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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- (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.)
- (a) 45% — Method: percentage increase = increase ÷ original amount × 100. Working: the increase is £116 − £80 = £36, and 36 ÷ 80 = 0.45, so 0.45 × 100 = 45%. Answer: 45%. The distractors: 36% comes from quoting the £36 increase as though pounds and per cent were the same thing; 31% comes from dividing the £36 increase by the new price £116 instead of by the original £80, which gives 31% to the nearest per cent; 145% is the new price written as a percentage of the original price, which is the whole of the new price rather than the increase.
- (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) 12 cm — Method: height:width = 3:2 means both lengths are built from parts of the same size, so write 3:2 = 18:x, find one part and multiply by the number of parts in the width. Working: the height is 3 parts and measures 18 cm, so one part is 18 ÷ 3 = 6 cm; the width is 2 parts, so 2 × 6 = 12 cm. Answer: 12 cm. The distractors: 27 cm comes from using the ratio the wrong way round, 18 ÷ 2 × 3, which makes the width longer than the height even though 2 is the smaller part; 6 cm is the value of one part, given as the width instead of being doubled; 17 cm comes from treating the ratio as a difference — 3 − 2 = 1, so 1 cm is taken off the height — but a ratio scales the lengths, it does not subtract from them.
- (d) 75 pages — Method: find the number of pages printed in one minute, then scale up to 10 minutes. Working: 45 ÷ 6 = 7.5 pages per minute, so 7.5 × 10 = 75 pages. Answer: 75 pages. 27 pages comes from using the ratio upside down, 45 × 6 ÷ 10, instead of finding the rate per minute first. 55 pages comes from simply adding the extra minutes, 10, onto the original number of pages, 45. 70 pages comes from rounding the rate down to 7 pages per minute before multiplying by 10, instead of using the exact rate of 7.5.
- (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.
- (d) £82.50 — Find the hourly rate: £52.50 ÷ 7 = £7.50 per hour. For 11 hours: 11 × £7.50 = £82.50. £30 comes from working out the pay for only the extra 4 hours (4 × £7.50), and forgetting to include the original £52.50. £99 comes from misremembering the hourly rate as £9 instead of £7.50, then 11 × £9. £56.50 comes from adding the extra number of hours (4) straight onto the pay in pounds (52.5 + 4), confusing hours with pounds.
- (c) 5 : 9 — Write the ratio mass : cost = 3 : 5.4. Multiply both parts by 10 to clear the decimal: 30 : 54. Both numbers share a factor of 6, so 30 ÷ 6 = 5 and 54 ÷ 6 = 9, giving 5 : 9. 3 : 5 comes from ignoring the decimal point and treating £5.40 as £5. 9 : 5 comes from writing the ratio the wrong way round, cost to mass instead of mass to cost. 1 : 18 comes from multiplying only the cost by 10 instead of both parts, giving 3 : 54, and then cancelling that correctly to 1 : 18 — the cancelling is fine, but the ratio being cancelled is not the right one.
- (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.
- (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) 75% — Method: change a fraction to a percentage by dividing the numerator by the denominator and multiplying by 100. Working: 3 ÷ 4 = 0.75, and 0.75 × 100 = 75. Answer: 75%. The distractors: 34% comes from reading the digits 3 and 4 straight off as a percentage; 0.75% comes from dividing but forgetting to multiply by 100; 133% comes from inverting the fraction and working out 4 ÷ 3 instead.
- (d) 1/4 — Method: work out both amounts first, then put the amount named before the word 'of' on top and the amount named after it on the bottom. Working: there are 4 × 5 = 20 apples, so the fraction is 5/20; dividing the numerator and the denominator by 5 gives 1/4. Answer: 1/4. The distractors: 4/1 comes from quoting the multiplier given in the question, which compares the apples with the pears rather than the pears with the apples; 4/5 comes from writing the 20 apples over the 25 fruits altogether, 20/25, a part-to-whole fraction; 5/9 comes from reading '4 times as many' as '4 more than', giving 5 + 4 = 9 apples and the fraction 5/9.
- (b) 20 — The gradient is the change in p divided by the change in t: (90 − 30) ÷ (5 − 2) = 60 ÷ 3 = 20. Choosing 0.05 comes from dividing the change in t by the change in p, the wrong way round (3 ÷ 60). Choosing −20 comes from subtracting the coordinates in the wrong order for one part of the calculation, for example (30 − 90) ÷ (5 − 2), giving a negative value. Choosing 18 comes from dividing the second p-coordinate by the second t-coordinate directly (90 ÷ 5) instead of using the change between the two points.
- (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) 4 — Method: find the length scale factor by taking the cube root of the volume scale factor, then square it to get the area scale factor. Working: 8 = 2³, so the length scale factor is 2, and the area scale factor is 2² = 4. Answer: 4. 8 comes from using the volume scale factor itself as if it were the area scale factor. 64 comes from squaring the volume scale factor, 8² = 64, instead of first taking its cube root. 2 comes from correctly finding the length scale factor but then forgetting to square it.
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