Using Number to Solve Problems

A flapjack recipe for 12 bars uses 250 g of oats, 150 g of butter and 100 g of syrup, and can be scaled to any number of bars. A café has 1 kg of oats, 500 g of butter and plenty of syrup. What is the greatest number of bars it can make?

• A40✓
• B88
• C36
• D48

Answer: Oats: 1,000 ÷ 250 = 4 batches, enough for 48 bars. Butter: 500 ÷ 150 = 3 1/3 batches, enough for 3 1/3 × 12 = 40 bars. The butter runs out first, so the most is 40 bars. 36 assumes whole batches only.

An L-shaped room is made of two rectangles: one 5 m by 4 m and one 3 m by 2 m. Carpet costs £18.50 per m² and there is a £45 fitting charge. What is the total cost?

• A£526✓
• B£600
• C£481
• D£415

Answer: Floor area: 5 × 4 + 3 × 2 = 20 + 6 = 26 m². Carpet: 26 × £18.50 = £481. Adding the £45 fitting charge gives £526. Carpeting only the larger rectangle gives £370 + £45 = £415.

A lorry driver is paid £15 an hour for the first 40 hours he works in a week and time-and-a-half for every hour after that. One week he works 46 hours. What is his pay for that week?

• A£690
• B£1,035
• C£645
• D£735✓

Answer: The first 40 hours pay 40 × £15 = £600. Time-and-a-half is 1.5 × £15 = £22.50 an hour, so the 6 extra hours pay 6 × £22.50 = £135. Total: £600 + £135 = £735. Paying all 46 hours at £15 gives £690.

A van has an 80-litre fuel tank that is 3/8 full. Diesel costs £1.52 per litre. How much does it cost to fill the tank completely?

• A£76.00✓
• B£45.60
• C£60.80
• D£121.60

Answer: The tank is 3/8 full, so 5/8 of it is empty. 5/8 of 80 litres is 50 litres. Cost: 50 × £1.52 = £76.00. £45.60 is the cost of the 30 litres already in the tank.

A block of oak measures 20 cm by 10 cm by 5 cm and has a mass of 750 g. What is the density of the oak?

• A0.075 g/cm³
• B1.33 g/cm³
• C0.75 g/cm³✓
• D7.5 g/cm³

Answer: Volume = 20 × 10 × 5 = 1,000 cm³. Density = mass ÷ volume = 750 ÷ 1,000 = 0.75 g/cm³. Dividing the volume by the mass gives 1.33, which is upside down.

Aluminium has a density of 2.7 g/cm³. What is the mass of an aluminium bar with a volume of 400 cm³?

• A148 g
• B1,080 g✓
• C108 g
• D402.7 g

Answer: Mass = density × volume = 2.7 × 400 = 1,080 g (1.08 kg). Dividing 400 by 2.7 gives about 148 g, which would make the bar lighter than the same volume of water.

Concrete has a density of 2,400 kg/m³. A concrete base for a shed is 3 m long, 2 m wide and 0.1 m thick. What is its mass in tonnes? (1 tonne = 1,000 kg)

• A0.25 tonnes
• B144 tonnes
• C1.44 tonnes✓
• D14.4 tonnes

Answer: Volume = 3 × 2 × 0.1 = 0.6 m³. Mass = 2,400 × 0.6 = 1,440 kg, which is 1.44 tonnes. Leaving out the 0.1 m thickness gives 6 m³ and 14.4 tonnes.

Two metal ornaments each have a mass of 540 g. Ornament A has a volume of 60 cm³ and ornament B has a volume of 200 cm³. Which statement is correct?

• AB is denser, because it takes up more space
• BA is denser, at 9 g/cm³ against 2.7 g/cm³✓
• CThey are equally dense, as their masses match
• DA is denser, at 0.11 g/cm³ against 0.37 g/cm³

Answer: Density = mass ÷ volume. A: 540 ÷ 60 = 9 g/cm³. B: 540 ÷ 200 = 2.7 g/cm³. The same mass packed into a smaller volume means A is denser. Equal masses do not mean equal densities, and dividing volume by mass gives the upside-down values.

A 5-litre tin of paint contains 6.5 kg of paint. What is the density of the paint in g/cm³? (1 litre = 1,000 cm³)

• A0.77 g/cm³
• B1,300 g/cm³
• C13 g/cm³
• D1.3 g/cm³✓

Answer: Convert the units first: 6.5 kg = 6,500 g and 5 litres = 5,000 cm³. Density = 6,500 ÷ 5,000 = 1.3 g/cm³. 1,300 comes from mixing grams with litres, and 0.77 divides volume by mass.

A gold-coloured ring has a mass of 38.6 g and a volume of 2 cm³. Pure gold has a density of 19.3 g/cm³ and brass about 8.5 g/cm³. What does the ring's density suggest?

• AIts density is 0.05 g/cm³, far below either metal
• BIts density is 9.65 g/cm³, which is closer to brass
• CIts density is 77.2 g/cm³, far above either metal
• DIts density is 19.3 g/cm³, which matches pure gold✓

Answer: Density = mass ÷ volume = 38.6 ÷ 2 = 19.3 g/cm³, the same as the value given for pure gold. Multiplying mass by volume gives 77.2, and dividing volume by mass gives about 0.05.

A garden pond holds 1.5 m³ of water. A hose fills it at 12 litres per minute. How long does it take to fill the empty pond? (1 m³ = 1,000 litres)

• A1 hour 25 minutes
• B12 minutes 30 seconds
• C20 hours 50 minutes
• D2 hours 5 minutes✓

Answer: 1.5 m³ = 1,500 litres. Time = 1,500 ÷ 12 = 125 minutes. 125 minutes = 120 + 5, which is 2 hours 5 minutes. Reading 125 minutes as 1 hour 25 minutes is a common slip, and 20 hours 50 minutes uses 15,000 litres.

Priya is paid £13.60 an hour. Last week she worked three 7.5-hour shifts and one 4-hour shift. What was her gross pay for the week?

• A£408.00
• B£353.60
• C£360.40✓
• D£306.00

Answer: Hours worked: 3 × 7.5 + 4 = 22.5 + 4 = 26.5 hours. Gross pay: 26.5 × £13.60 = £360.40. £306.00 leaves out the 4-hour shift, and £408.00 counts four 7.5-hour shifts.

A cyclist rides 18 km in 45 minutes. What is her average speed in kilometres per hour?

• A24 km/h✓
• B40 km/h
• C0.4 km/h
• D13.5 km/h

Answer: 45 minutes is 0.75 hours. Speed = distance ÷ time = 18 ÷ 0.75 = 24 km/h. Another way: 45 minutes is 3/4 of an hour, so in one hour she would ride 18 ÷ 3 × 4 = 24 km. Dividing by 45 gives 0.4, which is kilometres per minute.