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Exam questions · Maths · Ratio and Proportion

Ratio: Simplifying and Sharing

  • 6 exam questions
  • 18 marks
  • 9 quick checks
  1. 1 Write [2 marks]

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

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    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.

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    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.

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    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.

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    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.

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    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.

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    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\(6 : 9\)
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    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\(5 : 3\)
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    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\(1 : 3\)
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    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\(\pounds 40\)
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    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
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    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\(\dfrac{2}{3}\)
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    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
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    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\(\pounds 20\)
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    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\(6 : 12 : 15\)
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    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 \(A : B : C = 8 : 12 : 15\).