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Exam questions · Maths

Ratio and Proportion

  • 30 exam questions
  • 90 marks
  • 45 quick checks

Ratio: Simplifying and Sharing

Just this lesson
  1. 1 Write [2 marks]

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

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    Model answer

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

    Mark scheme

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

    \(\pounds 90\) is shared in the ratio \(2 : 7\). Calculate the larger share.

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    Model answer

    There are \(2 + 7 = 9\) parts, and one part is \(90 \div 9 = 10\). The larger share is \(7 \times 10 = \pounds 70\).

    Mark scheme

    • \(2 + 7 = 9\) or \(90 \div 9\) — M1
    • \(7 \times 10\) — M1
    • \(\pounds 70\) — A1
  3. 3 Calculate [3 marks]

    A fruit drink is made from orange juice, mango juice and pineapple juice in the ratio \(3 : 2 : 1\). A bottle holds 1.2 litres of the drink. Calculate the volume of mango juice in the bottle. Give your answer in millilitres.

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    Model answer

    There are \(3 + 2 + 1 = 6\) parts. \(1.2\text{ litres} = 1200\) ml, so one part is \(1200 \div 6 = 200\) ml. The mango juice is \(2 \times 200 = 400\) ml.

    Mark scheme

    • \(1200 \div 6 = 200\) — M1
    • \(2 \times 200\) — M1
    • 400 ml — A1
  4. 4 Calculate [3 marks]

    At a concert the ratio of adults to children is \(3 : 7\). There are 80 more children than adults. Calculate the total number of people at the concert.

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    Model answer

    The difference is \(7 - 3 = 4\) parts, and 4 parts \(= 80\), so one part is 20. The total is \(10 \times 20 = 200\).

    Mark scheme

    • \(80 \div 4 = 20\) — M1
    • \(10 \times 20\) or \(60 + 140\) — M1
    • 200 — A1
  5. 5 Calculate [3 marks]

    The ages of Sam and Tom are in the ratio \(4 : 5\). The sum of their ages is 36 years. Calculate Tom's age.

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    Model answer

    There are \(4 + 5 = 9\) parts, so one part is \(36 \div 9 = 4\) years. Tom is \(5 \times 4 = 20\) years old.

    Mark scheme

    • \(36 \div 9 = 4\) — M1
    • \(5 \times 4\) — M1
    • 20 — A1
  6. 6 Find [4 marks]

    \(a : b = 5 : 2\) and \(b : c = 3 : 4\). (a) Find \(a : b : c\). [2 marks] (b) \(a + b + c = 87\). Calculate the value of \(c\). [2 marks]

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    Model answer

    (a) Make \(b\) the same in both ratios, using the lowest common multiple of 2 and 3, which is 6. Then \(a : b = 15 : 6\) and \(b : c = 6 : 8\), so \(a : b : c = 15 : 6 : 8\). (b) There are \(15 + 6 + 8 = 29\) parts, so one part is \(87 \div 29 = 3\) and \(c = 8 \times 3 = 24\).

    Mark scheme

    • (a) \(15 : 6\) or \(6 : 8\) seen — M1
    • (a) \(15 : 6 : 8\) — A1
    • (b) \(87 \div 29 = 3\) — M1
    • (b) 24 — 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 Calculate [3 marks]

    A recipe for 12 biscuits uses 180 g of flour, 90 g of sugar and 60 g of butter. Calculate the amount of each ingredient needed for 30 biscuits.

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    Model answer

    The scale factor is \(30 \div 12 = 2.5\). Flour: \(180 \times 2.5 = 450\) g. Sugar: \(90 \times 2.5 = 225\) g. Butter: \(60 \times 2.5 = 150\) g.

    Mark scheme

    • Scale factor 2.5, or the amounts for 6 biscuits found — M1
    • Two of the three amounts correct — A1
    • 450 g, 225 g and 150 g — A1
  2. 2 Calculate [2 marks]

    8 pencils cost \(\pounds 5.20\). Calculate the cost of 5 pencils.

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    Model answer

    One pencil costs \(5.20 \div 8 = \pounds 0.65\), so 5 pencils cost \(5 \times 0.65 = \pounds 3.25\).

    Mark scheme

    • \(5.20 \div 8 = 0.65\) — M1
    • \(\pounds 3.25\) — A1
  3. 3 Calculate [3 marks]

    A shop sells potatoes in two bags. 3 kg bag: \(\pounds 2.40\) 5 kg bag: \(\pounds 3.75\) Which bag is the better value for money? You must show your working.

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    Model answer

    3 kg bag: \(2.40 \div 3 = \pounds 0.80\) per kg. 5 kg bag: \(3.75 \div 5 = \pounds 0.75\) per kg. The 5 kg bag is cheaper per kilogram, so it is better value.

    Mark scheme

    • A method to compare, such as the cost per kg — M1
    • \(\pounds 0.80\) and \(\pounds 0.75\) per kg, or equivalent — A1
    • 5 kg bag, with a correct comparison — A1
  4. 4 Calculate [2 marks]

    The exchange rate is \(\pounds 1 = \$1.30\). Calculate how many pounds you get for \(\$390\).

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    Model answer

    \(390 \div 1.3 = 300\), so you get \(\pounds 300\).

    Mark scheme

    • \(390 \div 1.3\) — M1
    • \(\pounds 300\) — A1
  5. 5 Calculate [2 marks]

    6 people take 8 hours to build a shed. All the people work at the same rate. Calculate how long it takes 4 people to build the shed.

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    Model answer

    The job needs \(6 \times 8 = 48\) person-hours, and \(48 \div 4 = 12\) hours.

    Mark scheme

    • \(6 \times 8 = 48\) — M1
    • 12 hours — A1
  6. 6 Find [4 marks]

    \(y\) is directly proportional to \(x\). When \(x = 4\), \(y = 10\). (a) Find a formula for \(y\) in terms of \(x\). [2 marks] (b) Find the value of \(x\) when \(y = 25\). [2 marks]

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    Model answer

    (a) \(y = kx\), so \(10 = 4k\) and \(k = 2.5\). The formula is \(y = 2.5x\). (b) \(25 = 2.5x\), so \(x = 25 \div 2.5 = 10\).

    Mark scheme

    • (a) \(y = kx\) with \(k = 2.5\) found — M1
    • (a) \(y = 2.5x\) — A1
    • (b) \(25 \div 2.5\) — M1
    • (b) 10 — 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 Calculate [3 marks]

    A van travels 180 km in 2 hours 24 minutes. Calculate the average speed of the van in kilometres per hour.

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    Model answer

    2 hours 24 minutes is \(2 + \dfrac{24}{60} = 2.4\) hours. Speed \(= 180 \div 2.4 = 75\) km/h.

    Mark scheme

    • 24 minutes written as 0.4 hours, or 2.4 hours seen — M1
    • \(180 \div 2.4\) — M1
    • 75 km/h — A1
  2. 2 Calculate [2 marks]

    A plane flies at an average speed of 600 km/h for 1 hour 30 minutes. Calculate the distance the plane flies.

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    Model answer

    1 hour 30 minutes is 1.5 hours, so the distance is \(600 \times 1.5 = 900\) km.

    Mark scheme

    • \(600 \times 1.5\) — M1
    • 900 km — A1
  3. 3 Calculate [5 marks]

    The graph shows Mia's journey from home to her cousin's house. (a) How far did Mia travel in the last 3 hours of her journey? [1 mark] (b) Calculate Mia's speed in the last 3 hours. [2 marks] (c) Mia says, “I travelled faster in the first hour than I did in the last 3 hours.” Is Mia correct? You must show your working. [2 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) At 2 hours she was 45 km from home and at 5 hours 150 km, so she travelled \(150 - 45 = 105\) km. (b) \(105 \div 3 = 35\) km/h. (c) Her speed in the first hour was \(45 \div 1 = 45\) km/h. Since \(45 > 35\), Mia is correct.

    Mark scheme

    • (a) 105 km — B1
    • (b) \(105 \div 3\) — M1
    • (b) 35 km/h — A1
    • (c) 45 km/h for the first hour — M1
    • (c) Yes, with 45 compared with 35 — A1
  4. 4 Calculate [2 marks]

    A coach travels 150 km at an average speed of 60 km/h. Calculate the time taken. Give your answer in hours and minutes.

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    Model answer

    Time \(= 150 \div 60 = 2.5\) hours, which is 2 hours 30 minutes.

    Mark scheme

    • \(150 \div 60\) — M1
    • 2 hours 30 minutes — A1
  5. 5 Calculate [3 marks]

    A solid metal cube has sides of length 5 cm and a mass of 1 kg. Calculate the density of the metal in g/cm\(^3\).

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    Model answer

    The volume is \(5 \times 5 \times 5 = 125\) cm\(^3\) and the mass is 1000 g. Density \(= 1000 \div 125 = 8\) g/cm\(^3\).

    Mark scheme

    • \(5^3 = 125\) — M1
    • \(1000 \div 125\) — M1
    • 8 g/cm\(^3\) — A1
  6. 6 Calculate [4 marks]

    (a) A box exerts a force of 900 N on a surface of area 1.5 m\(^2\). Calculate the pressure in N/m\(^2\). [2 marks] (b) Convert a speed of 108 km/h to metres per second. [2 marks]

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    Model answer

    (a) Pressure \(= 900 \div 1.5 = 600\) N/m\(^2\). (b) \(108 \times 1000 \div 3600 = 30\) m/s.

    Mark scheme

    • (a) \(900 \div 1.5\) — M1
    • (a) 600 — A1
    • (b) \(108 \times 1000 \div 3600\) or \(108 \div 3.6\) — M1
    • (b) 30 — 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 Convert [2 marks]

    (a) Convert 0.45 km to metres. [1 mark] (b) Convert 5.2 kg to grams. [1 mark]

    Show answerHide answer

    Model answer

    (a) \(0.45 \times 1000 = 450\) m. (b) \(5.2 \times 1000 = 5200\) g.

    Mark scheme

    • (a) 450 m — B1
    • (b) 5200 g — B1
  2. 2 Calculate [2 marks]

    A bag of rice has a mass of 1.5 kg. Calculate how many 60 g portions can be made from the bag.

    Show answerHide answer

    Model answer

    \(1.5\text{ kg} = 1500\) g, and \(1500 \div 60 = 25\) portions.

    Mark scheme

    • 1500 g seen, or \(1500 \div 60\) — M1
    • 25 — A1
  3. 3 Calculate [5 marks]

    The diagram shows a scale drawing of a living room. The scale is 1 cm represents 2 m. (a) Calculate the real length of the room. [1 mark] (b) Calculate the real width of the room. [1 mark] A rug has an area of 21 m\(^2\). (c) What percentage of the floor of the room is covered by the rug? [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 floor area is \(12 \times 7 = 84\) m\(^2\), so the rug covers \(\dfrac{21}{84} = \dfrac{1}{4} = 25\%\).

    Mark scheme

    • (a) 12 m — B1
    • (b) 7 m — B1
    • (c) \(12 \times 7 = 84\) — M1
    • (c) \(\dfrac{21}{84}\) or \(21 \div 84\) — M1
    • (c) 25% — A1
  4. 4 Calculate [3 marks]

    A plan of a field has a scale of \(1 : 5000\). The field is 4.5 cm long on the plan. Calculate the real length of the field in metres.

    Show answerHide answer

    Model answer

    \(4.5 \times 5000 = 22\,500\) cm, which is 225 m.

    Mark scheme

    • \(4.5 \times 5000\) — M1
    • \(22\,500\) cm — M1
    • 225 m — A1
  5. 5 Calculate [3 marks]

    A water tank has a volume of 2.4 m\(^3\). 1 m\(^3\) is 1000 litres. Calculate the volume of the tank in litres, and the number of 20-litre buckets needed to fill it.

    Show answerHide answer

    Model answer

    \(2.4 \times 1000 = 2400\) litres. The number of buckets is \(2400 \div 20 = 120\).

    Mark scheme

    • \(2.4 \times 1000\) — M1
    • 2400 litres — A1
    • 120 buckets — B1
  6. 6 Convert [4 marks]

    (a) Convert 5 m\(^2\) to cm\(^2\). [2 marks] (b) Convert \(15\,000\) mm\(^2\) to cm\(^2\). [2 marks]

    Show answerHide answer

    Model answer

    (a) \(1\text{ m}^2 = 10\,000\) cm\(^2\), so \(5 \times 10\,000 = 50\,000\) cm\(^2\). (b) \(1\text{ cm}^2 = 100\) mm\(^2\), so \(15\,000 \div 100 = 150\) cm\(^2\).

    Mark scheme

    • (a) \(5 \times 10\,000\) — M1
    • (a) \(50\,000\) — A1
    • (b) \(15\,000 \div 100\) — M1
    • (b) 150 — 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
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    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\(^3\)
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    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 Calculate [2 marks]

    Decrease \(\pounds 250\) by 12%.

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    Model answer

    10% of 250 is 25 and 1% is 2.50, so 12% is \(25 + 5 = 30\). The new amount is \(250 - 30 = \pounds 220\).

    Mark scheme

    • 12% of 250 \(= 30\), or \(250 \times 0.88\) — M1
    • \(\pounds 220\) — A1
  2. 2 Calculate [3 marks]

    Ed invests \(\pounds 6000\) for 2 years at 5% per year compound interest. Calculate the value of the investment after 2 years.

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    Model answer

    After year 1: \(6000 \times 1.05 = 6300\). After year 2: \(6300 \times 1.05 = 6615\). The value is \(\pounds 6615\).

    Mark scheme

    • \(6000 \times 1.05\) or \(6000 \times 1.05^2\) — M1
    • \(6300 \times 1.05\) — M1
    • \(\pounds 6615\) — A1
  3. 3 Calculate [3 marks]

    A laptop is worth \(\pounds 800\). Its value falls by 25% each year. Calculate the value of the laptop after 2 years.

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    Model answer

    After year 1: \(800 \times 0.75 = 600\). After year 2: \(600 \times 0.75 = 450\). The value is \(\pounds 450\).

    Mark scheme

    • \(800 \times 0.75\) — M1
    • \(600 \times 0.75\) — M1
    • \(\pounds 450\) — A1
  4. 4 Calculate [4 marks]

    \(\pounds 2000\) is invested for 2 years at 4% per year. Calculate how much more interest is earned with compound interest than with simple interest.

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    Model answer

    Simple interest: \(4\%\) of 2000 is 80, so 2 years gives \(\pounds 160\). Compound interest: \(2000 \times 1.04 = 2080\) and \(2080 \times 1.04 = 2163.20\), so the interest is \(\pounds 163.20\). The difference is \(163.20 - 160 = \pounds 3.20\).

    Mark scheme

    • Simple interest \(\pounds 160\) — M1
    • \(2000 \times 1.04\) and \(2080 \times 1.04\) — M1
    • \(\pounds 163.20\) as the compound interest or \(\pounds 2163.20\) — A1
    • \(\pounds 3.20\) — A1
  5. 5 Calculate [3 marks]

    In a sale, the price of a coat is reduced by 20%. The sale price is \(\pounds 48\). Calculate the original price of the coat.

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    Model answer

    The sale price is 80% of the original, so the multiplier is 0.8. The original price is \(48 \div 0.8 = \pounds 60\).

    Mark scheme

    • 80% or 0.8 seen — M1
    • \(48 \div 0.8\) — M1
    • \(\pounds 60\) — A1
  6. 6 Calculate [3 marks]

    A colony of bacteria increases in number by 50% every hour. There are 800 bacteria at the start. Calculate the number of bacteria after 3 hours.

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    Model answer

    After 1 hour: \(800 \times 1.5 = 1200\). After 2 hours: \(1200 \times 1.5 = 1800\). After 3 hours: \(1800 \times 1.5 = 2700\).

    Mark scheme

    • \(800 \times 1.5\) or \(800 \times 1.5^3\) — M1
    • \(1200 \times 1.5\) and \(1800 \times 1.5\) — M1
    • 2700 — A1

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
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    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

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