Printable · GCSE Higher · ages 14-16
Ratio, proportion and rates of change worksheet — GCSE Higher
Fifteen questions across the ratio, proportion and rates of change statements at Higher 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 Higher
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- 1.Two quantities are connected by the formula y = 0.8x. Write down the ratio y : x in its simplest form.y = 0.8x
- 2.A rope of length A is 1.5 times as long as a rope of length B. Write the ratio A : B in its simplest form.
- 3.A loan of £3000 has interest added at 2% each month, and then a fixed repayment of £200 is made. This is modelled by the recurrence B_{n+1} = 1.02B_n − 200, where B_n is the balance after n months and B_0 = 3000. Work out the balance after 2 months.
- 4.y is inversely proportional to x², and x is positive. When x = 2, y = 45. Construct the equation connecting x and y, then work out the value of x when y = 5.
- 5.y is always the same multiple of x. When x = 6, y = 15. Work out the value of y when x = 10.
- 6.An architect builds a scale model of a staircase before construction. The model is mathematically similar to the real staircase, at a scale of 1 : 10. The model covers a floor area of 0.4 m² on the plan. Work out the floor area the real staircase will cover, giving your answer in m².
- 7.A rectangular room is drawn on a plan with a scale of 1 : 50. On the plan, the room measures 8 cm by 6 cm. Work out the real area of the room, in square metres.
- 8.A greenhouse's temperature, in °C, is recorded every hour during the day: 09:00, 15.0; 10:00, 18.4; 11:00, 20.1; 12:00, 19.8; 13:00, 17.2. Work out between which two consecutive readings the instantaneous rate of change of the temperature is most likely to have been zero.
- 9.Tap A fills a swimming pool in 6 hours. Tap B pours water twice as fast as tap A. The time taken to fill the pool is inversely proportional to the rate of flow. Work out how long tap B takes to fill the pool.
- 10.A car uses fuel at a constant rate. It uses 24 litres of fuel to travel 300 km. Assuming the same rate, work out how many litres of fuel are needed for a journey of 175 km.
- 11.The cost, in pounds, of hiring a minibus is modelled against the number of passengers booked. A tangent to the cost graph at 20 passengers passes through the points (16, 184) and (24, 216). Work out the instantaneous rate at which the cost increases with each extra passenger, at 20 passengers.
- 12.Two mathematically similar cubes have edge lengths 2 cm and 6 cm. Write the ratio of the volume of the smaller cube to the volume of the larger cube in its simplest form.
- 13.A salt solution has a concentration of 10%. The solution contains 50 g of salt. Work out the total mass of the solution.
- 14.A cyclist rides at a steady speed of 20 mph. Given that 1 mile ≈ 1.6 km, work out the cyclist's speed in km/h.
- 15.Write 350 ml : 1.4 l as a ratio in its simplest form.
Answer key
- (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.
- (c) 3 : 2 — Method: 'A is k times B' means that if B is taken as 1 part then A is k parts, and a ratio in its simplest form is written with whole numbers that have no common factor. Working: taking B as 1, A is 1.5, so A : B = 1.5 : 1; multiplying both parts by 2 clears the decimal and gives 3 : 2, and 3 and 2 share no factor. Answer: 3 : 2. The distractors: 2 : 3 comes from writing B before A, reversing the order the question asks for; 1.5 : 1 is the right relationship but is not in its simplest form, because a ratio in its simplest form uses whole numbers; 3 : 5 comes from comparing A with the combined length rather than with B, since A : (A + B) = 1.5 : 2.5 = 3 : 5.
- (c) £2717.20 — The recurrence B_{n+1} = 1.02B_n − 200 must be applied once for each month, using the previous month's balance each time. Starting from B_0 = 3000: 3000 × 1.02 = 3060, so B_1 = 3060 − 200 = 2860. Then 2860 × 1.02 = 2917.2, so B_2 = 2917.2 − 200 = 2717.2. Stopping after one month leaves B_1 = £2860.00, not the balance after two months. Applying two months of interest together, 1.02² = 1.0404, and 3000 × 1.0404 = 3121.2, and then subtracting 400 in one go, 3121.2 − 400 = 2721.2, does not reproduce the recurrence, because the second month's interest should be earned on the balance after the first repayment, not on the original £3000. Subtracting £200 twice from B_1 without adding a second month of interest, 2860 − 200 = 2660, drops the interest for the second month altogether. The balance after 2 months is £2717.20.
- (a) 6 — Since y is inversely proportional to x², y = k/x². Using x = 2, y = 45: 2² = 4, so 45 = k ÷ 4, giving k = 45 × 4 = 180. The equation is y = 180/x². When y = 5: x² = 180 ÷ 5 = 36, so x = 6, taking the positive root because the question states that x is positive. Treating the relationship as inversely proportional to x itself gives k = 45 × 2 = 90 and then x = 90 ÷ 5 = 18, a different relationship. Multiplying by y instead of dividing by it when isolating x² gives x² = 180 × 5 = 900 and x = 30, the wrong operation. Stopping at x² = 36 without taking the square root leaves 36, not the value of x. When y = 5, x = 6.
- (c) 25 — Find the constant multiplier from the given pair: 15 ÷ 6 = 2.5, so y is always 2.5 times x. When x = 10, y = 10 × 2.5 = 25. 19 comes from assuming an additive relationship instead of a multiplicative one — adding the difference 15 − 6 = 9 onto 10. 4 comes from using the multiplier the wrong way round (6 ÷ 15 = 0.4) and then multiplying by 10. 15 comes from simply repeating the given value of y, without applying the multiplier to the new value of x at all.
- (a) 40 m² — A scale of 1 : 10 is a length scale factor of 10 from model to real. Areas scale with the square of the length scale factor: 10² = 100. 0.4 × 100 = 40, so the real staircase covers 40 m². Giving 4 m² uses the length scale factor, 10, without squaring it (0.4 × 10 = 4). Giving 0.04 m² divides by the scale factor instead of multiplying by its square (0.4 ÷ 10 = 0.04). Giving 400 m² cubes the scale factor, 10³ = 1000, as if area scaled like a volume (0.4 × 1000 = 400).
- (a) 12 m² — Real length = 8 × 50 = 400 cm = 4 m. Real width = 6 × 50 = 300 cm = 3 m. Real area = 4 × 3 = 12 m². Scaling the plan area (8 × 6 = 48 cm²) by 50 instead of by 50 squared gives 48 × 50 = 2400 cm² = 0.24 m² — area scales by the square of the length scale factor, not the scale factor itself. Multiplying the real dimensions in centimetres, 400 × 300 = 120 000, and calling the result 120 000 m² mistakes square centimetres for square metres. Converting only the length to metres and leaving the width as 6 (treating centimetres as metres), 4 × 6 = 24, gives 24 m², from a scaling that was never finished.
- (a) 11:00 to 12:00 — Method: the instantaneous rate of change is zero at a turning point, where a rising trend becomes a falling trend; that lies within the first interval whose difference has changed sign from the interval before it. Working: the differences between consecutive readings are +3.4 °C (09:00 to 10:00), +1.7 °C (10:00 to 11:00), −0.3 °C (11:00 to 12:00) and −2.6 °C (12:00 to 13:00); the sign changes from positive to negative within 11:00 to 12:00, since the temperature is still rising up to 11:00 (20.1 °C, the highest recorded value) and has fallen by 12:00, so the instantaneous rate of change was zero somewhere within that interval. Choosing 09:00 to 10:00 picks out the interval with the largest positive difference, +3.4 °C, confusing the fastest rise with no change at all. Choosing 10:00 to 11:00 picks the last interval where the temperature was still rising, one interval too early, without checking that the very next interval turns negative. Choosing 12:00 to 13:00 picks out the interval with the largest-magnitude difference, −2.6 °C, the fastest fall, not where the change is zero. Zero instantaneous rate of change happens at a turning point, where the readings stop rising and start falling — find the FIRST interval whose difference has flipped sign from the one before it, not the biggest change or an interval where the old sign still held.
- (a) 3 hours — Method: for a fixed pool the rate of flow multiplied by the time taken is constant, so multiplying the rate by a factor divides the time by that same factor. Working: tap B's rate is 2 times tap A's rate, so tap B's time is 6 ÷ 2 = 3 hours. Answer: 3 hours. The distractors: 12 hours comes from multiplying the time by 2 as well, which treats the time as directly proportional to the rate and has the faster tap taking longer; 4 hours comes from reading ‘twice as fast’ additively, as two hours quicker, and working out 6 − 2 instead of scaling the time by a factor of 2; 1.5 hours comes from applying the factor of 2 twice, halving 6 to 3 and then halving again.
- (c) 14 litres — Method: find the amount of fuel used per km first, then use it to find the fuel needed for 175 km. Working: 24 ÷ 300 = 0.08 litres per km, and 0.08 × 175 = 14 litres. So 14 litres are needed. Distractor 24 litres comes from assuming the same amount of fuel is used no matter the distance, without scaling. Distractor 21 litres comes from misreading the original distance as 200 km instead of 300 km. Distractor 1.4 litres comes from a decimal-point slip, giving an answer ten times too small.
- (d) £4 — The gradient of the tangent gives the instantaneous rate of change of cost with respect to the number of passengers, in pounds per passenger. The tangent passes through (16, 184) and (24, 216), so the change in cost is 216 − 184 = 32 and the change in passengers is 24 − 16 = 8. The gradient is 32 ÷ 8 = 4. Stopping after finding the change in cost, without dividing by the change in passengers, leaves 32, not a rate. Adding the two changes instead of dividing gives 32 + 8 = 40, which is not a rate either. Reading off only the change in passengers, 8, is not a rate at all — a rate needs the change in cost as well. The instantaneous rate is £4 per extra passenger.
- (c) 1 : 27 — The edge lengths are in the ratio 2 : 6, which simplifies to 1 : 3. Volumes scale with the cube of the length ratio, so the volume ratio is 1³ : 3³ = 1 : 27. Giving 1 : 3 uses the length ratio without cubing it. Giving 1 : 9 squares the length ratio, which is the rule for areas, instead of cubing it, which is the rule for volumes. Giving 27 : 1 has the ratio the right way round for larger to smaller, not smaller to larger as the question asks.
- (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) 32 km/h — Method: to change mph into km/h, multiply by the number of kilometres in a mile. Working: 20 × 1.6 = 32 km/h. So the cyclist's speed is 32 km/h. Distractor 12.5 km/h comes from dividing by 1.6 instead of multiplying. Distractor 21.6 km/h comes from adding 1.6 instead of multiplying by it. Distractor 20 km/h comes from not converting the units at all.
- (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.
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