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.
Answer key: Ratio, proportion and rates of change worksheet — GCSE Foundation
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- (c) 1 : 16 — Work out each area before comparing. Square P has area 3 × 3 = 9 cm². Square Q has area 12 × 12 = 144 cm². The ratio of the areas is 9 : 144, and both parts divide by 9: 9 ÷ 9 = 1 and 144 ÷ 9 = 16, giving 1 : 16. Notice that the sides are in the ratio 1 : 4, and the areas are in the ratio of the squares of those parts, which is what always happens when a length is scaled. Stopping at 1 : 4 would compare the sides and never the areas, and cubing the parts to reach 1 : 64 is the rule for volumes rather than for areas. The order matters too: the question names square P first, so its area must be the first part of the ratio.
- (b) 12 — Method: work out what one part of the ratio is worth, then multiply that by the number of parts in the other share. Working: the smaller number matches the 2 parts, so 8 ÷ 2 = 4 for one part, and 3 × 4 = 12. Answer: 12. The distractors: 16 comes from multiplying 8 by 2, the ratio part that belongs to the smaller number; 24 comes from multiplying 8 by 3 without first finding the value of one part; 9 comes from adding the difference between the ratio parts, 3 − 2 = 1, to 8.
- (c) y is trebled — In direct proportion, y = kx, so multiplying x by 3 multiplies y by 3 as well: y is trebled. Dividing y by 3 is what would happen for inverse proportion, not direct. Saying y stays the same ignores the proportional relationship entirely. Saying y increases by 3 mistakes multiplying by 3 for adding 3.
- (b) 7.2 m — Multiply the model wingspan by the scale factor: 15 × 48 = 720. This is in centimetres, and 720 cm = 7.2 m, since 1 m = 100 cm. Giving 0.31 m divides by the scale factor instead of multiplying (15 ÷ 48 ≈ 0.31), scaling the model down rather than the real aircraft up. Giving 72 m converts centimetres to metres by dividing by 10 instead of 100. Giving 0.72 m converts by dividing by 1000 instead of 100.
- (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.
- (d) 7/5 — The number of unbroken crayons is 60 − 25 = 35. The comparison is with the broken crayons, so the broken crayons are the denominator: 35/25. Both parts divide by 5: 35 ÷ 5 = 7 and 25 ÷ 5 = 5. The fraction is 7/5, which is greater than 1 because there are more unbroken crayons than broken ones.
- (d) 36 mph — First convert 1 hour 30 minutes to hours: 30 minutes is half an hour, so the time is 1.5 hours. Then divide the distance by the time: 54 ÷ 1.5 = 36 mph. Reading 1 hour 30 minutes as 1.3 hours (writing the minutes after the decimal point instead of as a fraction of 60) gives 54 ÷ 1.3 ≈ 41.54 mph. Working out 54 ÷ 30 = 1.8 divides by the number of minutes only, ignoring the hour. Working out 54 × 1.5 = 81 multiplies by the time instead of dividing. The coach's average speed is 36 mph.
- (c) £32 — First find the gradient: (26 − 14) ÷ (50 − 20) = 12 ÷ 30 = £0.40 per minute. Using the point (20, 14), the charge for 65 minutes is 14 + 0.40 × (65 − 20) = 14 + 18 = £32. Choosing £26 comes from treating the charge as directly proportional to the time, multiplying the gradient by 65 minutes and ignoring the fixed part of the charge (0.40 × 65 = 26). Choosing £40 comes from treating £14 as if it were the charge at 0 minutes, then adding the gradient multiplied by the full 65 minutes (14 + 0.40 × 65 = 40), instead of multiplying by the extra time past 20 minutes. Choosing £33.80 comes from assuming the charge is directly proportional to the minutes already known, scaling up from the point (50, 26) in the ratio 65:50 (65 ÷ 50 × 26 = 33.80).
- (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) 2 : 5 — A ratio is written in the order the question names the two shapes, so triangle A's length comes first: 4 : 10. Both parts divide by 2: 4 ÷ 2 = 2 and 10 ÷ 2 = 5, giving 2 : 5. Lengths are compared using the lengths themselves, so nothing is squared here; squaring both parts would give the ratio of the areas instead.
- (c) 448.00 US dollars — Method: multiply the amount in pounds by the exchange rate. Working: £350 × 1.28 = 448.00 US dollars. Wrong options: 273.44 US dollars comes from dividing by the rate instead of multiplying (350 ÷ 1.28); 351.28 US dollars comes from adding the rate to the amount instead of multiplying; 4,480.00 US dollars comes from a decimal-point slip, using 12.8 instead of 1.28.
- (d) £117.60 — Add the hours worked over the two days: 6 + 4.5 = 10.5 hours. Multiply by the rate of pay: 10.5 × £11.20 = £117.60. (£67.20 is Monday's pay only. £50.40 is Tuesday's pay only. £106.40 comes from mistakenly adding the hours as 6 + 3.5 = 9.5 — misreading Tuesday's 4.5 hours as 3.5 — and then multiplying by £11.20.)
- (a) £1,000 — Method: find the increase, then add it to the original price; the multiplier 1.25 does both steps at once. Working: 25% is one quarter, so 25% of £800 = £800 ÷ 4 = £200, and £800 + £200 = £1,000. Answer: £1,000. The distractors: £200 is the increase on its own, not the new price; £825 comes from adding £25 to £800, treating the 25% as £25; £600 comes from taking the 25% off the price instead of adding it on.
- (c) 2/3 — Method: 'as a fraction of' puts the quantity named after the word 'of' on the bottom, so the red counters go on top and the blue counters go on the bottom. Working: the fraction is 12/18; the highest common factor of 12 and 18 is 6, and 12 ÷ 6 = 2 while 18 ÷ 6 = 3. Answer: 2/3. The distractors: 2/5 comes from comparing the red counters with the total number of counters, 12/30, a part-to-whole fraction when the question asks for one part compared with the other part; 3/2 comes from writing the blue counters over the red counters, 18/12, reversing the order the question gives; 1/2 comes from simplifying 12/18 by subtracting the common factor instead of dividing by it, 12 − 6 = 6 and 18 − 6 = 12, giving 6/12, which then cancels to 1/2.
- (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.
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