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.
Ratio, proportion and rates of change worksheet — GCSE Foundation
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- 1.A force of 126 N acts on an area of 3.5 m². Work out the pressure on the area, in N/m².
- 2.Write the ratio 18:22 in its simplest form.
- 3.Write 350 ml : 1.4 l as a ratio in its simplest form.
- 4.Write the ratio 20 : 30 in its simplest form.
- 5.A shop buys a bike for £150 and sells it, making a profit of £45. Write the selling price as a fraction of the cost price, giving your answer in its simplest form.
- 6.Two numbers are in the ratio 5 : 8. The larger number is 40. Work out the smaller number.
- 7.A scale drawing of a car park has a scale of 1 : 300. On the drawing, the car park has an area of 15 cm². Work out the real area of the car park, in square metres.
- 8.A water butt is being filled from a hosepipe at a constant rate while a small leak drains water out at a constant rate, giving a constant net rate of change. The volume of water in the butt, V litres, is shown on a straight-line graph against time, t minutes. The line passes through the points (5, 20) and (15, 60). The butt is empty at t = 0. Work out how many minutes it takes to reach a volume of 100 litres.
- 9.A solution is made by dissolving 50 g of sugar in 200 g of water. Work out the concentration of sugar in the solution as a percentage.
- 10.Two mathematically similar oil drums have a length scale factor of 3 from the smaller drum to the larger drum. Work out the volume scale factor from the smaller drum to the larger drum.
- 11.A dog walker's charge, C pounds, for walking a dog for m minutes is shown on a straight-line graph. The line passes through the points (20, 14) and (50, 26). Work out the charge for a 65-minute walk.
- 12.Two mathematically similar polygons have perimeters in the ratio 2 : 5. Write the ratio of their areas in its simplest form.
- 13.A recipe needs 1.2 kg of flour. Dan has 750 g of flour in his cupboard. Write the mass of flour the recipe needs as a fraction of the mass of flour Dan has. Give your answer in its simplest form.
- 14.Triangle ABC is mathematically similar to triangle PQR, with AB corresponding to PQ and BC corresponding to QR. AB = 6 cm, BC = 8 cm and PQ = 12 cm. Work out the length of QR.
- 15.A cookery class lasts 45 minutes. A double lesson at school lasts 2 hours. Write the length of the cookery class as a fraction of the length of the double lesson. Give your answer in its simplest form.
Answer key
- (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) 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.
- (d) 1:4 — Convert 1.4 l to millilitres: 1.4 l = 1400 ml. The ratio is 350 : 1400. Divide both parts by 350: 350 ÷ 350 = 1 and 1400 ÷ 350 = 4, giving 1 : 4. Misreading 1.4 l as 14 (moving the decimal point) gives 350 : 14, which simplifies to 25 : 1 — a very different, implausible ratio. Dividing by 175 instead of 350 gives 2 : 8, which still shares a common factor of 2, so it is not fully simplified. Swapping the order gives 4 : 1, litres to millilitres the wrong way round.
- (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.
- (d) 13/10 — First find the selling price: £150 + £45 = £195. Put the selling price over the cost price: 195/150. Divide both numbers by their highest common factor, 15: 195÷15 = 13, 150÷15 = 10, giving 13/10. (10/13 comes from writing the prices the wrong way round. 3/10 is just the profit written as a fraction of the cost price, 45/150, not the selling price. 13/23 comes from comparing the selling price to the combined total of the cost price and the selling price, 195/345.)
- (c) 25 — Method: divide the larger number by its ratio part to find the value of one part, then multiply by the smaller number's ratio part. Working: 40 ÷ 8 = 5 (value of one part). Smaller number = 5 × 5 = 25. Wrong options: 64 comes from dividing by the smaller ratio part instead of the larger (40 ÷ 5 × 8); 45 comes from adding the value of one part onto 40 instead of scaling down (40 + 5); 35 comes from subtracting the value of one part from 40 (40 − 5) instead of multiplying it by the smaller ratio part.
- (d) 135 m² — Method: for area, the scale factor must be squared. Working: area scale = 300² = 90 000. 15 × 90 000 = 1,350,000 cm². Convert to m² by dividing by 10 000: 1,350,000 ÷ 10 000 = 135 m². Wrong options: 0.45 m² comes from using the linear scale factor (×300) instead of squaring it; 1,350,000 m² comes from forgetting to convert the answer from cm² to m²; 13,500 m² comes from dividing by 100 instead of 10 000 when converting units.
- (d) 25 — Gradient = (60 − 20) ÷ (15 − 5) = 40 ÷ 10 = 4 litres per minute. Since the butt is empty at t = 0, V = 4t. Setting V = 100 gives t = 100 ÷ 4 = 25 minutes.
- (a) 20% — Method: a percentage concentration compares the sugar with the whole solution, so add the two masses to get the mass of solution and then scale the ratio of sugar to solution to a denominator of 100. Working: the solution has a mass of 200 + 50 = 250 g; sugar:solution = 50:250, and scaling to per 100 gives 50 ÷ 250 × 100 = 20, so the ratio is 20:100. Answer: 20%. The distractors: 25% comes from comparing the sugar with the 200 g of water, 50:200, rather than with the whole solution; 80% is the percentage of the solution that is water, 200:250, which answers for the wrong part of the mixture; 0.2% comes from working out 50 ÷ 250 = 0.2 and writing that decimal down as a percentage without multiplying by 100.
- (d) 27 — Method: for similar shapes, the volume scale factor is the length scale factor cubed. Working: 3³ = 27. Answer: 27. 9 comes from squaring the length scale factor, which gives the area scale factor, not the volume scale factor. 3 comes from using the length scale factor itself as if it were the volume scale factor. 6 comes from doubling the length scale factor instead of cubing it.
- (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).
- (b) 4 : 25 — For similar shapes, the ratio of areas is the ratio of lengths squared: 2² : 5² = 4 : 25. 2 : 5 comes from using the perimeter ratio itself as the area ratio, without squaring it at all. 8 : 125 comes from cubing each part instead of squaring (2³ : 5³) — cubing is the rule for volume, not area. 4 : 5 comes from squaring only the first part of the ratio (2² = 4), and leaving the second part unsquared.
- (d) 8/5 — Two masses can only be compared once they are in the same unit. Since 1 kg is 1000 g, the recipe needs 1200 g. The recipe's mass is being written as a fraction of Dan's mass, so 1200 goes on the top and 750 on the bottom, giving 1200/750. The highest common factor of the two is 150: 1200 ÷ 150 = 8 and 750 ÷ 150 = 5. The fraction is 8/5, which is greater than 1 because the recipe needs more flour than Dan has.
- (c) 16 cm — Corresponding sides of similar triangles are all in the same ratio. Use the pair whose lengths are both known: the scale factor from triangle ABC to triangle PQR is 12 ÷ 6 = 2. Since QR corresponds to BC, multiply BC by that scale factor: 8 × 2 = 16, so QR = 16 cm.
- (a) 3/8 — Convert 2 hours to minutes: 2 hours = 120 minutes. Form the fraction 45/120. Both numbers share a factor of 15, so 45 ÷ 15 = 3 and 120 ÷ 15 = 8, giving 3/8. 8/3 comes from writing the fraction the wrong way round, as 120/45. 9/40 comes from converting 2 hours using ×100 instead of ×60, treating it as 200 minutes, then simplifying 45/200. 45/2 comes from not converting the hours to minutes at all, and writing 45 over 2.
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