AQA GCSE Mathematics 8300: Ratio, proportion and rates of change – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover Topic 3, Ratio, proportion and rates of change (references R1–R16), of the AQA GCSE Mathematics (8300) specification, Version 1.0, for exams in May/June 2017 onwards and every May/June and November series after that. Questions 8, 9 and 12 use content the specification lists as Higher tier only; the rest is on both tiers. Each question is labelled “(non-calculator)” for Paper 1 style or “(calculator allowed)” for Paper 2 and 3 style.

Learn the content first in the study guide and the revision notes. The course hub is AQA GCSE Mathematics, and the printable checklist lists every statement.

Questions

1. (non-calculator)

(a) Write 45 cm : 1.2 m in its simplest form. [2] (b) Write 1.2 m as a fraction of 45 cm. Give your answer in its simplest form. [1]

2. (non-calculator) Amir and Bea share some marbles in the ratio 5 : 3. Amir gets 14 more marbles than Bea. Work out how many marbles there are altogether. [3]

3. (calculator allowed) Pack A holds 750 g of rice and costs £2.85. Pack B holds 1.2 kg of the same rice and costs £4.44. Which pack is better value? You must show your working. [3]

4. (non-calculator) A walking map has a scale of 1 : 40 000.

(a) Two villages are 6.3 cm apart on the map. Work out the real distance between them in kilometres. [2] (b) A footpath is 3.4 km long. Work out its length on the map in centimetres. [1]

5. (calculator allowed)

(a) In a sale, a jacket is reduced by 35%. The sale price is £57.20. Work out the price before the sale. [2] (b) The population of a town rose from 12 400 to 13 516. Work out the percentage increase. [2]

6. (calculator allowed)

(a) Change 54 km/h into m/s. [2] (b) An alloy is made by melting together 300 cm³ of metal P, density 8.9 g/cm³, and 200 cm³ of metal Q, density 2.7 g/cm³. Assume the volume of the alloy is 500 cm³. Work out the density of the alloy. [3] (c) A box exerts a force of 480 N on an area of 0.6 m². Using pressure = force ÷ area, work out the pressure in N/m². [1]

7. (non-calculator) It takes 6 identical pumps 15 hours to empty a reservoir.

(a) Work out how long it would take 9 of these pumps. [2] (b) State one assumption you made in part (a). [1]

8. (calculator allowed, Higher tier only) d is inversely proportional to the square of r. When r = 3, d = 20.

(a) Find a formula for d in terms of r. [3] (b) Work out the value of d when r = 5. [1] (c) Work out the value of r when d = 45, given that r > 0. [2]

9. (non-calculator, Higher tier only) The depth, h cm, of water in a tank t minutes after a tap is turned on is given by h = 0.5t² for 0 ≤ t ≤ 10.

(a) Work out the average rate of change of the depth between t = 2 and t = 6. [2] (b) The tangent to the curve at t = 5 passes through the point (3, 2.5). Use it to work out the rate of change of the depth when t = 5. [2] (c) Is the water level rising faster or more slowly as time goes on? Give a reason. [1]

10. (calculator allowed)

(a) A car is bought for £18 500. Its value falls by 12% each year. Work out its value after 3 years. [2] (b) After how many complete years will the car first be worth less than half of £18 500? [2] (c) Nadia has £3000 to invest for 4 years. Account X pays 2.5% compound interest per year. Account Y pays 2.7% simple interest per year. Which account gives more money after 4 years, and by how much? [3]

11. (calculator allowed) A potting mix is made from loam, grit and bark. The ratio of loam to grit is 3 : 4. The ratio of grit to bark is 6 : 5.

(a) Write the ratio loam : grit : bark in its simplest form. [2] (b) A gardener makes 465 kg of the mix. Work out the mass of bark. [2] (c) What percentage of the mix is loam? Give your answer to 1 decimal place. [1] (d) How many more kilograms of loam must be added so that loam makes up 40% of the mix? [2]

12. (calculator allowed, Higher tier only) The number of fish in a lake at the start of year n is uₙ. A model gives uₙ₊₁ = 0.9uₙ + 400, with u₀ = 2000.

(a) Work out u₁, u₂ and u₃. [2] (b) Show that a population of 4000 would stay the same from one year to the next. [2] (c) The mass of food needed each day is directly proportional to the number of fish. 2000 fish need 50 kg of food a day. Work out the food needed per day for u₃ fish. [2]

Answers

1. (a) Same units: 45 : 120 [1]; ÷ 15 gives 3 : 8 [1] (b) 120/45 = 8/3 [1] Examiner insight: A ratio written with mixed units, such as 45 : 1.2, scores nothing until it is converted; the first mark is for making the units match.

2. Difference = 5 − 3 = 2 parts, so 2 parts = 14 [1]; 1 part = 7 [1]; total = 8 × 7 = 56 marbles [1] Examiner insight: Sharing 14 in the ratio 5 : 3 is a common wrong start and scores no method mark, because 14 is a difference, not a total.

3. Compare the cost of the same quantity [1]; A: 285 ÷ 7.5 = 38p per 100 g and B: 444 ÷ 12 = 37p per 100 g [1]; Pack B is better value [1] Examiner insight: The final mark needs a conclusion supported by two correct comparable values; “B” with no figures earns nothing.

4. (a) 6.3 × 40 000 = 252 000 cm [1]; = 2.52 km [1] (b) 3.4 km = 340 000 cm; 340 000 ÷ 40 000 = 8.5 cm [1] Examiner insight: The method mark in (a) is for the multiplication; the accuracy mark needs the correct unit conversion from cm to km (÷ 100 000).

5. (a) 57.20 is 65% of the original, so 57.20 ÷ 0.65 [1] = £88 [1] (b) Increase = 1116 [1]; 1116 ÷ 12 400 × 100 = 9% [1] Examiner insight: Adding 35% of £57.20 to £57.20 gives £77.22; this is the wrong base, so it scores no marks.

6. (a) 54 × 1000 ÷ 3600 [1] = 15 m/s [1] (b) Masses: 300 × 8.9 = 2670 g and 200 × 2.7 = 540 g [1]; total mass = 3210 g for 500 cm³ [1]; 3210 ÷ 500 = 6.42 g/cm³ [1] (c) 480 ÷ 0.6 = 800 N/m² [1] Examiner insight: Averaging the two densities in (b) gives 5.8 g/cm³ and earns no marks; the method mark is for finding each mass.

7. (a) Total work = 6 × 15 = 90 pump-hours [1]; 90 ÷ 9 = 10 hours [1] (b) All pumps work at the same constant rate the whole time [1] Examiner insight: More pumps must mean less time; an answer of 22.5 hours treats this as direct proportion and scores 0 in (a).

8. (a) d = k/r² [1]; 20 = k/9, so k = 180 [1]; d = 180/r² [1] (b) 180 ÷ 25 = 7.2 [1] (c) r² = 180 ÷ 45 = 4 [1]; r = 2 [1] Examiner insight: In (a) the final mark needs the full equation with k replaced; stopping at k = 180 earns only the first two marks.

9. (a) h(2) = 2 and h(6) = 18 [1]; (18 − 2) ÷ (6 − 2) = 4 cm per minute [1] (b) At t = 5, h = 12.5, so the tangent passes through (5, 12.5) [1]; gradient = (12.5 − 2.5) ÷ (5 − 3) = 5 cm per minute [1] (c) Faster, because the curve gets steeper, so tangent gradients increase with t [1] Examiner insight: Part (a) asks for an average rate, so it needs the chord; giving the gradient at t = 6 (6 cm per minute) answers a different question and scores 0.

10. (a) 18 500 × 0.88³ [1] = £12 607.23 [1] (b) Half is £9250. After 5 years: 18 500 × 0.88⁵ = £9763.04, still above [1]; after 6 years: £8591.48, below, so 6 years [1] (c) X: 3000 × 1.025⁴ = £3311.44 [1]; Y: 3000 + 3000 × 0.027 × 4 = £3324.00 [1]; Account Y, by £12.56 [1] Examiner insight: In (b) you need to show values either side of the target (years 5 and 6); an answer of 6 with no supporting value cannot earn the method mark.

11. (a) Make grit the same: 9 : 12 and 12 : 10 [1]; 9 : 12 : 10 [1] (b) 9 + 12 + 10 = 31 parts; 465 ÷ 31 = 15 kg per part [1]; bark = 10 × 15 = 150 kg [1] (c) 9/31 × 100 = 29.0% [1] (d) Loam = 135 kg; (135 + x) = 0.4(465 + x) [1]; 0.6x = 51, so x = 85 kg [1] Examiner insight: In (d) the total changes as loam is added; working out 40% of 465 = 186 and adding 51 kg ignores this and scores 0.

12. (a) u₁ = 0.9 × 2000 + 400 = 2200 [1]; u₂ = 2380, u₃ = 2542 [1] (b) 0.9 × 4000 = 3600 [1]; 3600 + 400 = 4000, the same value [1] (c) Food = 50 ÷ 2000 = 0.025 kg per fish [1]; 2542 × 0.025 = 63.55 kg [1] Examiner insight: In (a) a follow-through is normally allowed for u₂ and u₃ from a wrong u₁, but only if each value is fed back into the rule.

Where marks are usually lost

• Mixing units inside a ratio or a speed calculation, for example cm with m, or minutes written as a decimal of an hour.
• Treating a difference between shares as a total to be shared.
• Taking a percentage of the sale price in reverse-percentage questions instead of dividing by the multiplier.
• Averaging two densities or two speeds instead of using totals.
• Treating an inverse proportion situation as direct, so more workers take longer.
• Stopping after finding k and not writing the final proportion equation.
• Using the simple interest method for compound growth, or rounding to the nearest pound when pence are needed.
• Reading gradients from a tangent without stating the units of the rate.
• Not showing the values either side of a target in a “how many years” question.

Next steps

• Recap with the revision notes.
• Re-learn anything you got wrong in the study guide.
• See the whole course on the AQA GCSE Mathematics hub.
• Tick off statements on the printable checklist.
• Try all free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

AQA GCSE Mathematics (8300) specification, Version 1.0 (12 September 2014), for teaching from September 2015 and exams in May/June 2017 onwards, published by AQA. Section 3.3, Ratio, proportion and rates of change, R1–R16.