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) 4.5 litres — Method: a litre is larger than a cm³, so changing cm³ into litres means dividing by the conversion factor 1000. Working: 4500 ÷ 1000 = 4.5. Answer: 4.5 litres. The distractors: 45 litres comes from dividing by 100; 450 litres comes from dividing by 10; 0.45 litres comes from dividing by 10 000.
- (c) 2 hours — Method: time = distance ÷ speed, and the units of the speed fix the unit of the time. Working: 120 ÷ 60 = 2, and because the speed is in kilometres per hour the time is in hours. Answer: 2 hours. The distractors: 0.5 hours comes from dividing the speed by the distance, 60 ÷ 120, the division the wrong way round; 2 minutes comes from the correct division but the wrong unit, reading kilometres per hour as kilometres per minute; 180 minutes comes from adding 120 and 60 instead of dividing.
- (a) 36 — Pressure = force ÷ area, so 126 ÷ 3.5 = 36 N/m². (0.03 comes from dividing the area by the force instead of the force by the area, the wrong way round. 129.5 comes from adding 126 and 3.5 instead of dividing. 441 comes from multiplying 126 by 3.5 instead of dividing.)
- (a) 20% — Method: a percentage is a number of parts per hundred, so divide the numerator by the denominator and multiply by 100. Working: 1 ÷ 5 = 0.2, and 0.2 × 100 = 20; the same result comes from scaling 1/5 to 20/100. Answer: 20%. The distractors: 5% comes from reading the denominator 5 straight off as the percentage; 25% comes from confusing 1/5 with 1/4, the fraction met most often in percentage work; 50% comes from a place value slip in the division, writing 1 ÷ 5 as 0.5 instead of 0.2.
- (b) 460 — To increase by 15%, multiply by 1.15 (100% + 15%). 400 × 1.15 = 460. 60 comes from working out only the increase, 400 × 0.15 = 60, and forgetting to add it to the original number. 415 comes from adding 15 directly to 400 instead of 15% of 400. 340 comes from multiplying by 0.85, decreasing instead of increasing: 400 × 0.85 = 340.
- (c) £672 — Simple interest per year = 3% of £600 = £18. Over 4 years the interest is 18 × 4 = £72. Total in the account = £600 + £72 = £672. A student who gives just the interest, without adding it to the principal, writes £72. A student who adds only one year's interest instead of four gets £600 + £18 = £618. A student who wrongly compounds the interest each year gets 600 × 1.03⁴ = £675.31.
- (a) 28.8 km/h — Method: first change metres per second into metres per hour, then change metres into kilometres. Working: 8 × 3600 = 28800 metres per hour, then 28800 ÷ 1000 = 28.8 km/h. So the runner's speed is 28.8 km/h. Distractor 28800 km/h comes from stopping after the first step and forgetting to change metres into kilometres. Distractor 2.22 km/h comes from dividing by 3600 instead of multiplying, then multiplying by 1000. Distractor 2.88 km/h comes from using 360 instead of 3600 seconds in an hour, missing a zero.
- (b) £52 — Method: the difference between the two ratio numbers tells you how many parts the £39 difference represents. Working: the difference in parts is 7 − 4 = 3, and this represents £39, so one part is £39 ÷ 3 = £13. Josh's savings are 4 × £13 = £52. So Josh has £52. Distractor £91 is Mia's savings, not Josh's. Distractor £39 comes from using the given £39 difference as the final answer, without scaling it to Josh's number of parts. Distractor £13 is the value of one part, found correctly but never multiplied by 4.
- (a) 18/25 — First find the new number of rose bushes: 90 × 1.2 = 108 (a 20% increase multiplies by 1.2). Then write 108 over 150 and divide top and bottom by 6 to get 18/25. Choosing 3/5 comes from using the original 90 rose bushes without applying the 20% increase (90/150 = 3/5). Choosing 25/18 comes from writing the number of lavender bushes over the new number of rose bushes, the wrong way round. Choosing 3/25 comes from multiplying 90 by 0.2 instead of 1.2, finding only the increase (18) rather than the new total, then writing 18/150 = 3/25.
- (d) £144 — Method: find the length (perimeter) scale factor by taking the square root of the area ratio, then apply it to the cost. Working: 12 : 27 simplifies to 4 : 9, and the square root of each part gives the length ratio 2 : 3, so the scale factor from the smaller to the larger pond is 3 ÷ 2 = 1.5. Cost = £96 × 1.5 = £144. Answer: £144. £216 comes from using the area ratio itself as the cost ratio, £96 × (27 ÷ 12) = £216, without taking the square root. £64 comes from using the length ratio the wrong way round, £96 × (2 ÷ 3) = £64. £111 comes from simply adding the difference in area, 27 − 12 = 15, onto the original cost, £96 + £15 = £111, instead of scaling proportionally.
- (c) 4 : 5 — Method: a formula of the form y = kx says that for every 1 of x there are k of y, so the ratio y : x is k : 1, which is then written with whole numbers and cancelled. Working: here k = 0.8, so y : x = 0.8 : 1; multiplying both parts by 10 gives 8 : 10, and dividing both parts by their highest common factor 2 gives 4 : 5. Answer: 4 : 5. The distractors: 5 : 4 comes from writing x : y, reversing the order asked for; 8 : 10 has the decimal cleared but the ratio left unsimplified, and the question asks for the simplest form; 8 : 1 comes from multiplying only the left-hand part by 10 and leaving the 1 unchanged, which changes the relationship instead of keeping it.
- (b) 1500 m — Method: multiply the map length by the scale to get the real length in centimetres, then divide by 100 to change centimetres into metres. Working: 3 × 50 000 = 150 000 cm, and 150 000 ÷ 100 = 1500. Answer: 1500 m. The distractors: 15000 m comes from dividing the 150 000 cm by 10 instead of 100; 150 m comes from dividing by 1000, the conversion for kilometres; 50000 m comes from writing the scale itself as the answer and ignoring the 3 cm measured on the map.
- (d) 3.75 — 45 minutes is 45 ÷ 60 = 0.75 of an hour, so 3 hours 45 minutes = 3.75 hours. Getting 3.45 comes from writing the minutes directly after the decimal point instead of converting them to a fraction of an hour. Getting 3.67 comes from misreading 45 minutes as 40 minutes and converting 40 ÷ 60 instead. Getting 4.15 comes from rounding 45 minutes up to the next whole hour and adding the remainder as if it were more minutes past that hour.
- (d) 2:3 — The highest common factor of 20 and 30 is 10. Divide both parts of the ratio by 10: 20 ÷ 10 = 2 and 30 ÷ 10 = 3, so 20 : 30 = 2 : 3. Dividing by 5 instead of the highest common factor gives 4 : 6, which still shares a common factor of 2, so it is not fully simplified. Dividing only the first part by 10 and leaving the second part unchanged gives 2 : 30, which is not equivalent to the original ratio. Swapping the order of the two parts gives 3 : 2, the ratio the wrong way round.
- (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.)
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