OpenRevise

Exam questions · Maths

Ratio and Proportion

  • 30 exam questions
  • 98 marks
  • 45 quick checks

Ratio: Simplifying and Sharing

Just this lesson
  1. 1 Write [2 marks]

    Write the ratio \(45 : 60\) in its simplest form.

    Show answerHide answer

    Model answer

    The highest common factor of 45 and 60 is 15, so \(45 : 60 = 3 : 4\).

    Mark scheme

    • Divides both numbers by a common factor, such as \(15 : 20\) or \(9 : 12\) — M1
    • \(3 : 4\) — A1
  2. 2 Work out [3 marks]

    Ali and Beth share \(\pounds 72\) in the ratio \(5 : 4\). Work out how much each person gets.

    Show answerHide answer

    Model answer

    There are \(5 + 4 = 9\) parts, and one part is \(72 \div 9 = 8\). Ali gets \(5 \times 8 = \pounds 40\) and Beth gets \(4 \times 8 = \pounds 32\).

    Mark scheme

    • \(72 \div 9\) or \(5 + 4 = 9\) seen — M1
    • \(5 \times 8\) or \(4 \times 8\) — M1
    • \(\pounds 40\) and \(\pounds 32\) — A1
  3. 3 Work out [3 marks]

    Concrete is made from cement, sand and gravel in the ratio \(1 : 2 : 4\) by mass. Aaron uses 28 kg of gravel. Work out the total mass of concrete he makes.

    Show answerHide answer

    Model answer

    The gravel is 4 parts, so one part is \(28 \div 4 = 7\) kg. The total is \(1 + 2 + 4 = 7\) parts, so the mass is \(7 \times 7 = 49\) kg.

    Mark scheme

    • \(28 \div 4 = 7\) — M1
    • \(7 \times 7\) or \(1 + 2 + 4 = 7\) used with their 7 — M1
    • 49 kg — A1
  4. 4 Work out [3 marks]

    In a club, the ratio of boys to girls is \(3 : 5\). There are 12 more girls than boys. Work out the total number of members in the club.

    Show answerHide answer

    Model answer

    The difference is \(5 - 3 = 2\) parts, and 2 parts \(= 12\), so one part is 6. The total is \(3 + 5 = 8\) parts, which is \(8 \times 6 = 48\) members.

    Mark scheme

    • \(5 - 3 = 2\) parts or \(12 \div 2 = 6\) — M1
    • \(8 \times 6\) or \(18 + 30\) — M1
    • 48 — A1
  5. 5 Show that [3 marks]

    The ratio of red counters to blue counters in a bag is \(3 : 2\). There are 40 counters in the bag. Show that there are 8 more red counters than blue counters.

    Show answerHide answer

    Model answer

    There are \(3 + 2 = 5\) parts, so one part is \(40 \div 5 = 8\) counters. Red is \(3 \times 8 = 24\) and blue is \(2 \times 8 = 16\), and \(24 - 16 = 8\), as required. A quicker route is that the difference is 1 part, which is 8.

    Mark scheme

    • \(40 \div 5 = 8\) — M1
    • \(24\) and \(16\) found — M1
    • \(24 - 16 = 8\) with a conclusion — C1
  6. 6 Work out [4 marks]

    The ratio of the number of apples to the number of pears in a shop is \(2 : 3\). The ratio of the number of pears to the number of oranges is \(4 : 5\). There are 175 pieces of fruit altogether. Work out the number of oranges.

    Show answerHide answer

    Model answer

    Make the number of pears the same in both ratios. The lowest common multiple of 3 and 4 is 12, so apples : pears \(= 8 : 12\) and pears : oranges \(= 12 : 15\). That gives apples : pears : oranges \(= 8 : 12 : 15\), which is 35 parts. One part is \(175 \div 35 = 5\), so the number of oranges is \(15 \times 5 = 75\).

    Mark scheme

    • Uses 12 as a common number of pears, such as \(8 : 12\) or \(12 : 15\) — M1
    • \(8 : 12 : 15\) — A1
    • \(175 \div 35 = 5\) — M1
    • 75 — A1

Quick check

  1. 1

    Simplify the ratio \(24 : 36\).

    1. A\(4 : 6\)
    2. B\(2 : 3\)
    3. C\(3 : 2\)
    4. D\(6 : 9\)
    Show answerHide answer

    B: \(2 : 3\)

    The highest common factor of 24 and 36 is 12. Dividing both parts by 12 gives \(2 : 3\).

  2. 2

    Which ratio is equivalent to \(3 : 5\)?

    1. A\(9 : 15\)
    2. B\(9 : 12\)
    3. C\(6 : 8\)
    4. D\(5 : 3\)
    Show answerHide answer

    A: \(9 : 15\)

    Multiplying both parts of \(3 : 5\) by 3 gives \(9 : 15\). The others do not keep the same relationship.

  3. 3

    Write \(40\) cm to \(1.2\) m as a ratio in its simplest form.

    1. A\(40 : 1.2\)
    2. B\(1 : 30\)
    3. C\(3 : 1\)
    4. D\(1 : 3\)
    Show answerHide answer

    D: \(1 : 3\)

    Change to the same unit: \(1.2\) m \(= 120\) cm. Then \(40 : 120 = 1 : 3\).

  4. 4

    \(\pounds 60\) is shared in the ratio \(2 : 3\). How much is the larger share?

    1. A\(\pounds 24\)
    2. B\(\pounds 30\)
    3. C\(\pounds 36\)
    4. D\(\pounds 40\)
    Show answerHide answer

    C: \(\pounds 36\)

    There are \(2 + 3 = 5\) parts, and one part is \(60 \div 5 = 12\). The larger share is \(3 \times 12 = 36\).

  5. 5

    The ratio of boys to girls in a class is \(3 : 5\). There are 12 boys. How many girls are there?

    1. A15
    2. B20
    3. C36
    4. D4
    Show answerHide answer

    B: 20

    One part is \(12 \div 3 = 4\), so the number of girls is \(5 \times 4 = 20\).

  6. 6

    Red and blue beads are in the ratio \(2 : 5\). What fraction of the beads are red?

    1. A\(\dfrac{2}{7}\)
    2. B\(\dfrac{2}{5}\)
    3. C\(\dfrac{5}{7}\)
    4. D\(\dfrac{2}{3}\)
    Show answerHide answer

    A: \(\dfrac{2}{7}\)

    The whole is \(2 + 5 = 7\) parts, so red is \(\dfrac{2}{7}\).

  7. 7

    Orange juice and water are mixed in the ratio \(1 : 4\). There is 250 ml of orange juice. How much water is there?

    1. A750 ml
    2. B1250 ml
    3. C4 ml
    4. D1000 ml
    Show answerHide answer

    D: 1000 ml

    The water is 4 times the juice, so \(4 \times 250 = 1000\) ml.

  8. 8

    Amy and Ben share money in the ratio \(3 : 7\). Ben gets \(\pounds 20\) more than Amy. How much money is shared?

    1. A\(\pounds 35\)
    2. B\(\pounds 80\)
    3. C\(\pounds 50\)
    4. D\(\pounds 20\)
    Show answerHide answer

    C: \(\pounds 50\)

    The difference is \(7 - 3 = 4\) parts, so 4 parts \(= 20\) and one part is 5. The total is \(10 \times 5 = 50\).

  9. 9

    \(A : B = 2 : 3\) and \(B : C = 4 : 5\). Write \(A : B : C\) in its simplest form.

    1. A\(2 : 3 : 5\)
    2. B\(8 : 12 : 15\)
    3. C\(8 : 12 : 20\)
    4. D\(6 : 12 : 15\)
    Show answerHide answer

    B: \(8 : 12 : 15\)

    Make the \(B\) values match. The lowest common multiple of 3 and 4 is 12. Then \(A : B = 8 : 12\) and \(B : C = 12 : 15\), so \(A : B : C = 8 : 12 : 15\).

Direct and Inverse Proportion

Just this lesson
  1. 1 Work out [3 marks]

    Here are the ingredients for 8 pancakes: 200 g of flour, 2 eggs and 300 ml of milk. Work out the amounts needed for 20 pancakes.

    Show answerHide answer

    Model answer

    The scale factor is \(20 \div 8 = 2.5\). Flour: \(200 \times 2.5 = 500\) g. Eggs: \(2 \times 2.5 = 5\). Milk: \(300 \times 2.5 = 750\) ml.

    Mark scheme

    • Scale factor \(2.5\) or the amounts for 4 pancakes found (100 g, 1 egg, 150 ml) — M1
    • Two of the three amounts correct — A1
    • 500 g, 5 eggs and 750 ml — A1
  2. 2 Work out [3 marks]

    5 identical tins of beans cost \(\pounds 3.40\) altogether. Work out the cost of 8 of these tins.

    Show answerHide answer

    Model answer

    One tin costs \(3.40 \div 5 = \pounds 0.68\), so 8 tins cost \(8 \times 0.68 = \pounds 5.44\).

    Mark scheme

    • \(3.40 \div 5 = 0.68\) — M1
    • \(8 \times 0.68\) or \(8 \times 68\) — M1
    • \(\pounds 5.44\) — A1
  3. 3 Work out [3 marks]

    Shampoo is sold in two sizes. Large bottle (750 ml) costs \(\pounds 4.50\) Small bottle (400 ml) costs \(\pounds 2.60\) Which bottle is the better value for money? You must show how you get your answer.

    Show answerHide answer

    Model answer

    Large bottle: \(450 \div 750 = 0.6\) pence per ml. Small bottle: \(260 \div 400 = 0.65\) pence per ml. The large bottle costs less per millilitre, so it is better value.

    Mark scheme

    • A method to compare, such as the cost of 1 ml or the amount for 1 penny — M1
    • \(0.6\) and \(0.65\) pence per ml, or equivalent figures — A1
    • Large bottle, with a correct comparison — C1
  4. 4 Work out [3 marks]

    The exchange rate is \(\pounds 1 = \$1.25\). A jacket costs \(\$60\) in New York and \(\pounds 50\) in London. In which city is the jacket cheaper? You must show your working.

    Show answerHide answer

    Model answer

    Change the New York price to pounds: \(60 \div 1.25 = 48\), so it costs \(\pounds 48\). Alternatively \(\pounds 50 = 50 \times 1.25 = \$62.50\). The jacket is cheaper in New York.

    Mark scheme

    • \(60 \div 1.25\) or \(50 \times 1.25\) — M1
    • \(\pounds 48\) or \(\$62.50\) — A1
    • New York, with a correct comparison — C1
  5. 5 Work out [3 marks]

    It takes 4 workers 6 days to paint a house. All the workers paint at the same rate. Work out how long it would take 3 workers to paint the house.

    Show answerHide answer

    Model answer

    The job needs \(4 \times 6 = 24\) worker-days. With 3 workers it takes \(24 \div 3 = 8\) days.

    Mark scheme

    • \(4 \times 6 = 24\) — M1
    • \(24 \div 3\) — M1
    • 8 days — A1
  6. 6 Work out [4 marks]

    \(y\) is inversely proportional to \(x\). When \(x = 5\), \(y = 8\). (a) Find a formula for \(y\) in terms of \(x\). (2 marks) (b) Work out the value of \(y\) when \(x = 10\). (2 marks)

    Show answerHide answer

    Model answer

    (a) \(y = \dfrac{k}{x}\), so \(8 = \dfrac{k}{5}\) and \(k = 40\). The formula is \(y = \dfrac{40}{x}\). (b) \(y = \dfrac{40}{10} = 4\).

    Mark scheme

    • (a) \(y = \dfrac{k}{x}\) with \(k = 40\) found — M1
    • (a) \(y = \dfrac{40}{x}\) — A1
    • (b) \(\dfrac{40}{10}\) — M1
    • (b) 4 — A1

Quick check

  1. 1

    5 pens cost \(\pounds 3.50\). How much do 8 pens cost?

    1. A\(\pounds 4.50\)
    2. B\(\pounds 6.30\)
    3. C\(\pounds 5.60\)
    4. D\(\pounds 2.80\)
    Show answerHide answer

    C: \(\pounds 5.60\)

    One pen costs \(3.50 \div 5 = 0.70\). Then 8 pens cost \(8 \times 0.70 = 5.60\).

  2. 2

    A recipe for 4 people uses 300 g of flour. How much flour is needed for 10 people?

    1. A600 g
    2. B750 g
    3. C1200 g
    4. D450 g
    Show answerHide answer

    B: 750 g

    For one person, \(300 \div 4 = 75\) g. For 10 people, \(75 \times 10 = 750\) g.

  3. 3

    Which of these shows that \(y\) is directly proportional to \(x\)?

    1. A\(y = 3x\)
    2. B\(y = 3 + x\)
    3. C\(y = \dfrac{3}{x}\)
    4. D\(y = x^2\)
    Show answerHide answer

    A: \(y = 3x\)

    In direct proportion \(y\) is a constant multiple of \(x\), so \(y = 3x\), which makes a straight line through the origin.

  4. 4

    6 workers take 10 days to build a wall. How long would 15 workers take, working at the same rate?

    1. A25 days
    2. B9 days
    3. C1.5 days
    4. D4 days
    Show answerHide answer

    D: 4 days

    The job takes \(6 \times 10 = 60\) worker-days. With 15 workers it takes \(60 \div 15 = 4\) days.

  5. 5

    Which pack of cereal is the best value?

    1. A500 g for \(\pounds 2.00\)
    2. B1 kg for \(\pounds 3.95\)
    3. C750 g for \(\pounds 2.85\)
    4. D2 kg for \(\pounds 7.80\)
    Show answerHide answer

    C: 750 g for \(\pounds 2.85\)

    The cost per 100 g is 40p, 38p, 39.5p and 39p. The 750 g pack is the cheapest per 100 g.

  6. 6

    \(\pounds 1 = \euro 1.20\). How many euros do you get for \(\pounds 50\)?

    1. A\(\euro 41.67\)
    2. B\(\euro 60\)
    3. C\(\euro 51.20\)
    4. D\(\euro 600\)
    Show answerHide answer

    B: \(\euro 60\)

    Multiply by the exchange rate: \(50 \times 1.2 = 60\).

  7. 7

    Two taps fill a tank in 30 minutes. How long would 5 identical taps take?

    1. A12 minutes
    2. B75 minutes
    3. C6 minutes
    4. D15 minutes
    Show answerHide answer

    A: 12 minutes

    This is inverse proportion. The total is \(2 \times 30 = 60\) tap-minutes, so 5 taps need \(60 \div 5 = 12\) minutes.

  8. 8

    \(y\) is directly proportional to \(x\). \(y = 12\) when \(x = 3\). Work out \(y\) when \(x = 7\).

    1. A16
    2. B84
    3. C36
    4. D28
    Show answerHide answer

    D: 28

    \(y = kx\) and \(12 = 3k\), so \(k = 4\). Then \(y = 4 \times 7 = 28\).

  9. 9

    \(y\) is inversely proportional to \(x\). \(y = 6\) when \(x = 4\). Work out \(y\) when \(x = 3\).

    1. A4.5
    2. B18
    3. C8
    4. D2
    Show answerHide answer

    C: 8

    \(y = \dfrac{k}{x}\) and \(6 = \dfrac{k}{4}\), so \(k = 24\). Then \(y = \dfrac{24}{3} = 8\).

Speed, Distance, Time and Density

Just this lesson
  1. 1 Work out [3 marks]

    A car travels 135 miles in 2 hours 30 minutes. Work out the average speed of the car in miles per hour.

    Show answerHide answer

    Model answer

    2 hours 30 minutes is 2.5 hours. Speed \(= 135 \div 2.5 = 54\) mph.

    Mark scheme

    • Time converted to 2.5 hours, or \(135 \div 5 = 27\) seen — M1
    • \(135 \div 2.5\) or \(135 \times 2 \div 5\) — M1
    • 54 mph — A1
  2. 2 Work out [3 marks]

    A cyclist travels at an average speed of 18 km/h for 2 hours 20 minutes. Work out the distance she travels.

    Show answerHide answer

    Model answer

    2 hours 20 minutes is \(2\dfrac{1}{3}\) hours, so the distance is \(18 \times 2\dfrac{1}{3} = 36 + 6 = 42\) km.

    Mark scheme

    • Time as \(2\dfrac{1}{3}\) or \(\dfrac{7}{3}\) hours — M1
    • \(18 \times \dfrac{7}{3}\) or \(36 + 6\) — M1
    • 42 km — A1
  3. 3 Work out [5 marks]

    Anna drives from home to her gran's house. The graph shows her journey. (a) For how long did Anna stop? (1 mark) (b) Work out Anna's speed during the first hour of her journey. (1 mark) (c) Work out Anna's average speed for the whole journey. (3 marks)

    A distance-time graph of a journey: 45 km in the first hour, a stop of one hour, then a steady journey reaching 150 km after 5 hours.
    Show answerHide answer

    Model answer

    (a) The flat part runs from 1 hour to 2 hours, so she stopped for 1 hour. (b) She travelled 45 km in 1 hour, so her speed was 45 km/h. (c) The whole journey was 150 km in 5 hours, so the average speed is \(150 \div 5 = 30\) km/h.

    Mark scheme

    • (a) 1 hour — B1
    • (b) 45 km/h — B1
    • (c) Total distance 150 km and total time 5 hours seen — M1
    • (c) \(150 \div 5\) — M1
    • (c) 30 km/h — A1
  4. 4 Explain [4 marks]

    Raj drives 90 km at an average speed of 45 km/h. He then drives a further 90 km at an average speed of 90 km/h. Raj says, “My average speed for the whole journey was 67.5 km/h.” Is Raj correct? You must show your working.

    Show answerHide answer

    Model answer

    The first part takes \(90 \div 45 = 2\) hours and the second part takes \(90 \div 90 = 1\) hour. The total is 180 km in 3 hours, so the average speed is \(180 \div 3 = 60\) km/h. Raj has averaged the two speeds, which is wrong, so he is not correct.

    Mark scheme

    • Times \(2\) hours and \(1\) hour found — M1
    • Total distance \(180\) km and total time \(3\) hours — M1
    • \(180 \div 3 = 60\) — A1
    • No, with the correct average speed shown — C1
  5. 5 Work out [2 marks]

    A block of wood has a volume of 400 cm\(^3\). The density of the wood is 0.6 g/cm\(^3\). Work out the mass of the block.

    Show answerHide answer

    Model answer

    Mass \(=\) density \(\times\) volume \(= 0.6 \times 400 = 240\) g.

    Mark scheme

    • \(0.6 \times 400\) — M1
    • 240 g — A1
  6. 6 Work out [4 marks]

    (a) Change a speed of 54 km/h to metres per second. (2 marks) (b) A box exerts a force of 450 N on a floor. The area of the box in contact with the floor is 0.9 m\(^2\). Work out the pressure on the floor in N/m\(^2\). (2 marks)

    Show answerHide answer

    Model answer

    (a) \(54 \times 1000 \div 3600 = 15\) m/s. (b) Pressure \(=\) force \(\div\) area \(= 450 \div 0.9 = 500\) N/m\(^2\).

    Mark scheme

    • (a) \(54 \times 1000 \div 3600\) or \(54 \div 3.6\) — M1
    • (a) 15 — A1
    • (b) \(450 \div 0.9\) — M1
    • (b) 500 — A1

Quick check

  1. 1

    A car travels 150 km in 2 hours 30 minutes. What is its average speed?

    1. A375 km/h
    2. B75 km/h
    3. C50 km/h
    4. D60 km/h
    Show answerHide answer

    D: 60 km/h

    Write 2 hours 30 minutes as 2.5 hours. Then \(150 \div 2.5 = 60\) km/h.

  2. 2

    What is 45 minutes as a decimal of an hour?

    1. A0.45 hours
    2. B4.5 hours
    3. C0.75 hours
    4. D1.25 hours
    Show answerHide answer

    C: 0.75 hours

    \(45 \div 60 = 0.75\).

  3. 3

    A train travels at 80 km/h for 3 hours 15 minutes. How far does it go?

    1. A252 km
    2. B260 km
    3. C240 km
    4. D320 km
    Show answerHide answer

    B: 260 km

    3 hours 15 minutes is 3.25 hours, so the distance is \(80 \times 3.25 = 260\) km. Writing the time as 3.15 hours is a common error.

  4. 4

    A cyclist rides 60 km at 30 km/h and then 60 km at 60 km/h. What is the average speed for the whole journey?

    1. A40 km/h
    2. B45 km/h
    3. C50 km/h
    4. D90 km/h
    Show answerHide answer

    A: 40 km/h

    The times are 2 hours and 1 hour. The total distance is 120 km and the total time is 3 hours, so \(120 \div 3 = 40\) km/h.

  5. 5

    What does a horizontal line on a distance-time graph show?

    1. AThe object has constant speed
    2. BThe object is speeding up
    3. CThe object is returning to the start
    4. DThe object is stationary
    Show answerHide answer

    D: The object is stationary

    A flat line means the distance is not changing with time, so the speed is zero.

  6. 6

    A metal block has mass 240 g and volume 30 cm\(^3\). What is its density?

    1. A7200 g/cm\(^3\)
    2. B0.125 g/cm\(^3\)
    3. C8 g/cm\(^3\)
    4. D270 g/cm\(^3\)
    Show answerHide answer

    C: 8 g/cm\(^3\)

    Density \(=\) mass \(\div\) volume \(= 240 \div 30 = 8\) g/cm\(^3\).

  7. 7

    The density of a metal is 8 g/cm\(^3\). What is the mass of 25 cm\(^3\) of it?

    1. A3.125 g
    2. B200 g
    3. C33 g
    4. D0.32 g
    Show answerHide answer

    B: 200 g

    Mass \(=\) density \(\times\) volume \(= 8 \times 25 = 200\) g.

  8. 8

    Change 72 km/h into metres per second.

    1. A20 m/s
    2. B259.2 m/s
    3. C72 000 m/s
    4. D1.2 m/s
    Show answerHide answer

    A: 20 m/s

    \(72 \times 1000 = 72\,000\) m per hour, and \(72\,000 \div 3600 = 20\) m/s.

  9. 9

    A force of 600 N acts on an area of 0.5 m\(^2\). What is the pressure?

    1. A300 N/m\(^2\)
    2. B600.5 N/m\(^2\)
    3. C0.0008 N/m\(^2\)
    4. D1200 N/m\(^2\)
    Show answerHide answer

    D: 1200 N/m\(^2\)

    Pressure \(=\) force \(\div\) area \(= 600 \div 0.5 = 1200\) N/m\(^2\).

Units, Conversions and Scale

Just this lesson
  1. 1 Change [2 marks]

    (a) Change 4.5 km to metres. (1 mark) (b) Change 2750 ml to litres. (1 mark)

    Show answerHide answer

    Model answer

    (a) \(4.5 \times 1000 = 4500\) m. (b) \(2750 \div 1000 = 2.75\) litres.

    Mark scheme

    • (a) 4500 m — B1
    • (b) 2.75 litres — B1
  2. 2 Work out [3 marks]

    A cake needs 125 g of butter. Maria has a 2 kg block of butter. Work out how many cakes she can make.

    Show answerHide answer

    Model answer

    \(2\text{ kg} = 2000\) g, and \(2000 \div 125 = 16\) cakes.

    Mark scheme

    • 2 kg changed to 2000 g — M1
    • \(2000 \div 125\) — M1
    • 16 — A1
  3. 3 Work out [5 marks]

    The diagram shows a scale drawing of the floor of a living room. The scale is 1 cm represents 2 m. (a) Work out the real length of the room. (1 mark) (b) Work out the real width of the room. (1 mark) Carpet costs \(\pounds 15\) per square metre. (c) Work out the cost of carpet to cover the whole floor. (3 marks)

    A floor plan of a living room measuring 6 cm by 3.5 cm on the drawing, with the scale 1 cm represents 2 m.
    Show answerHide answer

    Model answer

    (a) \(6 \times 2 = 12\) m. (b) \(3.5 \times 2 = 7\) m. (c) The area is \(12 \times 7 = 84\) m\(^2\), so the cost is \(84 \times 15 = \pounds 1260\).

    Mark scheme

    • (a) 12 m — B1
    • (b) 7 m — B1
    • (c) \(12 \times 7 = 84\) — M1
    • (c) \(84 \times 15\) — M1
    • (c) \(\pounds 1260\) — A1
  4. 4 Work out [3 marks]

    A map has a scale of \(1 : 25\,000\). Two villages are 6.4 cm apart on the map. Work out the real distance between the villages in kilometres.

    Show answerHide answer

    Model answer

    \(6.4 \times 25\,000 = 160\,000\) cm. This is \(1600\) m, which is 1.6 km.

    Mark scheme

    • \(6.4 \times 25\,000\) — M1
    • \(160\,000\) cm or \(1600\) m — M1
    • 1.6 km — A1
  5. 5 Work out [3 marks]

    A rectangle measures 2.5 m by 40 cm. Work out the area of the rectangle in cm\(^2\).

    Show answerHide answer

    Model answer

    \(2.5\text{ m} = 250\) cm, so the area is \(250 \times 40 = 10\,000\) cm\(^2\).

    Mark scheme

    • 2.5 m changed to 250 cm, or 40 cm changed to 0.4 m — M1
    • \(250 \times 40\) — M1
    • \(10\,000\) cm\(^2\) — A1
  6. 6 Change [4 marks]

    (a) Change 3.5 m\(^2\) to cm\(^2\). (2 marks) (b) Change \(4\,000\,000\) cm\(^3\) to m\(^3\). (2 marks)

    Show answerHide answer

    Model answer

    (a) \(1\text{ m}^2 = 100 \times 100 = 10\,000\) cm\(^2\), so \(3.5 \times 10\,000 = 35\,000\) cm\(^2\). (b) \(1\text{ m}^3 = 1\,000\,000\) cm\(^3\), so \(4\,000\,000 \div 1\,000\,000 = 4\) m\(^3\).

    Mark scheme

    • (a) \(3.5 \times 10\,000\) or \(350 \times 100\) — M1
    • (a) \(35\,000\) — A1
    • (b) \(4\,000\,000 \div 1\,000\,000\) — M1
    • (b) 4 — A1

Quick check

  1. 1

    Change 3.6 km to metres.

    1. A3600 m
    2. B360 m
    3. C36 m
    4. D36 000 m
    Show answerHide answer

    A: 3600 m

    Multiply by 1000: \(3.6 \times 1000 = 3600\).

  2. 2

    Change 2450 g to kilograms.

    1. A24.5 kg
    2. B0.245 kg
    3. C2450 kg
    4. D2.45 kg
    Show answerHide answer

    D: 2.45 kg

    Divide by 1000: \(2450 \div 1000 = 2.45\).

  3. 3

    Change 0.75 litres to millilitres.

    1. A75 ml
    2. B7.5 ml
    3. C750 ml
    4. D7500 ml
    Show answerHide answer

    C: 750 ml

    Multiply by 1000: \(0.75 \times 1000 = 750\).

  4. 4

    Change 250 cm to metres.

    1. A25 m
    2. B2.5 m
    3. C0.25 m
    4. D2500 m
    Show answerHide answer

    B: 2.5 m

    Divide by 100: \(250 \div 100 = 2.5\).

  5. 5

    A map has a scale of \(1 : 50\,000\). Two towns are 4 cm apart on the map. What is the real distance?

    1. A2 km
    2. B200 km
    3. C20 km
    4. D0.2 km
    Show answerHide answer

    A: 2 km

    \(4 \times 50\,000 = 200\,000\) cm. Divide by 100 to get 2000 m, then by 1000 to get 2 km.

  6. 6

    A scale drawing uses 1 cm to represent 2 m. A line on the drawing is 7 cm. How long is it in real life?

    1. A3.5 m
    2. B9 m
    3. C140 m
    4. D14 m
    Show answerHide answer

    D: 14 m

    Multiply by the scale: \(7 \times 2 = 14\) m.

  7. 7

    A cake uses 250 g of flour. How many cakes can be made from a 2 kg bag?

    1. A80
    2. B125
    3. C8
    4. D500
    Show answerHide answer

    C: 8

    \(2\text{ kg} = 2000\) g, and \(2000 \div 250 = 8\).

  8. 8

    How many cm\(^2\) are in 1 m\(^2\)?

    1. A100
    2. B10 000
    3. C1000
    4. D1 000 000
    Show answerHide answer

    B: 10 000

    A square metre is a square of side 100 cm, so its area is \(100 \times 100 = 10\,000\) cm\(^2\).

  9. 9

    Change 5 m\(^3\) to cm\(^3\).

    1. A5 000 000 cm\(^3\)
    2. B500 000 cm\(^3\)
    3. C500 cm\(^3\)
    4. D5000 cm\(^3\)
    Show answerHide answer

    A: 5 000 000 cm\(^3\)

    \(1\text{ m}^3 = 100^3 = 1\,000\,000\) cm\(^3\), so \(5\text{ m}^3 = 5\,000\,000\) cm\(^3\).

Repeated Percentage Change and Compound Interest

Just this lesson
  1. 1 Work out [3 marks]

    A coat costs \(\pounds 80\). In a sale the price is reduced by 15%. Work out the sale price of the coat.

    Show answerHide answer

    Model answer

    10% of 80 is 8 and 5% is 4, so 15% is 12. The sale price is \(80 - 12 = \pounds 68\).

    Mark scheme

    • 10% \(= 8\) or 5% \(= 4\) or 15% \(= 12\) seen — M1
    • \(80 - 12\) or \(80 \times 0.85\) — M1
    • \(\pounds 68\) — A1
  2. 2 Work out [3 marks]

    Sam invests \(\pounds 5000\) for 2 years at 4% per year compound interest. Work out the value of his investment at the end of 2 years.

    Show answerHide answer

    Model answer

    After year 1: \(5000 \times 1.04 = 5200\). After year 2: \(5200 \times 1.04 = 5408\). The value is \(\pounds 5408\).

    Mark scheme

    • \(5000 \times 1.04\) or \(5000 \times 1.04^2\) — M1
    • \(5200 \times 1.04\) or \(5200 + 208\) — M1
    • \(\pounds 5408\) — A1
  3. 3 Work out [3 marks]

    The value of a car is \(\pounds 9000\). Each year the value of the car falls by 10% of its value at the start of that year. Work out the value of the car after 2 years.

    Show answerHide answer

    Model answer

    After year 1: \(9000 \times 0.9 = 8100\). After year 2: \(8100 \times 0.9 = 7290\). The value is \(\pounds 7290\).

    Mark scheme

    • \(9000 \times 0.9\) or \(9000 - 900\) — M1
    • \(8100 \times 0.9\) or \(8100 - 810\) — M1
    • \(\pounds 7290\) — A1
  4. 4 Work out [4 marks]

    Mia invests \(\pounds 1000\) for 3 years at 5% per year simple interest. Ben invests \(\pounds 1000\) for 3 years at 5% per year compound interest. Who has more money at the end of the 3 years? You must show your working.

    Show answerHide answer

    Model answer

    Mia: \(5\%\) of 1000 is 50, so after 3 years she has \(1000 + 3 \times 50 = \pounds 1150\). Ben: \(1000 \times 1.05 = 1050\), then \(1050 \times 1.05 = 1102.50\), then \(1102.50 \times 1.05 = 1157.625\), so about \(\pounds 1157.63\). Ben has more.

    Mark scheme

    • Simple interest \(\pounds 50\) a year or \(\pounds 1150\) — M1
    • \(1000 \times 1.05\) and \(1050 \times 1.05\) or \(1000 \times 1.05^3\) — M1
    • \(\pounds 1157.63\) or \(\pounds 1157.625\) — A1
    • Ben, with a correct comparison — C1
  5. 5 Work out [3 marks]

    In a sale, all prices are reduced by 20%. The sale price of a bag is \(\pounds 68\). Work out the original price of the bag.

    Show answerHide answer

    Model answer

    The sale price is 80% of the original, so the multiplier is 0.8. The original price is \(68 \div 0.8 = \pounds 85\).

    Mark scheme

    • 80% or 0.8 seen — M1
    • \(68 \div 0.8\) or \(68 \div 80 \times 100\) — M1
    • \(\pounds 85\) — A1
  6. 6 Work out [4 marks]

    The population of a town increases by 10% each year. At the start of 2024 the population was 20 000. (a) Work out the population at the start of 2026. (3 marks) (b) Show that the population increased by 21% over the two years. (1 mark)

    Show answerHide answer

    Model answer

    (a) After 1 year: \(20\,000 \times 1.1 = 22\,000\). After 2 years: \(22\,000 \times 1.1 = 24\,200\). (b) \(1.1^2 = 1.21\), which is an increase of 21%. Check: \(24\,200 \div 20\,000 = 1.21\).

    Mark scheme

    • (a) \(20\,000 \times 1.1\) — M1
    • (a) \(22\,000 \times 1.1\) — M1
    • (a) \(24\,200\) — A1
    • (b) \(1.1^2 = 1.21\) or \(24\,200 \div 20\,000 = 1.21\) — C1

Quick check

  1. 1

    What is the multiplier for an increase of 15%?

    1. A0.15
    2. B1.15
    3. C0.85
    4. D1.015
    Show answerHide answer

    B: 1.15

    An increase of 15% gives \(100\% + 15\% = 115\%\), which is 1.15.

  2. 2

    What is the multiplier for a decrease of 12%?

    1. A0.88
    2. B0.12
    3. C1.12
    4. D0.012
    Show answerHide answer

    A: 0.88

    \(100\% - 12\% = 88\%\), which is 0.88.

  3. 3

    A phone costs \(\pounds 400\). The price is reduced by 15%. What is the new price?

    1. A\(\pounds 385\)
    2. B\(\pounds 60\)
    3. C\(\pounds 460\)
    4. D\(\pounds 340\)
    Show answerHide answer

    D: \(\pounds 340\)

    \(400 \times 0.85 = 340\).

  4. 4

    \(\pounds 1000\) increases by 10% each year for 2 years. What is it worth after 2 years?

    1. A\(\pounds 1200\)
    2. B\(\pounds 1100\)
    3. C\(\pounds 1210\)
    4. D\(\pounds 1110\)
    Show answerHide answer

    C: \(\pounds 1210\)

    \(1000 \times 1.1 = 1100\) and \(1100 \times 1.1 = 1210\). The second increase is 10% of the new amount.

  5. 5

    \(\pounds 2000\) is invested at 5% compound interest per year. What is it worth after 2 years?

    1. A\(\pounds 2200\)
    2. B\(\pounds 2205\)
    3. C\(\pounds 2100\)
    4. D\(\pounds 2210\)
    Show answerHide answer

    B: \(\pounds 2205\)

    \(2000 \times 1.05^2 = 2000 \times 1.1025 = 2205\). The simple interest answer would be \(\pounds 2200\).

  6. 6

    A car worth \(\pounds 12\,000\) loses 20% of its value each year. What is it worth after 2 years?

    1. A\(\pounds 7680\)
    2. B\(\pounds 7200\)
    3. C\(\pounds 9600\)
    4. D\(\pounds 6144\)
    Show answerHide answer

    A: \(\pounds 7680\)

    \(12\,000 \times 0.8 = 9600\) and \(9600 \times 0.8 = 7680\). Taking 40% off in one go would give \(\pounds 7200\).

  7. 7

    Which statement about interest is correct?

    1. ASimple interest grows faster than compound interest
    2. BSimple and compound interest are always equal
    3. CCompound interest adds the same amount every year
    4. DAfter the first year, compound interest is greater than simple interest
    Show answerHide answer

    D: After the first year, compound interest is greater than simple interest

    Compound interest is paid on interest already earned, so it grows faster than simple interest, which is always paid on the original amount.

  8. 8

    After a 25% increase, a house is worth \(\pounds 150\,000\). What was it worth before the increase?

    1. A\(\pounds 112\,500\)
    2. B\(\pounds 187\,500\)
    3. C\(\pounds 120\,000\)
    4. D\(\pounds 125\,000\)
    Show answerHide answer

    C: \(\pounds 120\,000\)

    The multiplier is 1.25, so the original is \(150\,000 \div 1.25 = 120\,000\).

  9. 9

    A population of 8000 falls by 10% each year. What is the population after 2 years?

    1. A6400
    2. B6480
    3. C7200
    4. D6600
    Show answerHide answer

    B: 6480

    \(8000 \times 0.9^2 = 8000 \times 0.81 = 6480\).