Ratio and Proportion

It takes 4 decorators 9 days to paint a school. How many days would it take 6 decorators working at the same rate?

• A11 days
• B13.5 days
• C6 days✓
• D7 days

Answer: This is inverse proportion: more workers means fewer days. The job needs 4 × 9 = 36 decorator-days. With 6 decorators it takes 36 ÷ 6 = 6 days. Scaling up as if it were direct proportion gives 13.5 days, which is the wrong direction.

On a college course, the ratio of men to women is 3:5. There are 18 more women than men. How many learners are on the course altogether?

• A27
• B72✓
• C144
• D45

Answer: The difference between the parts is 5 - 3 = 2 parts, and that is 18 learners, so 1 part = 9. The whole course is 3 + 5 = 8 parts: 8 × 9 = 72 learners. 27 is the number of men and 45 is the number of women.

Red and blue counters in a bag are in the ratio 2:5. There are 10 red counters. After 6 more red counters are added, what is the new ratio of red to blue, in its simplest form?

• A8:25
• B4:5
• C16:31
• D16:25✓

Answer: With 10 red counters, 1 part = 5, so there are 5 × 5 = 25 blue counters. Adding 6 red gives 16 red. The ratio 16:25 cannot be simplified because 16 and 25 have no common factor. Adding the 6 to both colours gives 16:31.

A tank of drinking water lasts 5 horses for 12 days. For how many days would the same tank last 6 horses, if each horse drinks the same amount?

• A14.4 days
• B13 days
• C10 days✓
• D11 days

Answer: The tank holds 5 × 12 = 60 horse-days of water. Shared by 6 horses, it lasts 60 ÷ 6 = 10 days. More horses means the water runs out sooner, so the answer must be less than 12 days; 14.4 days treats it as direct proportion.

A decorator mixes grey paint from black and white paint in the ratio 1:4. She needs 7.5 litres of grey. Black costs £12 per litre and white costs £8 per litre. What does the paint for the mix cost?

• A£84
• B£66✓
• C£90
• D£60

Answer: There are 1 + 4 = 5 parts, so 1 part = 7.5 ÷ 5 = 1.5 litres. Black: 1.5 × £12 = £18. White: 4 × 1.5 = 6 litres, costing 6 × £8 = £48. Total: £18 + £48 = £66. Swapping the ratio round gives £84.

A builder mixes cement, sand and gravel in the ratio 1:2:3 by mass. He needs 480 kg of the mix. Cement is sold in 25 kg bags. How many bags of cement must he buy?

• A20
• B4✓
• C3
• D7

Answer: There are 1 + 2 + 3 = 6 parts, so 1 part = 480 ÷ 6 = 80 kg of cement. 80 ÷ 25 = 3.2 bags, and part bags are not sold, so he must buy 4 bags. Rounding down to 3 bags would leave him 5 kg short.

A map has a scale of 1:100,000. If a road is 2 cm on the map, what is the actual distance in km?

• A2 km✓
• B200 km
• C20 km
• D2,000 km

Answer: Scale 1:100,000 means 1 cm on map = 100,000 cm actual = 1 km. So 2 cm on map = 2 km actual.

Apples and oranges in a fruit box are in the ratio 5:3. A box holds 64 pieces of fruit. How many more apples than oranges are in the box?

• A16✓
• B40
• C24
• D8

Answer: There are 5 + 3 = 8 parts, so 1 part = 64 ÷ 8 = 8 pieces. Apples: 5 × 8 = 40. Oranges: 3 × 8 = 24. The difference is 40 - 24 = 16, which is 2 parts.

A recipe for 6 people uses 450 g of mince. Mince is sold in 500 g packs. How many packs are needed to make the recipe for 15 people?

• A3 packs✓
• B5 packs
• C2 packs
• D4 packs

Answer: For 15 people the recipe must be scaled by 15 ÷ 6 = 2.5, so it needs 450 × 2.5 = 1,125 g of mince. 1,125 ÷ 500 = 2.25 packs, so 3 packs must be bought. 2 packs would only give 1,000 g.

Three business partners share a profit of £84,000 in the ratio 2:3:7. How much more does the partner with the largest share receive than the partner with the smallest share?

• A£28,000
• B£35,000✓
• C£14,000
• D£49,000

Answer: There are 2 + 3 + 7 = 12 parts, so 1 part = £84,000 ÷ 12 = £7,000. The largest share is 7 parts and the smallest is 2 parts; the difference is 5 parts = 5 × £7,000 = £35,000. £49,000 is the largest share itself.

A rectangular garden is 18 m long and 12 m wide. A designer draws it to a scale of 1:200. What are the length and width on the drawing?

• A36 m by 24 m
• B90 cm by 60 cm
• C9 cm by 6 cm✓
• D0.9 cm by 0.6 cm

Answer: Convert to centimetres: 18 m = 1,800 cm and 12 m = 1,200 cm. A scale of 1:200 means dividing by 200: 1,800 ÷ 200 = 9 cm and 1,200 ÷ 200 = 6 cm. Multiplying by 2 instead gives 36 m by 24 m.

A journey takes 2 hours 15 minutes at an average speed of 48 mph. How long would the same journey take at an average speed of 60 mph?

• A2 hours 49 minutes
• B1 hour 8 minutes
• C1 hour 48 minutes✓
• D2 hours 20 minutes

Answer: Distance = speed × time = 48 × 2.25 = 108 miles. Time at 60 mph = 108 ÷ 60 = 1.8 hours. 0.8 of an hour is 0.8 × 60 = 48 minutes, so the time is 1 hour 48 minutes. Reading 1.8 hours as 1 hour 80 minutes leads to 2 hours 20 minutes.