Best buys, recipes, exchange rates & unit conversions – GCSE Maths exam questions with mark scheme

1Grade 1–3FoundationCalculator3 marks

Kitchen roll is sold in two sizes.

Pack A: 6 rolls for £2.70
Pack B: 9 rolls for £3.96

Which pack is the better value for money? You must show your working. [3]

a correct comparable value for one pack, e.g. $2.70 \div 6 = 0.45$ or $3.96 \div 9 = 0.44$ (or cost of 18 rolls)M1
both values correct, e.g. 45p and 44p per roll (or £8.10 and £7.92 for 18 rolls)A1
Pack B, with correct supporting figuresC1
Answer: Pack B (44p per roll compared with 45p)
Examiner tip: A bare answer of 'Pack B' scores nothing – show a comparable figure for both packs.

2Grade 1–3FoundationNon-calculator4 marks

Here are the ingredients needed to make 12 biscuits.

150 g flour
90 g butter
60 g sugar

(a) Work out how much butter is needed to make 30 biscuits. [2]

(b) Leo has 400 g of flour and plenty of the other ingredients. What is the greatest number of biscuits he can make? [2]

(a) $90 \div 12 \times 30$ or $90 \times 2.5$ or $90 + 90 + 45$M1
225 gA1
(b) $400 \div 150 \times 12$ or $150 \div 12 = 12.5$ g per biscuitM1
32A1
Answer: (a) 225 g (b) 32 biscuits
Examiner tip: Scale using the number of biscuits (12 → 30), not the mass of flour.

3Grade 4–5Calculator4 marks

The exchange rate is £1 = €1.16

(a) Sophie changes £350 into euros. How many euros does she get? [1]

(b) On holiday she spends €290. She changes the euros she has left back into pounds at the same rate. How many pounds does she get? [3]

(a) €406B1
(b) $406 - 290 = 116$M1
their $116 \div 1.16$M1
£100A1
Answer: (a) €406 (b) £100
Examiner tip: Pounds → euros: multiply by 1.16; euros → pounds: divide by 1.16.

4Grade 4–5Calculator4 marks

A pair of headphones costs £84 in the UK.

The same headphones cost \$109 in the USA.

The exchange rate is £1 = \$1.27

In which country are the headphones cheaper, and by how much in pounds? [4]

$109 \div 1.27$ or $84 \times 1.27$M1
£85.83 (85.826…) or \$106.68A1
UK is cheaper, supported by correct figuresC1
£1.83 (accept £1.82–£1.83)A1
Answer: UK cheaper by £1.83
Examiner tip: The difference must be in pounds, so convert the US price into pounds rather than finding a difference in dollars.

5Grade 4–5Non-calculator4 marks

(a) Change 3.5 m² into cm². [1]

(b) Change 0.6 m³ into cm³. [1]

(c) Water flows from a tap at 18 litres per minute. Work out the rate of flow in cm³ per second. (1 litre = 1000 cm³) [2]

(a) 35 000 cm²B1
(b) 600 000 cm³B1
(c) $18 \times 1000 \div 60$M1
300 cm³/sA1
Answer: (a) 35 000 cm² (b) 600 000 cm³ (c) 300 cm³/s
Examiner tip: 1 m² = 100² = 10 000 cm² and 1 m³ = 100³ = 1 000 000 cm³ – not 100.

6Grade 4–5Calculator5 marks

Sam needs 6 litres of orange juice for a party.

Shop A sells 2-litre cartons for £3.20 each, with the offer: buy 2, get a third carton half price.

Shop B sells 750 ml bottles for £1.10 each.

Sam wants to spend as little as possible. Which shop should he use? You must show your working. [5]

$6 \div 2 = 3$ cartons from Shop AM1
$3.20 \times 2 + 1.60$ (= £8.00)M1
$6000 \div 750 = 8$ bottles from Shop BM1
$8 \times 1.10$ (= £8.80)M1
Shop A, with £8.00 and £8.80C1
Answer: Shop A (£8.00 compared with £8.80)
Examiner tip: Apply the offer carefully: only the third carton is half price.

7Grade 6–7HigherCalculator4 marks

A car uses 6.2 litres of fuel for every 100 km it travels.

1 gallon = 4.546 litres 1 mile = 1.609 km

Work out the fuel consumption of the car in miles per gallon. Give your answer to 3 significant figures. [4]

inverts to distance per unit of fuel, e.g. $100 \div 6.2$ (= 16.1… km per litre), or uses $\dfrac{\text{distance}}{\text{fuel}}$ at the final stepM1
correct litres → gallons conversion, e.g. $16.1… \times 4.546$ (= 73.3… km per gallon) or $6.2 \div 4.546$ (= 1.363… gallons per 100 km)M1
correct km → miles conversion, e.g. $\div 1.609$ (100 km = 62.15… miles)M1
45.6 (accept 45.5–45.7 from rounded working)A1
Answer: 45.6 miles per gallon
Examiner tip: Litres per 100 km is upside down compared with miles per gallon – find distance per unit of fuel first.

8Grade 1–3FoundationNon-calculator5 marks

The conversion graph can be used to change between pounds (£) and euros (€).

(a) Use the graph to change £20 into euros. [1]

(b) Use the graph to change €36 into pounds. [1]

(d) Explain how the graph shows that the number of euros is directly proportional to the number of pounds. [1]

(a) €24B1
(b) £30B1
(c) reads a correct pair from the graph and scales up, e.g. £50 = €60 then $60 \times 5$, or £10 = €12 then $12 \times 25$, or finds the rate 1.2 and does $250 \times 1.2$M1
€300A1
(d) it is a straight line and it passes through the origin (0, 0) (both needed; ignore reference to the gradient)C1
Answer: (a) €24 (b) £30 (c) €300 (d) straight line through the origin
Examiner tip: £250 is off the graph – read an easy pair such as £50 = €60 and scale it, never extend the axes by guesswork.

9Grade 1–3FoundationNon-calculator3 marks

Mia earns £12.40 per hour.

Last Saturday she started work at 08:45 and finished at 12:30.

(a) How long did Mia work? Give your answer in hours and minutes. [1]

(b) Work out how much Mia earned last Saturday. [2]

(a) 3 hours 45 minutesB1
(b) correct method using their time correctly converted, e.g. $3.75 \times 12.40$, or $3 \times 12.40 + \tfrac{3}{4} \times 12.40$ (= 37.20 + 9.30)M1
£46.50 (accept £46.5)A1
Answer: (a) 3 hours 45 minutes (b) £46.50
Examiner tip: 45 minutes is 0.75 of an hour, not 0.45 – using 3.45 gives £42.78 and loses the accuracy mark.

10Grade 4–5Calculator3 marks

A shop sells porridge oats in three sizes.

Size	Mass	Price
Small	500 g	£2.35
Medium	750 g	£3.40
Large	1.2 kg	£5.52

Which size is the best value for money? You must show your working. [3]

correct method to find a comparable value for at least two sizes, e.g. price per 100 g, price per gram or grams per £1 (1.2 kg must be treated as 1200 g if used)M1
comparable values for all three sizes, e.g. 47p, 45.3p, 46p per 100 g or 212.7…, 220.5…, 217.3… g per £1M1
Medium, with correct supporting values for all three sizesA1
Answer: Medium (45.3p per 100 g, compared with 47p and 46p)
Examiner tip: Change 1.2 kg to 1200 g before dividing – comparing 1.2 with 500 and 750 gives nonsense values.

11Grade 4–5Non-calculator4 marks

Here are the ingredients needed to make lentil soup for 6 people.

900 ml vegetable stock
3 onions
240 g red lentils
150 g carrots

Paula has
2 litres of vegetable stock
8 onions
500 g of red lentils
plenty of carrots.

What is the greatest number of people Paula can make this soup for? You must show your working. [4]

finds the amount per person (or a scale factor) for at least one ingredient, e.g. 150 ml stock per person, ½ onion per person or 40 g lentils per personM1
finds the number of people for at least two ingredients, e.g. stock $2000 \div 150 = 13.3\ldots$ (2 litres = 2000 ml), onions 16, lentils 12.5
Alternative: scales the whole recipe for a trial number of people, e.g. for 12 people 1800 ml, 6 onions, 480 gM1
finds the number of people for all three ingredients (stock, onions, lentils) and identifies lentils as the limiting ingredient
Alternative: shows 12 people is possible and 13 people is not, e.g. 13 people need 520 g lentils > 500 gM1
12 (do not accept 12.5 or 13)A1
Answer: 12 people
Examiner tip: Round down: the lentils only stretch to 12.5 people, so the answer is 12, not 13.

12Grade 4–5Calculator4 marks

Olivia wants to change £500 into euros.

Bureau X: £1 = €1.14 No commission.

Bureau Y: £1 = €1.18 3% commission is taken from the pounds before they are changed.

Which bureau gives Olivia more euros, and by how many euros? You must show your working. [4]

$500 \times 1.14$ (= 570)M1
$500 \times 0.97$ (= 485), or $500 - 15$, or 3% of 500 = 15M1
their $485 \times 1.18$ (= 572.30) (also award for $500 \times 1.18 \times 0.97$ in either order)M1
Bureau Y by €2.30 (both 570 and 572.30 seen)A1
Answer: Bureau Y, by €2.30 (€572.30 compared with €570)
Examiner tip: Take the 3% off before comparing – the better-looking rate is only worth it if it beats the commission.

13Grade 6–7HigherNon-calculator3 marks

A brand of coffee is sold in jars.

The company makes a new jar. The new jar contains 25% more coffee than the old jar. The price of the new jar is 15% more than the price of the old jar.

Work out the percentage change in the price per gram of coffee. State whether the new jar is better or worse value for money. [3]

uses multipliers 1.15 and 1.25 (or chooses values, e.g. old jar 100 g for £1, new jar 125 g for £1.15)M1
$1.15 \div 1.25$ (= 0.92), or price per gram compared, e.g. 1p per gram and 0.92p per gramM1
8% decrease and better valueA1
Answer: Price per gram decreases by 8%, so the new jar is better value
Examiner tip: Percentages here are multipliers, not things to subtract: 25% − 15% = 10% is wrong – divide 1.15 by 1.25.

14Grade 8–9HigherNon-calculator3 marks

A tank holds $V$ m³ of water when full.

The tank is filled from empty at a constant rate of $r$ cm³ per second.

Show that the time taken, in hours, to fill the tank is $\dfrac{2500V}{9r}$ [3]

converts volume correctly: $V$ m³ = $1\,000\,000V$ cm³ (or converts rate to $\dfrac{r}{1\,000\,000}$ m³ per second)M1
correct expression for time in seconds, e.g. $\dfrac{1\,000\,000V}{r}$, and divides by 3600 (or converts the rate to per hour, $3600r$ cm³ per hour)M1
completes to $\dfrac{2500V}{9r}$ with all steps shown, e.g. $\dfrac{1\,000\,000V}{3600r} = \dfrac{2500V}{9r}$A1
Answer: $\dfrac{1\,000\,000V}{3600r} = \dfrac{2500V}{9r}$
Examiner tip: 1 m³ is 100 × 100 × 100 = 1 000 000 cm³, not 100 cm³ – cube the length conversion.

15Grade 6–7HigherCalculator5 marks

In the UK, petrol costs 138.9p per litre.

In the USA, petrol costs \$3.62 per US gallon.

1 US gallon = 3.785 litres £1 = \$1.27

Work out the percentage by which the UK price is greater than the USA price. Give your answer to 3 significant figures. [5]

correct currency conversion, e.g. $3.62 \div 1.27$ (= 2.850…) or $1.389 \times 1.27$ (= 1.764…)M1
correct volume conversion, e.g. $\div 3.785$ or $1.389 \times 3.785$ (= 5.257…)M1
both prices in the same units, e.g. 75.3p and 138.9p per litre, or £2.85 and £5.26 per US gallon, or \$0.956 and \$1.764 per litreA1
$\dfrac{\text{UK} - \text{USA}}{\text{USA}} \times 100$ or $\dfrac{\text{UK}}{\text{USA}}$ (= 1.844…) using their comparable pricesM1
84.4 (accept 84.4–84.6 from rounded working)A1
Answer: 84.4%
Examiner tip: The comparison is with the USA price, so divide the difference by the USA price – dividing by the UK price gives 45.8%.

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