 0.99, so the final price is 99% of the original, which is 1% less. Answer: an overall change of −1%. The distractors: 0% comes from assuming a 10% fall and a 10% rise cancel — they do not, because the rise is 10% of a smaller amount; +1% has the size right but the sign wrong, from reading the multiplier 0.99 as 1% above 1 instead of 1% below it; −2% comes from finding the 1% fall and then counting it once for each of the two changes.
- (a) 2/3 — Put the laptop bag's mass over the school bag's mass: 2.4/3.6. Multiply both numbers by 10 to clear the decimals: 24/36. Divide both by their highest common factor, 12: 24÷12 = 2, 36÷12 = 3, giving 2/3. (3/2 comes from writing the masses the wrong way round. 1/3 comes from finding the difference in the masses, 3.6 − 2.4 = 1.2 kg, and writing it as a fraction of the school bag's mass, 1.2/3.6. 2/5 comes from comparing the laptop bag's mass to the total mass of both bags, 2.4/6.)
- (b) £52,000 — Method: convert 130% to a decimal multiplier and multiply it by last year's profit. Working: 130% = 1.3, so this year's profit is £40,000 × 1.3 = £52,000. Answer: £52,000. £12,000 comes from using only the extra 30% (130% − 100%) and forgetting to include the original 100%, £40,000 × 0.3 = £12,000. £40,130 comes from simply adding 130 onto £40,000, treating the percentage as an amount of money rather than a multiplier. £5,200 comes from misreading 130% as 13%, giving £40,000 × 0.13 = £5,200.
- (d) 1 : 2500 — Method: convert the real-world measurement to the same unit as the drawing (centimetres) before writing the ratio. Working: 25 m = 2500 cm, so the scale is 1 : 2500. Wrong options: 1 : 25 comes from not converting metres to centimetres at all; 1 : 250 comes from converting metres to centimetres using ×10 instead of ×100; 1 : 2.5 comes from converting in the wrong direction (treating 25 m as 2.5 cm).
- (d) The line is straight and passes through the origin. — Direct proportion means y = kx for a constant k. This is a straight line, and when x = 0, y = 0, so it passes through the origin. For a positive k it slopes upward from left to right.
- (a) 20% — Method: percentage increase = increase ÷ original amount × 100. Working: the increase is 84 − 70 = 14 marks, and 14 ÷ 70 = 0.2, so 0.2 × 100 = 20. Answer: an increase of 20%. The distractors: 14% comes from quoting the 14 mark increase as though marks and per cent were the same thing; 17% comes from dividing the 14 by the new mean 84 instead of by the original 70, which gives 17% to the nearest per cent; 120% is the new mean written as a percentage of the old one, which is the whole of the new mean rather than the increase.
- (b) 8:12 — Method: two ratios are equivalent when one is obtained from the other by multiplying, or dividing, both parts by the same number. Working: multiplying both parts of 2:3 by 4 gives 2 × 4 = 8 and 3 × 4 = 12, and the check runs the other way too, since the highest common factor of 8 and 12 is 4 and dividing both parts by 4 returns 2:3. Answer: 8:12. The distractors: 8:3 comes from multiplying only the first part by 4 and leaving the second part alone; 12:8 comes from multiplying both parts by 4 correctly but then writing the two parts the wrong way round; 4:5 comes from adding 2 to each part instead of multiplying, and adding the same amount to both parts changes the ratio.
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