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.
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Ratio, proportion and rates of change worksheet — GCSE Higher
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- 1.Two numbers A and B are in the ratio 3:5. When 6 is added to B, the ratio of A to the new value of B is 1:2. Work out the value of A.
- 2.Two mathematically similar logos are printed on a poster. They have areas 20 cm² and 45 cm². Nadia says that the length scale factor from the smaller logo to the larger logo is 45 ÷ 20 = 2.25. Give a reason why Nadia is incorrect, and work out the correct length scale factor.
- 3.Work out the value of x when 4/x = 2/5.
- 4.Butter costs £2.40 per kg. Work out the cost of the butter in pence per 100 g.
- 5.The rent on a flat increases by 10% one year and by a further 10% the following year. Work out the overall percentage increase over the two years.
- 6.Cheng is paid an hourly rate that is proportional to the number of hours she works. She earns £52.50 for 7 hours. Work out how much she earns for 11 hours, assuming the same rate.
- 7.The total cost, in pounds, of manufacturing a batch of components is plotted against the number of components produced. A tangent to this graph at a particular point has gradient 3.60. What does the value 3.60 represent?
- 8.A cleaning solution is made by mixing disinfectant and water in the ratio 1 : 9. Priti wants to make 6 litres of the solution in total. Work out how much disinfectant she needs.
- 9.Two quantities, P and Q, are in inverse proportion. A graph is drawn with P on the vertical axis and Q on the horizontal axis. Write down which description fits the shape of this graph.
- 10.A curve has equation y = 2x² + 5. Work out an estimate for the gradient of the curve at x = 3, using a chord between the points where x = 2 and x = 4.y = 2x² + 5
- 11.Write the ratio 5 : 8 in the form 1 : n.
- 12.A car travels at a constant speed. It travels 150 miles in 3 hours. At the same speed, work out how far it travels in 5 hours.
- 13.A jumper normally costs £45. In a sale it is reduced by 20%. Work out the sale price, then write the sale price as a fraction of the normal price. Give your answer in its simplest form.
- 14.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.
- 15.Write 350 ml : 1.4 l as a ratio in its simplest form.
Answer key
- (a) 18 — Method: write both numbers with the same multiplier, turn the second ratio into an equation by cross-multiplying, solve for the multiplier and then build A from it. Working: let A = 3k and B = 5k, so 3k : (5k + 6) = 1 : 2; cross-multiplying gives 2 × 3k = 5k + 6, so 6k = 5k + 6 and k = 6; A = 3 × 6 = 18. Answer: 18, and the check works, because B = 30, B + 6 = 36 and 18:36 = 1:2. The distractors: 9 comes from reading the 6 as the difference between the two numbers — 5 − 3 = 2 parts, so one part is 3 and A is 3 × 3 — but the 6 is added to B, it is not the gap between A and B; 6 comes from solving A : (A + 6) = 1 : 2, adding the 6 to A instead of to B; 30 is the value of B, found from the correct multiplier but given in place of A.
- (d) 1.5 — Method: the length scale factor is the square root of the area scale factor, not the area scale factor itself. Working: the area scale factor is 45 ÷ 20 = 2.25, and the square root of 2.25 is 1.5. Answer: 1.5. Nadia's answer, 2.25, is the AREA scale factor — she never took the square root to get back to the length scale factor. 4.5 comes from doubling the area scale factor instead of taking its square root. 0.67 comes from taking the square root in the wrong direction, finding the scale factor from the larger rug to the smaller rug instead of the other way round.
- (a) 10 — Method: two equal fractions can be rearranged by cross-multiplying, multiplying each numerator by the other denominator. Working: 4 × 5 = 2 × x, so 2x = 20 and x = 20 ÷ 2 = 10. Answer: 10. The distractors: 20 comes from cross-multiplying to 4 × 5 = 20 and stopping there, without dividing by the 2; 8 comes from multiplying the two numerators, 4 × 2; 2.5 comes from working only with the right-hand fraction, 5 ÷ 2, and ignoring the 4.
- (c) 24p — Method: change the price to pence, then find what one tenth of 1 kg costs, since 100 g is one tenth of 1 kg. Working: £2.40 = 240p per kg, and 240 ÷ 10 = 24p per 100 g. So the cost is 24p per 100 g. Distractor 2.4p comes from dividing by 100 instead of 10. Distractor 2400p comes from multiplying by 10 instead of dividing. Distractor 240p comes from using the price per kg without scaling it down to 100 g.
- (c) 21% — Method: an increase of 10% is a multiplier of 1.1, and two successive increases are found by multiplying the multipliers. Working: 1.1 × 1.1 = 1.21, so the rent is 121% of the original, which is an increase of 21%. Answer: 21%. The distractors: 20% comes from adding the two percentages, which ignores that the second 10% is taken of a larger amount; 121% is the multiplier written as the change rather than the change itself; 11% comes from slipping in the multiplication and getting 1.11 instead of 1.21.
- (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.
- (d) £3.60 per component — The gradient of a cost-against-components graph has units of pounds per component, since cost is measured in pounds and the horizontal axis counts components. So 3.60 means it costs an extra £3.60 to produce one more component at that point. Calling it '£3.60 total cost' confuses the gradient, a rate, with the y-value on the graph, which is the total cost itself. Giving it as 3.60 components per pound swaps which axis is on top, giving the units of the reciprocal gradient, not the gradient itself. Turning 3.60 into a percentage invents a unit that has no basis in the graph's axes — a gradient here is a number of pounds, not a percentage. Always build the gradient's units from the two axes' own units, in the order y-axis over x-axis.
- (a) 0.6 litres — The ratio 1 : 9 means the solution has 1 + 9 = 10 equal parts in total. Each part is 6 ÷ 10 = 0.6 litres, and disinfectant is 1 part, so Priti needs 0.6 litres of disinfectant. Giving 0.667 litres divides by 9, the number of parts of water, instead of the total number of parts, 10 (6 ÷ 9 ≈ 0.667). Giving 6 litres is the total amount of solution, not just the disinfectant's share of it. Giving 5.4 litres works out the water's share (6 × 9 ÷ 10 = 5.4), not the disinfectant's.
- (d) A falling curve that never touches either axis — Method: inverse proportion means the product of the two quantities is constant, so P = k ÷ Q; as Q grows P shrinks, and P can never reach zero because k divided by a number is never zero. Working: taking k = 12 as an example, the pairs (1, 12), (2, 6), (3, 4), (6, 2) and (12, 1) drop steeply at first and then flatten out, so the graph is a curve that approaches both axes without meeting either of them. Answer: a falling curve that never touches either axis. The distractors: 'a straight line through the origin' is the graph of direct proportion, P = kQ, which is the opposite relationship; 'a straight line with a negative gradient' is the commonest error, reading 'P falls as Q rises' as a straight line, but on such a line P would drop by the same amount for every increase in Q and would cross the horizontal axis into negative values; 'a straight line crossing the vertical axis above zero' is a relationship of the form P = mQ + c, in which P and Q are not proportional at all.
- (d) 12 — First find the y-values at the two ends of the chord. At x = 2, y = 2 × 2² + 5 = 2 × 4 + 5 = 8 + 5 = 13. At x = 4, y = 2 × 4² + 5 = 2 × 16 + 5 = 32 + 5 = 37. The gradient of the chord is the change in y divided by the change in x: 37 − 13 = 24, then 24 ÷ (4 − 2) = 24 ÷ 2 = 12. Stopping after finding the change in y and not dividing by the change in x gives 24, which is not a gradient at all. Inverting the fraction, change in x divided by change in y, gives 2 ÷ 24 ≈ 0.08. Averaging the two y-values instead of finding the change between them gives (13 + 37) ÷ 2 = 50 ÷ 2 = 25. A gradient is always change in y over change in x — never the other way round, and never a single y-value.
- (b) 1:1.6 — To write 5 : 8 in the form 1 : n, divide both parts by 5, the first number, so that it becomes 1: 5 ÷ 5 = 1 and 8 ÷ 5 = 1.6, giving 1 : 1.6. Dividing both parts by 8 instead gives 0.6 : 1 (5 ÷ 8 = 0.625, rounded to 0.6) — the first part is no longer 1, so this is not in the required form. Dividing 5 by 8 but writing the result after the 1 gives 1 : 0.6, which divides in the wrong direction: n must come from 8 ÷ 5, not 5 ÷ 8. A slip in the division 8 ÷ 5, rounding it to 1.5 instead of the correct 1.6, gives 1 : 1.5.
- (c) 250 miles — Find the distance travelled in 1 hour: 150 ÷ 3 = 50 miles. Multiply by 5 hours: 50 × 5 = 250 miles. Giving 300 miles doubles the original distance (150 × 2 = 300) using a scale factor of 2 instead of the correct 5 ÷ 3. Giving 200 miles adds only one extra hour's distance, 50, instead of the two extra hours actually needed (150 + 50 = 200, rather than 150 + 100). Giving 90 miles divides by the scale factor instead of multiplying (150 × 3 ÷ 5 = 90).
- (c) 4/5 — Find 20% of £45: 10% is £4.50, so 20% is £9. The sale price is £45 − £9 = £36. Form the fraction 36/45; both numbers share a factor of 9, so 36 ÷ 9 = 4 and 45 ÷ 9 = 5, giving 4/5. 1/5 comes from writing the discount itself as a fraction of the normal price (9/45), instead of the sale price. 6/5 comes from adding the 20% instead of subtracting it, giving a sale price of £54, then 54/45 = 6/5. 5/9 comes from treating 'reduced by 20%' as 'reduced by £20', giving a sale price of £25, then 25/45 = 5/9.
- (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.
- (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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