AQA GCSE Maths · 3.3

Ratio, proportion and rates of change

50 maths questions with worked answers with answers for 3.3 Ratio, proportion and rates of change in AQA GCSE Maths (spec 8300). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official AQA material.

All 50 questions and answers

Question 1 · Units · R1
Convert 2.5 hours to minutes.
Answer:
1 hour = 60 minutes.
2.5 × 60 = 150 minutes.
Question 2 · Units · R1
Convert 3.2 kg to grams.
Answer:
1 kg = 1,000 g.
3.2 × 1,000 = 3,200 g.
Question 3 · Units · R1
Convert 7,500 cm² to m².
Answer:
1 m² = 10,000 cm².
7,500 ÷ 10,000 = 0.75 m².
Question 4 · Units · R1
Convert 36 km/h to m/s.
Answer:
36 km = 36,000 m and 1 hour = 3,600 seconds.
36,000 ÷ 3,600 = 10 m/s.
Question 5 · Scale · R2
A map scale is 1 cm to 5 km. What real distance does 4 cm represent?
Answer:
Each centimetre represents 5 km.
4 × 5 = 20 km.
Question 6 · Scale · R2
A model car is 1:50. Its model length is 8 cm. Find the real length.
Answer:
The real car is 50 times the model length.
8 × 50 = 400 cm = 4 m.
Question 7 · Scale · R2
A shape is enlarged by scale factor 3. A side was 4 cm. Find the new length.
Answer:
Multiply the original length by the scale factor.
4 × 3 = 12 cm.
Question 8 · Ratios · R4
Simplify the ratio 18:30.
Answer:
Divide both parts by their highest common factor, 6.
18:30 = 3:5.
Question 9 · Ratios · R4
Simplify the ratio 2.4:3.6.
Answer:
Multiply both parts by 10: 24:36.
Divide both by 12: 2:3.
Question 10 · Ratios · R4
Write 3 m:75 cm in simplest form.
Answer:
Convert 3 m to 300 cm: 300:75.
Divide both parts by 75: 4:1.
Question 11 · Sharing · R5
Share £84 in the ratio 2:5.
Answer:
Total parts = 2 + 5 = 7; one part is £84 ÷ 7 = £12.
Shares: 2 × £12 = £24 and 5 × £12 = £60.
Question 12 · Sharing · R5
Divide 96 in the ratio 1:3:4.
Answer:
Total parts = 1 + 3 + 4 = 8; one part is 96 ÷ 8 = 12.
Shares: 12, 36 and 48.
Question 13 · Sharing · R5
Red:blue is 3:4 and there are 28 blue counters. How many are red?
Answer:
Four blue parts are 28, so one part is 28 ÷ 4 = 7.
Three red parts = 3 × 7 = 21.
Question 14 · Sharing · R5
A:B = 5:2 and A exceeds B by 21. Find A and B.
Answer:
The difference is 5 − 2 = 3 parts; one part = 21 ÷ 3 = 7.
A = 5 × 7 = 35 and B = 2 × 7 = 14.
Question 15 · Sharing · R5
Paint uses red:white in the ratio 2:3. How much white paint is needed for 8 litres of red?
Answer:
Two red parts are 8 litres, so one part is 4 litres.
Three white parts = 3 × 4 = 12 litres.
Question 16 · Sharing · R5
A group has 18 adults and 12 children. Give adults:children in simplest form.
Answer:
Divide both parts of 18:12 by 6.
Adults:children = 3:2.
Question 17 · Ratio links · R6
If a:b = 2:3 and b:c = 6:5, find a:b:c.
Answer:
Double a:b = 2:3 to make b = 6: a:b = 4:6.
Combine with b:c = 6:5 to get 4:6:5.
Question 18 · Ratio links · R6
If x:y = 4:7, express xy as a fraction.
Answer:
x:y = 4:7 means x corresponds to 4 parts and y to 7.
Therefore xy = 47.
Question 19 · Ratio links · R7
If a:b = 3:8 and b = 40, find a.
Answer:
Eight parts are 40, so one part is 40 ÷ 8 = 5.
a = 3 × 5 = 15.
Question 20 · Ratio fractions · R8
Flour:sugar is 3:2. What fraction of their combined mass is sugar?
Answer:
Total parts = 3 + 2 = 5.
Sugar accounts for 2 parts: 25.
Question 21 · Ratio fractions · R8
A rope is cut in ratio 7:5. What fraction is the longer piece?
Answer:
Total parts = 7 + 5 = 12.
The longer piece is 7 parts, so its fraction is 712.
Question 22 · Percentages · R9
Increase £80 by 12.5%.
Answer:
12.5% = 18; 18 of £80 is £10.
£80 + £10 = £90.
Question 23 · Percentages · R9
Decrease 500 by 18%.
Answer:
An 18% decrease leaves 82%.
500 × 0.82 = 410.
Question 24 · Percentages · R9
A value changes from 80 to 100. Find the percentage increase.
Answer:
Increase = 100 − 80 = 20.
Percentage increase = 2080 × 100% = 25%.
Question 25 · Percentages · R9
After a 20% discount an item costs £48. Find the original price.
Answer:
A 20% discount leaves 80% of the original price.
Original price = £48 ÷ 0.80 = £60.
Question 26 · Growth · R16
£200 gains 10% interest each year, compounded. Find its value after two years.
Answer:
Each year's multiplier is 1.10.
After two years: £200 × 1.10 × 1.10 = £242.
Question 27 · Growth · R16
A quantity of 1,000 falls by 10% each year. Find it after two years.
Answer:
Each year's multiplier is 0.90.
After two years: 1,000 × 0.90 × 0.90 = 810.
Question 28 · Growth · R9
An £800 investment earns simple interest of 5% per year. Find the total after 3 years.
Answer:
Yearly interest = 0.05 × £800 = £40.
Three years' interest = 3 × £40 = £120.
Total = £800 + £120 = £920.
Question 29 · Best buys · R5
A 600 g pack costs £3 and a 1 kg pack costs £4. Which is cheaper per kg?
Answer:
600 g = 0.6 kg, so £3 ÷ 0.6 = £5 per kg.
The other pack is £4 per kg; the 1 kg pack is cheaper.
Question 30 · Direct proportion · R10
Five notebooks cost £15. At the same unit price, what do eight cost?
Answer:
One notebook costs £15 ÷ 5 = £3.
Eight cost 8 × £3 = £24.
Question 31 · Direct proportion · R10
y is directly proportional to x. When x = 3, y = 12. Find y when x = 5.
Answer:
Write y = kx. When x = 3, 12 = 3k, so k = 4.
When x = 5, y = 4 × 5 = 20.
Question 32 · Direct proportion · R10
y is directly proportional to x and y = 18 when x = 6. Find y when x = 2.
Answer:
Write y = kx. When x = 6, 18 = 6k, so k = 3.
When x = 2, y = 3 × 2 = 6.
Question 33 · Direct proportion · R10
A recipe for 4 people uses 300 g flour. How much is needed for 6 people?
Answer:
For one person: 300 ÷ 4 = 75 g.
For six people: 75 × 6 = 450 g.
Question 34 · Direct proportion · R13 · Higher only
y is directly proportional to x². When x = 2, y = 20. Find y when x = 3.
Answer:
Write y = kx². Then 20 = k × 2², so k = 5.
When x = 3, y = 5 × 3² = 45.
Question 35 · Inverse proportion · R10
y is inversely proportional to x. If y = 8 when x = 3, find y when x = 6.
Answer:
For inverse proportion, xy is constant: 3 × 8 = 24.
When x = 6, y = 24 ÷ 6 = 4.
Question 36 · Inverse proportion · R13 · Higher only
y is inversely proportional to x². If y = 12 when x = 2, find y when x = 4.
Answer:
Write y = kx². Then k = 12 × 2² = 48.
When x = 4, y = 48 ÷ 4² = 3.
Question 37 · Inverse proportion · R10
Six workers take 4 days at the same rate. How long would 3 workers take?
Answer:
The total work is 6 × 4 = 24 worker-days.
For 3 workers, time = 24 ÷ 3 = 8 days.
Question 38 · Speed · R11
A car travels 150 km in 3 hours. Find its average speed.
Answer:
Average speed = distance ÷ time.
150 ÷ 3 = 50 km/h.
Question 39 · Speed · R11
A runner moves at 5 m/s for 40 seconds. Find the distance.
Answer:
Distance = speed × time.
5 × 40 = 200 m.
Question 40 · Speed · R11
A journey of 72 km is made at 48 km/h. Find the time.
Answer:
Time = distance ÷ speed.
72 ÷ 48 = 1.5 hours = 1 hour 30 minutes.
Question 41 · Density · R11
A block has mass 240 g and volume 30 cm³. Find its density.
Answer:
Density = mass ÷ volume.
240 ÷ 30 = 8 g/cm³.
Question 42 · Density · R11
A liquid has density 2 g/cm³ and volume 15 cm³. Find its mass.
Answer:
Mass = density × volume.
2 × 15 = 30 g.
Question 43 · Pressure · R11
A force of 60 N acts over 3 m². Find the pressure.
Answer:
Pressure = force ÷ area.
60 ÷ 3 = 20 N/m².
Question 44 · Similarity · R12
Two similar shapes have length scale factor 4. A side is 7 cm on the smaller shape. Find its corresponding larger side.
Answer:
Multiply the smaller length by 4.
7 × 4 = 28 cm.
Question 45 · Similarity · R12 · Higher only
Similar shapes have length scale factor 3. The smaller area is 5 cm². Find the larger area.
Answer:
Area scale factor = 3² = 9.
Larger area = 5 × 9 = 45 cm².
Question 46 · Similarity · R12 · Higher only
Similar solids have length scale factor 2. The smaller volume is 12 cm³. Find the larger volume.
Answer:
Volume scale factor = 2³ = 8.
Larger volume = 12 × 8 = 96 cm³.
Question 47 · Rates · R14
A tank gains 18 litres in 6 minutes at a constant rate. What is its rate of filling?
Answer:
Rate = volume gained ÷ time.
18 ÷ 6 = 3 litres per minute.
Question 48 · Rates · R14
Velocity rises uniformly from 4 m/s to 14 m/s in 5 s. Find acceleration.
Answer:
Acceleration = change in velocity ÷ time.
(14 − 4) ÷ 5 = 2 m/s².
Question 49 · Ratios · R5
The ratio of cats to dogs is 4:5. There are 27 animals altogether. How many are dogs?
Answer:
Total parts = 4 + 5 = 9; one part = 27 ÷ 9 = 3.
Dogs = 5 × 3 = 15.
Question 50 · Error check · R9
A student says increasing then decreasing a price by 10% restores the original price. Is this true?
Answer:
Increase multiplier = 1.10; decrease multiplier = 0.90.
1.10 × 0.90 = 0.99, so the final price is 1% lower.
The statement is false.

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