Ratio Proportion and Rates

Simplify the ratio 12:8.

• A4:3
• B12:8
• C3:2✓
• D6:4

Answer: Divide both by HCF(12, 8) = 4: 12:8 = 3:2.

Simplify the ratio 15:25.

• A6:10
• B15:25
• C3:5✓
• D5:8

Answer: HCF(15, 25) = 5: 15:25 = 3:5.

Share £30 in the ratio 2:3.

• A£20 and £10
• B£12 and £18✓
• C£15 and £15
• D£10 and £20

Answer: Ratio 2:3 means 5 parts total. £30 ÷ 5 = £6 per part. 2 parts = £12, 3 parts = £18.

A car travels 120 miles in 2 hours. What is its average speed?

• A50 mph
• B70 mph
• C80 mph
• D60 mph✓

Answer: Speed = distance ÷ time = 120 ÷ 2 = 60 mph.

If 3 apples cost 90p, how much do 5 apples cost?

• A150p✓
• B180p
• C130p
• D120p

Answer: Cost per apple = 90p ÷ 3 = 30p. 5 apples = 5 × 30p = 150p.

A recipe uses flour and sugar in the ratio 4:1. If there are 400g of flour, how much sugar is needed?

• A1600g
• B200g
• C400g
• D100g✓

Answer: Ratio 4:1 means for every 4g flour, 1g sugar is needed. 400g ÷ 4 = 100g sugar.

A train travels 180 km in 3 hours. What is its speed?

• A50 km/h
• B70 km/h
• C60 km/h✓
• D40 km/h

Answer: Speed = 180 ÷ 3 = 60 km/h.

Share 24 sweets in the ratio 1:2.

• A6 and 18
• B12 and 12
• C8 and 16✓
• D10 and 14

Answer: Ratio 1:2 means 3 parts total. 24 ÷ 3 = 8 per part. 1 part = 8, 2 parts = 16.

If 10 litres of paint cover 50 m², how much paint is needed to cover 75 m²?

• A25 litres
• B10 litres
• C15 litres✓
• D20 litres

Answer: Paint needed per m² = 10 ÷ 50 = 0.2 litres. For 75 m² = 75 × 0.2 = 15 litres.

Simplify 20:30:10.

• A4:6:2
• B10:15:5
• C2:3:1✓
• D20:30:10

Answer: HCF(20, 30, 10) = 10: 20:30:10 = 2:3:1.

A cyclist travels 24 km in 2 hours. What is the speed?

• A12 km/h✓
• B10 km/h
• C14 km/h
• D24 km/h

Answer: Speed = 24 ÷ 2 = 12 km/h.

In a school, the ratio of boys to girls is 3:2. If there are 60 boys, how many girls are there?

• A30
• B40✓
• C50
• D90

Answer: Ratio 3:2 means 3 boys for every 2 girls. If 3 parts = 60, then 1 part = 20. Girls = 2 × 20 = 40.

A car uses 8 litres of fuel to travel 80 km. How far can it travel on 16 litres?

• A120 km
• B160 km✓
• C180 km
• D80 km

Answer: Fuel efficiency = 80 ÷ 8 = 10 km per litre. 16 litres × 10 = 160 km.

A shop sells apples at 3 for £1.50. What is the cost per apple?

• A£1.00
• B£1.50
• C£0.50✓
• D£0.75

Answer: £1.50 ÷ 3 = £0.50 per apple.

Divide £60 in the ratio 1:2:3.

• A£20, £20, £20
• B£15, £20, £25
• C£10, £15, £35
• D£10, £20, £30✓

Answer: Ratio 1:2:3 means 6 parts total. £60 ÷ 6 = £10 per part. Parts are £10, £20, £30.

A plane travels 600 miles in 5 hours. What is its average speed?

• A120 mph✓
• B100 mph
• C150 mph
• D200 mph

Answer: Speed = 600 ÷ 5 = 120 mph.

If you can buy 4 items for £12, what is the cost of 7 items?

• A£19
• B£21✓
• C£20
• D£18

Answer: Cost per item = £12 ÷ 4 = £3. 7 items = 7 × £3 = £21.

The ratio of red to blue sweets is 4:5. If there are 20 red sweets, how many blue sweets are there?

• A16
• B20
• C25✓
• D32

Answer: Ratio 4:5 means 4 red for every 5 blue. If 4 = 20, then 1 = 5. Blue = 5 × 5 = 25.

Simplify the ratio 12:18.

• A6:9
• B2:3✓
• C3:4
• D4:6

Answer: HCF of 12 and 18 is 6. 12:18 = (12÷6):(18÷6) = 2:3.

Simplify the ratio 20:15:25.

• A4:3:5✓
• B2:1.5:2.5
• C10:7.5:12.5
• D20:15:25

Answer: HCF of 20, 15, and 25 is 5. So 20:15:25 = 4:3:5.

If y is directly proportional to x and y = 12 when x = 3, find y when x = 7.

• A14
• B18
• C25
• D28✓

Answer: y = kx where k is the constant. 12 = k × 3, so k = 4. When x = 7, y = 4 × 7 = 28.

If y is inversely proportional to x and y = 6 when x = 4, find y when x = 3.

• A4
• B6
• C8✓
• D9

Answer: y = k/x where k is the constant. 6 = k/4, so k = 24. When x = 3, y = 24/3 = 8.

A recipe for 4 people requires 200g of flour. How much flour is needed for 6 people?

• A300g✓
• B250g
• C333.33g
• D266.67g

Answer: For 4 people: 200g. Per person: 200÷4 = 50g. For 6 people: 50 × 6 = 300g.

A recipe uses flour and sugar in the ratio 3:2. If 450g of flour is used, how much sugar is used?

• A350g
• B225g
• C150g
• D300g✓

Answer: Ratio 3:2 means flour:sugar = 3:2. If flour = 450g, then 450/3 = 150 per part. Sugar = 2 × 150 = 300g.

A car travels 150 km in 2.5 hours. What is its average speed?

• A50 km/h
• B60 km/h✓
• C100 km/h
• D75 km/h

Answer: Speed = distance/time = 150/2.5 = 60 km/h.

A train travels at 80 km/h for 3 hours. How far does it travel?

• A150 km
• B200 km
• C240 km✓
• D300 km

Answer: Distance = speed × time = 80 × 3 = 240 km.

£1 = €1.15. How many euros do you get for £50?

• A€115
• B€50
• C€57.50✓
• D€43.48

Answer: Multiply the amount in pounds by the exchange rate: £50 × 1.15 = €57.50.

The density of water is 1 g/cm³. What is the mass of 500 cm³ of water?

• A500 g✓
• B0.5 g
• C5000 g
• D50 g

Answer: Mass = density × volume = 1 × 500 = 500 g.

A block of metal has a mass of 240 g and a volume of 30 cm³. What is its density?

• A4 g/cm³
• B8 g/cm³✓
• C240 g/cm³
• D30 g/cm³

Answer: Density = mass/volume = 240/30 = 8 g/cm³.

Pressure = force/area. If force = 200 N and area = 0.5 m², what is the pressure?

• A400 Pa✓
• B200 Pa
• C0.25 Pa
• D100 Pa

Answer: Pressure = 200/0.5 = 400 Pa.

A map is drawn to a scale of 1:50000. If the distance on the map is 4 cm, what is the real distance?

• A0.2 km
• B200 km
• C2 km✓
• D20 km

Answer: Real distance = map distance × scale = 4 × 50000 = 200000 cm = 2 km.

A shop sells apples at 3 for £2. How much do 9 apples cost?

• A£4.50
• B£2
• C£18
• D£6✓

Answer: 3 apples cost £2. 9 apples = 3 groups of 3. So 9 apples cost 3 × £2 = £6.

A car uses 8 litres of petrol to travel 96 km. How much petrol is used to travel 120 km?

• A8 litres
• B10 litres✓
• C15 litres
• D12 litres

Answer: 8 litres → 96 km. Per km: 8/96 = 1/12 litres. For 120 km: 120/12 = 10 litres.

A recipe uses butter and sugar in the ratio 5:4. If 300g of butter is used, how much sugar is used?

• A200g
• B240g✓
• C375g
• D150g

Answer: Ratio 5:4 means butter:sugar = 5:4. 300/5 = 60 per part. Sugar = 4 × 60 = 240g.

Three friends share £120 in the ratio 2:3:5. How much does the first friend get?

• A£40
• B£24✓
• C£20
• D£60

Answer: Total parts = 2 + 3 + 5 = 10. Per part = £120/10 = £12. First friend gets 2 × £12 = £24.

The value of a house fell by 8% to £184,000. What was the house worth before the fall?

• A£230,000
• B£200,000✓
• C£198,720
• D£169,280

Answer: £184,000 is 92% of the earlier value, so that value = 184000 / 0.92 = £200,000. Adding 8% back on gives £198,720, which is short because the 8% is taken from the smaller amount.

A restaurant bill comes to £59.40 including a 10% service charge. What was the bill before the service charge was added?

• A£49.40
• B£53.46
• C£65.34
• D£54.00✓

Answer: £59.40 is 110% of the bill, so the bill = 59.40 / 1.1 = £54. Taking 10% off £59.40 gives £53.46, and subtracting £10 gives £49.40.

Ben puts £5,000 into an account paying 2.5% simple interest per year. What is the total in the account after 3 years?

• A£375.00
• B£5,125.00
• C£5,384.45
• D£5,375.00✓

Answer: Interest each year = 2.5% of 5000 = £125, so three years give 3 x 125 = £375. The total is 5000 + 375 = £5,375. The £375 on its own is the interest, and £5,384.45 assumes compound interest.

Rice is sold in a 500 g bag for £1.40 and in a 1.2 kg bag for £3.24. Which bag is better value, and what does it cost per 100 g?

• AThe 500 g bag, at 28p per 100 g
• BThe 500 g bag, at 27p per 100 g
• CThe 1.2 kg bag, at 27p per 100 g✓
• DThe 1.2 kg bag, at 28p per 100 g

Answer: The 500 g bag holds five lots of 100 g, so 140 / 5 = 28p per 100 g. The 1.2 kg bag holds twelve lots of 100 g, so 324 / 12 = 27p per 100 g. The larger bag is cheaper for the same weight.

A 500 ml bottle of juice costs £1.25 and a 2 litre bottle costs £4.40. How much cheaper per litre is the 2 litre bottle?

• A15p
• B30p✓
• C60p
• D£3.15

Answer: The small bottle works out at 1.25 x 2 = £2.50 per litre and the large bottle at 4.40 / 2 = £2.20 per litre, a saving of 30p per litre. The 60p figure is the saving across the whole 2 litres.