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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Ratio, proportion and rates of change worksheet — GCSE Foundation
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- 1.It takes 8 painters 6 days to paint a fence. Working at the same rate, work out how many days 4 painters would take to paint the same fence.
- 2.The price of a games console is reduced by 10%. In a later sale the reduced price is reduced by 10% again. Work out the overall percentage decrease.
- 3.A sunflower is 140 cm tall. A tomato plant is 80 cm tall. Write the height of the tomato plant as a fraction of the height of the sunflower, giving your answer in its simplest form.
- 4.The ratio of cats to dogs in a shelter is 3:4. There are 18 cats. Work out the total number of animals in the shelter.
- 5.In a car park there are 35 cars and 15 vans. Write the ratio of the number of vans to the number of cars in its simplest form.
- 6.Write the ratio 18:22 in its simplest form.
- 7.The ratio of Josh's savings to Mia's savings is 4:7. Mia has £39 more than Josh. Work out Josh's savings.
- 8.A section of a kitchen wall has an area of 45,000 cm². Work out the area of this section in m².
- 9.In a fruit drink, 3/8 of the total volume is orange juice and the rest is apple juice. Write down the ratio of orange juice to apple juice, in its simplest form.
- 10.The number of biscuits, b, that each guest at a party receives is inversely proportional to the number of guests, g. Which equation could show this relationship?
- 11.Two quantities are connected by the formula y = 0.8x. Write down the ratio y : x in its simplest form.y = 0.8x
- 12.Work out 50% of 60.
- 13.2.4 kg of cheese costs £36. Work out the cost of 1.5 kg of the same cheese.
- 14.The ratio of the number of pages Mia has read to the number of pages Leo has read is 7 : 4. Mia has read 84 pages. Work out how many pages they have read altogether.
- 15.A footpath is drawn 3 cm long on a map with a scale of 1 : 50 000. Work out the real length of the footpath, in metres.
Answer key
- (b) 12 days — This is inverse proportion: fewer painters take longer. Multiply the original numbers to find the total painter-days needed: 8 × 6 = 48 painter-days. Divide by the new number of painters: 48 ÷ 4 = 12 days. Working out 6 × 4 ÷ 8 = 3 days treats it as direct proportion, as if fewer painters needed less time. Stopping at 48 gives the total painter-days, not the number of days. Working out 6 + (8 − 4) = 10 days adds the change in the number of painters straight onto the number of days, treating painters and days as the same kind of quantity. 4 painters take 12 days.
- (d) 19% — Method: write each decrease as a multiplier, multiply the multipliers, then compare the result with 100%. Working: a 10% decrease is a multiplier of 0.9, so the two reductions together give 0.9 × 0.9 = 0.81; the final price is 81% of the original, so the price has fallen by 100% − 81% = 19%. Answer: an overall decrease of 19%. The distractors: 20% comes from adding the two reductions, 10% + 10%, which charges the second 10% against the original price instead of against the already reduced price; 21% comes from using the increase multiplier by mistake, since 1.1 × 1.1 = 1.21, and reading that 21% as a decrease; 81% is the percentage of the original price still being paid, not the percentage taken off.
- (d) 4/7 — Put the tomato plant's height over the sunflower's height: 80/140. Divide both numbers by their highest common factor, 20: 80÷20 = 4, 140÷20 = 7, giving 4/7. (7/4 comes from writing the heights the wrong way round. 3/7 comes from finding the difference in the heights, 140 − 80 = 60 cm, and writing it as a fraction of the sunflower's height, 60/140. 4/11 comes from comparing the tomato plant's height to the total height of both plants, 80/220.)
- (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.)
- (c) 3 : 7 — Method: a ratio 'of A to B' is written with A first, and it is simplified by dividing both parts by their highest common factor. Working: vans are named first, so the ratio starts as 15 : 35; the highest common factor of 15 and 35 is 5, and 15 ÷ 5 = 3 while 35 ÷ 5 = 7. Answer: 3 : 7. The distractors: 7 : 3 comes from writing the two numbers in the order they appear in the question, cars before vans, instead of the order asked for; 3 : 10 comes from comparing the vans with the total number of vehicles, 15 : 50, a part-to-whole ratio in place of the part-to-part ratio asked for; 15 : 35 is the right comparison left unsimplified, and the question asks for the simplest form.
- (a) 9:11 — Method: divide both parts of the ratio by their highest common factor. Working: the factors of 18 are 1, 2, 3, 6, 9 and 18, and the factors of 22 are 1, 2, 11 and 22, so the highest common factor is 2; 18 ÷ 2 = 9 and 22 ÷ 2 = 11. Answer: 9:11. The distractors: 11:9 comes from dividing both parts correctly but writing them the wrong way round, so it describes 22 to 18 rather than 18 to 22; 9:22 comes from dividing only the first part by 2 and leaving the second part untouched; 9:13 comes from subtracting 9 from each part instead of dividing, since 18 − 9 = 9 and 22 − 9 = 13, and subtracting the same amount from both parts does not give an equivalent ratio.
- (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.
- (d) 4.50 m² — 1 m² = 100 cm × 100 cm = 10,000 cm², so to convert cm² to m² you divide by 10,000: 45,000 ÷ 10,000 = 4.50 m². Dividing by 100 instead of 10,000, treating area like a length conversion, gives 45,000 ÷ 100 = 450.00 m², a common mistake since 1 m = 100 cm. Multiplying by 100 instead of dividing gives 45,000 × 100 = 4500000.00 m², using the conversion factor the wrong way round entirely. Dividing by 100,000 instead of 10,000 gives 0.45 m², an extra factor of 10 too far.
- (c) 3 : 5 — If orange juice is 3/8 of the total, apple juice is the remaining 1 − 3/8 = 5/8. The ratio of orange to apple is therefore 3 : 5. Inverting gives 5 : 3, apple to orange instead of orange to apple. Using the denominator 8 as the second part of the ratio, 3 : 8, compares orange juice to the whole drink rather than to the apple juice alone. Pairing the total 8 with the apple fraction's numerator 5 gives 8 : 5, which mixes a whole-total figure with a part figure.
- (d) b = k ÷ g — Inverse proportion means b × g stays constant (equal to k), so b = k ÷ g. 'b = kg' is the equation for direct proportion, not inverse. 'b = k + g' treats the relationship as additive, which is not proportion at all. 'b = g ÷ k' is direct proportion in disguise: rearranging it gives g = kb, so b still grows as g grows. For inverse proportion the g must be underneath the constant, not above it.
- (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.
- (d) 30 — Method: 50% is one half, so 50% of a quantity is the quantity divided by 2. Working: 60 ÷ 2 = 30. Answer: 30. The distractors: 120 comes from multiplying by 2 instead of dividing; 3000 comes from multiplying by 50 without dividing by 100; 6 comes from finding 10% instead of 50%.
- (c) £22.50 — Method: find the cost of 1 kg, then multiply by the mass wanted. Working: £36 ÷ 2.4 = £15 per kilogram, and 1.5 × £15 = £22.50. Answer: £22.50. The distractors: £15 is the price of 1 kg, which is the first step and not what the question asks for; £54 comes from multiplying the £36 by 1.5 without first reducing it to a price per kilogram; £57.60 comes from using the scale factor upside down, multiplying £36 by 2.4 ÷ 1.5 = 1.6 instead of by 1.5 ÷ 2.4.
- (d) 132 — Method: find the value of one part of the ratio, use it to find Leo's pages, then add both amounts together. Working: 84 ÷ 7 = 12 (value of one part). Leo's pages = 12 × 4 = 48. Total = 84 + 48 = 132. Wrong options: 48 gives only Leo's pages and forgets to add Mia's; 147 comes from reversing the ratio parts (84 ÷ 4 × 7 = 147) and stopping there; 231 comes from reversing the ratio parts and then adding Mia's pages (84 + 147).
- (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.
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