Maths: Ratio, Proportion & Rates

£360 is shared between two people in the ratio 3 : 5. How much does the person with the larger share receive?

• A£216
• B£135
• C£225✓
• D£240

Answer: The ratio has 3 + 5 = 8 parts, so one part is 360 ÷ 8 = £45. The larger share is 5 × 45 = £225 and the smaller is 3 × 45 = £135.

A recipe for 6 people uses 250 g of flour. How much flour is needed for 4 people, to the nearest gram?

• A375 g
• B150 g
• C167 g✓
• D188 g

Answer: Flour per person is 250 ÷ 6 = 41.67 g, so 4 people need 4 × 41.67 = 166.7 g, which rounds to 167 g. 375 g is the amount for 9 people, and the others come from using the wrong number of people.

A car travels 240 miles in 3 hours. What is its average speed?

• A90 mph
• B60 mph
• C80 mph✓
• D72 mph

Answer: Average speed is distance ÷ time: 240 ÷ 3 = 80 mph. The others do not divide 240 miles by 3 hours.

A map has a scale of 1 : 25 000. Two villages are 6 cm apart on the map. How far apart are they in real life, in kilometres?

• A15 km
• B0.6 km
• C2.5 km
• D1.5 km✓

Answer: 6 cm × 25 000 = 150 000 cm. There are 100 000 cm in a kilometre, so 150 000 cm = 1.5 km. Slipping a factor of ten gives 15 km; the other figures do not follow from the scale.

It takes 5 workers 8 days to lay a floor. Working at the same rate, how long would 4 workers take?

• A12 days
• B8 days
• C6.4 days
• D10 days✓

Answer: This is inverse proportion: the job is 5 × 8 = 40 worker-days, so 4 workers need 40 ÷ 4 = 10 days. Treating it as direct proportion gives 6.4 days, which would mean fewer workers finishing faster.

After a 25% price rise, a train fare is £30. What was the fare before the rise?

• A£24✓
• B£37.50
• C£22.50
• D£25

Answer: A 25% rise means the new price is 125% of the old one, so the old fare is 30 ÷ 1.25 = £24. Taking 25% off £30 gives £22.50, which is the common reverse-percentage mistake.

Simplify the ratio 24 : 36.

• A4 : 6
• B2 : 3✓
• C3 : 4
• D1 : 2

Answer: Divide both parts by their highest common factor, 12: 24 ÷ 12 = 2 and 36 ÷ 12 = 3. 4 : 6 is equivalent but not fully simplified.

A train travels 90 miles at an average speed of 60 mph. How long does the journey take?

• A1 hour 20 minutes
• B1 hour 50 minutes
• C2 hours
• D1 hour 30 minutes✓

Answer: Time = distance ÷ speed = 90 ÷ 60 = 1.5 hours, which is 1 hour 30 minutes. Reading 1.5 hours as 1 hour 50 minutes is the classic decimal-to-minutes mistake.

Exchange rate: £1 = €1.12. How many euros do you get for £450?

• A€401.79
• B€562
• C€450
• D€504✓

Answer: Multiply the pounds by the rate: 450 × 1.12 = €504. Dividing instead of multiplying gives €401.79, which would be the pounds you get for €450.

In a school the ratio of boys to girls is 3 : 2. What percentage of the students are girls?

• A33%
• B20%
• C40%✓
• D60%

Answer: The ratio has 5 parts, and girls are 2 of them: 2/5 = 0.4 = 40%. 60% is the boys' share, and 33% comes from reading the ratio as a fraction 2/3.

A cyclist covers 250 metres in 20 seconds. What is her speed in metres per second?

• A50 m/s
• B12.5 m/s✓
• C5 m/s
• D25 m/s

Answer: Speed = distance ÷ time = 250 ÷ 20 = 12.5 m/s. The alternatives come from dividing the wrong way or misplacing a decimal point.

Paint is mixed from blue and white in the ratio 1 : 4. How much white paint is needed with 3 litres of blue?

• A15 litres
• B12 litres✓
• C7 litres
• D4 litres

Answer: For every 1 part blue there are 4 parts white, so 3 litres of blue needs 3 × 4 = 12 litres of white. Adding 4 gives 7, and 15 is the total amount of paint, not the white alone.