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Exam questions · Maths

Number Without a Calculator

  • 34 exam questions
  • 93 marks
  • 48 quick checks

Place Value, Negatives and the Four Operations

Just this lesson
  1. 1 Put [2 marks]

    Put these numbers in order, smallest first. \(0.3\) \(0.29\) \(0.302\) \(0.0399\)

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    Model answer

    Writing each number to four decimal places gives \(0.3000\), \(0.2900\), \(0.3020\) and \(0.0399\). The order is \(0.0399,\ 0.29,\ 0.3,\ 0.302\).

    Mark scheme

    • At least three numbers in the correct relative order, or all written to the same number of decimal places — M1
    • \(0.0399, 0.29, 0.3, 0.302\) — A1
  2. 2 Work out [3 marks]

    Work out \(408 \times 53\).

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    Model answer

    \(408 \times 50 = 20\,400\) and \(408 \times 3 = 1224\). Adding gives \(20\,400 + 1224 = 21\,624\).

    Mark scheme

    • A complete method, such as \(408\times50\) and \(408\times3\), or a grid or column method, with at most one error — M1
    • Adds their partial products — M1
    • 21624 — A1
  3. 3 Work out [3 marks]

    (a) Work out \(15.6 \div 0.4\). [2 marks] (b) Work out \(0.2 \times 0.03\). [1 mark]

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    Model answer

    (a) Multiply both numbers by 10: \(156 \div 4 = 39\). (b) \(2 \times 3 = 6\), and there are 3 decimal places in total, so \(0.006\).

    Mark scheme

    • (a) \(156 \div 4\) or equivalent — M1
    • (a) 39 — A1
    • (b) 0.006 — B1
  4. 4 Work out [3 marks]

    Source: Lowest temperature: Aberdeen \(-5^\circ\text{C}\), Bath \(3^\circ\text{C}\), Cardiff \(-12^\circ\text{C}\), Derby \(0^\circ\text{C}\). The table shows the lowest temperature in four towns one night. (a) Write the four temperatures in order, starting with the lowest. [1 mark] (b) Work out the difference between the highest and the lowest of the temperatures. [1 mark] The temperature in Bath then falls by \(8^\circ\text{C}\). (c) Work out the new temperature in Bath. [1 mark]

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    Model answer

    (a) \(-12,\ -5,\ 0,\ 3\). (b) \(3 - (-12) = 15^\circ\text{C}\). (c) \(3 - 8 = -5^\circ\text{C}\).

    Mark scheme

    • (a) \(-12, -5, 0, 3\) in this order — B1
    • (b) \(15^\circ\text{C}\) — B1
    • (c) \(-5^\circ\text{C}\) — B1
  5. 5 Work out [3 marks]

    Work out \(2 \times (-3)^2 - 18 \div (-3)\).

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    Model answer

    \((-3)^2 = 9\), so \(2 \times 9 = 18\). \(18 \div (-3) = -6\), so \(18 - (-6) = 18 + 6 = 24\).

    Mark scheme

    • \((-3)^2 = 9\) or \(2 \times 9 = 18\) — M1
    • \(18 \div (-3) = -6\) — M1
    • 24 — A1
  6. 6 Show that [3 marks]

    Show that \(6 + 4 \times 2.5 \div 0.5 = 26\).

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    Model answer

    Multiplication and division come before addition, working from left to right: \(4 \times 2.5 = 10\), then \(10 \div 0.5 = 20\), then \(6 + 20 = 26\).

    Mark scheme

    • \(4 \times 2.5 = 10\) — M1
    • \(10 \div 0.5 = 20\) — M1
    • \(6 + 20 = 26\) with the conclusion shown — A1
  7. 7 Work out [3 marks]

    A school hall has 24 rows of 32 chairs. In every row, 3 chairs are reserved for teachers. How many chairs are not reserved?

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    Model answer

    Each row has \(32 - 3 = 29\) chairs that are not reserved. So the total is \(24 \times 29 = 24 \times 30 - 24 = 720 - 24 = 696\).

    Mark scheme

    • \(32 - 3 = 29\) or \(24 \times 3 = 72\) — M1
    • \(24 \times 29\) or \(24 \times 32 = 768\) — M1
    • 696 — A1

Quick check

  1. 1

    Estimate \(38 \times 21\) by rounding each number to 1 significant figure.

    1. A600
    2. B80
    3. C1000
    4. D800
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    D: 800

    \(38 \approx 40\) and \(21 \approx 20\), so the estimate is \(40 \times 20 = 800\).

  2. 2

    Tickets cost £8.50 each. How many tickets can be bought with £60?

    1. A7
    2. B8
    3. C6
    4. D70
    Show answerHide answer

    A: 7

    \(8.50 \times 7 = 59.50\), which fits, and \(8.50 \times 8 = 68\), which does not.

  3. 3

    What is the value of the digit 7 in the number 4.073?

    1. A70
    2. B7 hundredths
    3. C7 tenths
    4. D7 thousandths
    Show answerHide answer

    B: 7 hundredths

    The 7 is in the second column after the decimal point, so it is worth 7 hundredths.

  4. 4

    Which of these decimals is the largest?

    1. A0.75
    2. B0.809
    3. C0.098
    4. D0.8
    Show answerHide answer

    B: 0.809

    Writing each to 3 decimal places gives 0.800, 0.750, 0.809 and 0.098, so 0.809 is largest.

  5. 5

    What is 6.4 ÷ 100?

    1. A0.064
    2. B0.64
    3. C0.0064
    4. D640
    Show answerHide answer

    A: 0.064

    Dividing by 100 moves every digit two columns to the right, giving 0.064.

  6. 6

    Work out \(-3 - (-7)\).

    1. A\(10\)
    2. B\(4\)
    3. C\(-10\)
    4. D\(-4\)
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    B: \(4\)

    Subtracting a negative is the same as adding: \(-3 + 7 = 4\).

  7. 7

    Work out \((-6) \times 3 \div (-2)\).

    1. A\(-36\)
    2. B\(-9\)
    3. C\(9\)
    4. D\(36\)
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    C: \(9\)

    \((-6)\times 3 = -18\), then \(-18 \div (-2) = 9\) because the signs are the same.

  8. 8

    Work out \(2 + 3 \times 4^2\).

    1. A\(196\)
    2. B\(22\)
    3. C\(50\)
    4. D\(80\)
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    C: \(50\)

    Indices first: \(4^2 = 16\). Then multiply: \(3 \times 16 = 48\). Then add: \(2 + 48 = 50\).

  9. 9

    Work out \(0.3 \times 0.2\).

    1. A0.06
    2. B0.006
    3. C0.5
    4. D0.6
    Show answerHide answer

    A: 0.06

    \(3 \times 2 = 6\) and there are 2 decimal places in total, so the answer is 0.06.

Factors, Multiples and Primes

Just this lesson
  1. 1 Write down [2 marks]

    Here is a list of numbers. \(6\) \(9\) \(16\) \(17\) \(27\) \(30\) (a) Write down a cube number from the list. [1 mark] (b) Write down a prime number from the list. [1 mark]

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    Model answer

    (a) \(27 = 3^3\). (b) \(17\) is prime, because its only factors are 1 and 17.

    Mark scheme

    • (a) 27 — B1
    • (b) 17 — B1
  2. 2 Write [3 marks]

    Write 126 as a product of prime factors. Give your answer in index form.

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    Model answer

    \(126 = 2 \times 63 = 2 \times 7 \times 9 = 2 \times 3 \times 3 \times 7 = 2 \times 3^2 \times 7\).

    Mark scheme

    • A correct first step, such as a factor tree with a correct first pair — M1
    • \(2 \times 3 \times 3 \times 7\) — A1
    • \(2 \times 3^2 \times 7\) — A1
  3. 3 Work out [4 marks]

    (a) Find the highest common factor (HCF) of 56 and 84. [2 marks] (b) Find the lowest common multiple (LCM) of 56 and 84. [2 marks]

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    Model answer

    \(56 = 2^3 \times 7\) and \(84 = 2^2 \times 3 \times 7\). (a) HCF \(= 2^2 \times 7 = 28\). (b) LCM \(= 2^3 \times 3 \times 7 = 168\).

    Mark scheme

    • (a) A correct list of factors or prime factors of both numbers — M1
    • (a) 28 — A1
    • (b) A correct list of multiples or a correct product of primes — M1
    • (b) 168 — A1
  4. 4 Explain [2 marks]

    (a) Aisha says, ‘All prime numbers are odd.’ Explain why Aisha is wrong. [1 mark] (b) Sam says, ‘The number 51 is prime.’ Show that Sam is wrong. [1 mark]

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    Model answer

    (a) 2 is a prime number and it is even. (b) \(51 = 3 \times 17\), so 51 has factors other than 1 and itself. The digits add to 6, so 51 is divisible by 3.

    Mark scheme

    • (a) Identifies that 2 is an even prime — B1
    • (b) \(3 \times 17\), or shows that 51 is divisible by 3 — B1
  5. 5 Work out [3 marks]

    A cog with 24 teeth meshes with a cog with 40 teeth. A tooth on each cog is marked, and at the start the two marked teeth are touching. The cogs turn. How many complete turns does each cog make before the two marked teeth next touch?

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    Model answer

    The marked teeth next touch after a number of teeth that is a common multiple of 24 and 40. \(24 = 2^3 \times 3\) and \(40 = 2^3 \times 5\), so the LCM is \(2^3 \times 3 \times 5 = 120\). The smaller cog turns \(120 \div 24 = 5\) times and the larger cog turns \(120 \div 40 = 3\) times.

    Mark scheme

    • Identifies that 120 teeth have passed, or a correct list of multiples of 24 and 40 — M1
    • 120 — A1
    • 5 turns of the smaller cog and 3 turns of the larger cog — B1
  6. 6 Work out [4 marks]

    \(N = 2^3 \times 3 \times 5^2\) (a) Work out the value of \(N\). [2 marks] (b) Is \(N\) a multiple of 45? You must give a reason for your answer. [2 marks]

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    Model answer

    (a) \(2^3 = 8\) and \(5^2 = 25\), so \(N = 8 \times 3 \times 25 = 600\). (b) No. \(45 = 3^2 \times 5\), which needs two factors of 3, but \(N\) only has one. Also \(600 \div 45 = 13.33\ldots\), which is not a whole number.

    Mark scheme

    • (a) \(8 \times 3 \times 25\) or two of \(8, 3, 25\) multiplied — M1
    • (a) 600 — A1
    • (b) Writes \(45 = 3^2 \times 5\) or attempts \(600 \div 45\) — M1
    • (b) No, with a correct reason — A1

Quick check

  1. 1

    What is the smallest whole number that 72 must be multiplied by to give a square number?

    1. A2
    2. B9
    3. C6
    4. D3
    Show answerHide answer

    A: 2

    \(72 = 2^3 \times 3^2\). The index of 2 is odd, so multiply by 2 to get \(144 = 12^2\).

  2. 2

    A teacher has 48 red pens and 72 blue pens. What is the greatest number of identical packs that she can make using all of the pens?

    1. A6
    2. B12
    3. C144
    4. D24
    Show answerHide answer

    D: 24

    The answer is the HCF of 48 and 72, which is 24.

  3. 3

    Which of these numbers is prime?

    1. A63
    2. B51
    3. C59
    4. D57
    Show answerHide answer

    C: 59

    59 has no factors other than 1 and 59. 51 = 3 × 17, 57 = 3 × 19 and 63 = 7 × 9.

  4. 4

    What is the highest common factor (HCF) of 18 and 30?

    1. A6
    2. B9
    3. C3
    4. D90
    Show answerHide answer

    A: 6

    The common factors are 1, 2, 3 and 6, and the highest of these is 6.

  5. 5

    What is the lowest common multiple (LCM) of 6 and 8?

    1. A14
    2. B48
    3. C24
    4. D2
    Show answerHide answer

    C: 24

    Multiples of 8 are 8, 16, 24 and 24 is the first one that is also a multiple of 6.

  6. 6

    Which of these is 72 written as a product of its prime factors?

    1. A\(2 \times 36\)
    2. B\(2^2 \times 3^3\)
    3. C\(2^3 \times 3^2\)
    4. D\(8 \times 9\)
    Show answerHide answer

    C: \(2^3 \times 3^2\)

    72 = 8 × 9 = 2 × 2 × 2 × 3 × 3 = \(2^3 \times 3^2\).

  7. 7

    How many factors does 20 have?

    1. A8
    2. B5
    3. C6
    4. D4
    Show answerHide answer

    C: 6

    The factors are 1, 2, 4, 5, 10 and 20, which makes six.

  8. 8

    Which of these numbers is divisible by 9?

    1. A4520
    2. B4531
    3. C4527
    4. D4528
    Show answerHide answer

    C: 4527

    The digits of 4527 add up to 18, which is a multiple of 9.

  9. 9

    Two lights flash every 8 seconds and every 12 seconds. They flash together now. After how many seconds will they next flash together?

    1. A20
    2. B96
    3. C4
    4. D24
    Show answerHide answer

    D: 24

    The LCM of 8 and 12 is 24, so they next flash together after 24 seconds.

Fractions

Just this lesson
  1. 1 Put [2 marks]

    Put these fractions in order, smallest first. \(\dfrac{5}{8}\) \(\dfrac{3}{5}\) \(\dfrac{7}{10}\)

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    Model answer

    Use a common denominator of 40: \(\dfrac{25}{40}\), \(\dfrac{24}{40}\) and \(\dfrac{28}{40}\). The order is \(\dfrac{3}{5},\ \dfrac{5}{8},\ \dfrac{7}{10}\).

    Mark scheme

    • Converts to a common denominator, or to decimals (0.625, 0.6, 0.7) — M1
    • \(\dfrac{3}{5}, \dfrac{5}{8}, \dfrac{7}{10}\) — A1
  2. 2 Work out [2 marks]

    Work out \(\dfrac{5}{6} - \dfrac{3}{8}\).

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    Model answer

    The LCM of 6 and 8 is 24. \(\dfrac{20}{24} - \dfrac{9}{24} = \dfrac{11}{24}\).

    Mark scheme

    • \(\dfrac{20}{24}\) and \(\dfrac{9}{24}\) — M1
    • \(\dfrac{11}{24}\) — A1
  3. 3 Work out [3 marks]

    Work out \(1\dfrac{2}{3} + 2\dfrac{3}{5}\). Give your answer as a mixed number.

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    Model answer

    \(1\dfrac{2}{3} = \dfrac{5}{3} = \dfrac{25}{15}\) and \(2\dfrac{3}{5} = \dfrac{13}{5} = \dfrac{39}{15}\). Adding gives \(\dfrac{64}{15} = 4\dfrac{4}{15}\).

    Mark scheme

    • A common denominator of 15, or whole numbers and fractions dealt with separately using a common denominator — M1
    • \(\dfrac{25}{15} + \dfrac{39}{15}\) or \(3 + \dfrac{10}{15} + \dfrac{9}{15}\) — M1
    • \(4\dfrac{4}{15}\) — A1
  4. 4 Work out [3 marks]

    Work out \(3\dfrac{3}{4} \times 2\dfrac{2}{5}\).

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    Model answer

    \(3\dfrac{3}{4} = \dfrac{15}{4}\) and \(2\dfrac{2}{5} = \dfrac{12}{5}\). Cancel: \(\dfrac{15}{4} \times \dfrac{12}{5} = \dfrac{3}{1} \times \dfrac{3}{1} = 9\).

    Mark scheme

    • Both mixed numbers converted to improper fractions — M1
    • \(\dfrac{15}{4} \times \dfrac{12}{5}\) or \(\dfrac{180}{20}\) — M1
    • 9 — A1
  5. 5 Work out [3 marks]

    Work out \(2\dfrac{1}{4} \div 1\dfrac{1}{8}\).

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    Model answer

    \(2\dfrac{1}{4} = \dfrac{9}{4}\) and \(1\dfrac{1}{8} = \dfrac{9}{8}\). Then \(\dfrac{9}{4} \div \dfrac{9}{8} = \dfrac{9}{4} \times \dfrac{8}{9} = 2\).

    Mark scheme

    • Both mixed numbers converted to improper fractions — M1
    • \(\dfrac{9}{4} \times \dfrac{8}{9}\) — M1
    • 2 — A1
  6. 6 Work out [3 marks]

    In a class of 30 students, \(\dfrac{2}{5}\) walk to school and \(\dfrac{1}{3}\) come by bus. The rest are driven. How many students are driven to school? You must show your working.

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    Model answer

    Walk: \(\dfrac{2}{5} \times 30 = 12\). Bus: \(\dfrac{1}{3} \times 30 = 10\). Driven: \(30 - 12 - 10 = 8\) students.

    Mark scheme

    • \(\dfrac{2}{5} \times 30 = 12\) or \(\dfrac{1}{3} \times 30 = 10\) — M1
    • \(30 - 12 - 10\) — M1
    • 8 — A1
  7. 7 Explain [3 marks]

    Lena says that \(\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{2}{5}\). (a) Explain what Lena has done wrong. [1 mark] (b) Work out the correct answer. [2 marks]

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    Model answer

    (a) Lena has added the numerators and added the denominators, instead of finding a common denominator. (b) \(\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}\).

    Mark scheme

    • (a) States that she added the numerators and the denominators — B1
    • (b) \(\dfrac{3}{6}\) and \(\dfrac{2}{6}\) — M1
    • (b) \(\dfrac{5}{6}\) — A1

Quick check

  1. 1

    \(\dfrac{3}{5}\) of a number is 36. What is the number?

    1. A60
    2. B21.6
    3. C108
    4. D12
    Show answerHide answer

    A: 60

    \(36 \div 3 = 12\) is one fifth, so the whole number is \(12 \times 5 = 60\).

  2. 2

    What fraction of 2 hours is 35 minutes? Give your answer in its simplest form.

    1. A\(\dfrac{35}{2}\)
    2. B\(\dfrac{7}{4}\)
    3. C\(\dfrac{7}{12}\)
    4. D\(\dfrac{7}{24}\)
    Show answerHide answer

    D: \(\dfrac{7}{24}\)

    Convert to minutes: \(\dfrac{35}{120} = \dfrac{7}{24}\).

  3. 3

    Work out \(\dfrac{1}{2} + \dfrac{1}{3}\).

    1. A\(\dfrac{5}{6}\)
    2. B\(\dfrac{2}{6}\)
    3. C\(\dfrac{1}{6}\)
    4. D\(\dfrac{2}{5}\)
    Show answerHide answer

    A: \(\dfrac{5}{6}\)

    Use a common denominator of 6: \(\dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}\).

  4. 4

    What is \(\dfrac{3}{5}\) of 40?

    1. A8
    2. B24
    3. C16
    4. D120
    Show answerHide answer

    B: 24

    \(40 \div 5 = 8\), then \(8 \times 3 = 24\).

  5. 5

    Work out \(\dfrac{2}{3} \div 4\).

    1. A\(\dfrac{8}{3}\)
    2. B6
    3. C\(\dfrac{1}{6}\)
    4. D\(\dfrac{1}{3}\)
    Show answerHide answer

    C: \(\dfrac{1}{6}\)

    Dividing by 4 is multiplying by \(\dfrac{1}{4}\): \(\dfrac{2}{3} \times \dfrac{1}{4} = \dfrac{2}{12} = \dfrac{1}{6}\).

  6. 6

    Write \(3\dfrac{2}{5}\) as an improper fraction.

    1. A\(\dfrac{11}{5}\)
    2. B\(\dfrac{6}{5}\)
    3. C\(\dfrac{17}{5}\)
    4. D\(\dfrac{32}{5}\)
    Show answerHide answer

    C: \(\dfrac{17}{5}\)

    \(3 \times 5 + 2 = 17\), so the fraction is \(\dfrac{17}{5}\).

  7. 7

    Which fraction is equal to \(\dfrac{18}{24}\)?

    1. A\(\dfrac{4}{5}\)
    2. B\(\dfrac{2}{3}\)
    3. C\(\dfrac{9}{10}\)
    4. D\(\dfrac{3}{4}\)
    Show answerHide answer

    D: \(\dfrac{3}{4}\)

    The HCF of 18 and 24 is 6, and dividing both by 6 gives \(\dfrac{3}{4}\).

  8. 8

    Work out \(\dfrac{3}{4} \times \dfrac{2}{9}\).

    1. A\(\dfrac{27}{8}\)
    2. B\(\dfrac{5}{13}\)
    3. C\(\dfrac{2}{3}\)
    4. D\(\dfrac{1}{6}\)
    Show answerHide answer

    D: \(\dfrac{1}{6}\)

    Multiply across and simplify: \(\dfrac{6}{36} = \dfrac{1}{6}\).

  9. 9

    What is the reciprocal of \(\dfrac{5}{7}\)?

    1. A\(\dfrac{5}{7}\)
    2. B\(\dfrac{7}{5}\)
    3. C\(-\dfrac{5}{7}\)
    4. D35
    Show answerHide answer

    B: \(\dfrac{7}{5}\)

    The reciprocal is the fraction turned upside down, which is \(\dfrac{7}{5}\).

  10. 10

    Which of these fractions is the largest?

    1. A\(\dfrac{2}{3}\)
    2. B\(\dfrac{5}{8}\)
    3. C\(\dfrac{3}{5}\)
    4. D\(\dfrac{7}{12}\)
    Show answerHide answer

    A: \(\dfrac{2}{3}\)

    As decimals they are 0.625, 0.667, 0.583 and 0.6, so \(\dfrac{2}{3}\) is the largest.

Fractions, Decimals and Percentages

Just this lesson
  1. 1 Write [2 marks]

    Write \(0.08\) as a fraction in its simplest form.

    Show answerHide answer

    Model answer

    \(0.08 = \dfrac{8}{100}\). Dividing the top and bottom by 4 gives \(\dfrac{2}{25}\).

    Mark scheme

    • \(\dfrac{8}{100}\) — M1
    • \(\dfrac{2}{25}\) — A1
  2. 2 Work out [3 marks]

    Work out \(45\%\) of \(\pounds 620\).

    Show answerHide answer

    Model answer

    \(10\% = 62\), so \(40\% = 248\). \(5\% = 31\). Then \(45\% = 248 + 31 = \pounds 279\).

    Mark scheme

    • Finds 10% (£62) or 5% (£31) — M1
    • Adds correct parts, such as 248 + 31 — M1
    • \(\pounds 279\) — A1
  3. 3 Work out [3 marks]

    A restaurant bill is \(\pounds 84\) before a service charge. A service charge of \(12.5\%\) is added to the bill. Work out the total amount to pay.

    Show answerHide answer

    Model answer

    \(12.5\% = \dfrac{1}{8}\), and \(84 \div 8 = 10.50\). The total is \(84 + 10.50 = \pounds 94.50\).

    Mark scheme

    • Finds 12.5% of 84, such as \(84 \div 8\), or 10% + 2.5% — M1
    • \(84 + 10.50\) — M1
    • \(\pounds 94.50\) — A1
  4. 4 Work out [3 marks]

    Increase \(\pounds 350\) by \(14\%\).

    Show answerHide answer

    Model answer

    \(10\% = 35\) and \(4\% = 14\), so \(14\% = 49\). The new amount is \(350 + 49 = \pounds 399\).

    Mark scheme

    • Finds 10% (£35) and 1% (£3.50), or 14% as 49 — M1
    • \(350 + 49\) or \(350 \times 1.14\) — M1
    • \(\pounds 399\) — A1
  5. 5 Work out [3 marks]

    The price of a jacket is reduced from \(\pounds 64\) to \(\pounds 48\) in a sale. Work out the percentage reduction.

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    Model answer

    The reduction is \(64 - 48 = \pounds 16\). \(\dfrac{16}{64} = \dfrac{1}{4} = 25\%\).

    Mark scheme

    • \(64 - 48 = 16\) — M1
    • \(\dfrac{16}{64} \times 100\) or \(\dfrac{1}{4}\) — M1
    • 25% — A1
  6. 6 Work out [3 marks]

    Priya invests some money for one year at \(4\%\) interest. At the end of the year she has \(\pounds 1560\). Work out how much she invested.

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    Model answer

    After a \(4\%\) increase, the amount is \(104\%\) of the original. \(104\% = 1560\), so \(1\% = 15\) and \(100\% = \pounds 1500\).

    Mark scheme

    • Recognises that £1560 is 104% of the original — M1
    • \(1560 \div 104 = 15\) or \(1560 \div 1.04\) — M1
    • \(\pounds 1500\) — A1
  7. 7 Explain [3 marks]

    Dev says, ‘If I increase an amount by \(10\%\) and then decrease the result by \(10\%\), I will end up with the amount I started with.’ Is Dev correct? You must show how you decide.

    Show answerHide answer

    Model answer

    No. Take \(\pounds 200\) as an example. A \(10\%\) increase gives \(200 + 20 = 220\). A \(10\%\) decrease of \(220\) is \(22\), giving \(220 - 22 = \pounds 198\), which is not \(\pounds 200\). The second percentage is taken of a bigger number.

    Mark scheme

    • Chooses a starting amount and applies a 10% increase correctly — M1
    • Applies a 10% decrease to the new amount, not the original — M1
    • No, with a correct comparison such as £198 and £200 — A1

Quick check

  1. 1

    Which of these fractions is a recurring decimal?

    1. A\(\dfrac{7}{20}\)
    2. B\(\dfrac{1}{6}\)
    3. C\(\dfrac{9}{25}\)
    4. D\(\dfrac{3}{8}\)
    Show answerHide answer

    B: \(\dfrac{1}{6}\)

    6 has the prime factor 3, so \(\dfrac{1}{6} = 0.1\dot{6}\) recurs. The others have denominators with only the prime factors 2 and 5.

  2. 2

    £500 is invested for 4 years at 3% simple interest per year. How much interest is earned in total?

    1. A£62.75
    2. B£15
    3. C£60
    4. D£560
    Show answerHide answer

    C: £60

    3% of £500 is £15 a year, and \(15 \times 4 = £60\).

  3. 3

    Write 0.035 as a percentage.

    1. A350%
    2. B3.5%
    3. C0.35%
    4. D35%
    Show answerHide answer

    B: 3.5%

    Multiply by 100: \(0.035 \times 100 = 3.5\%\).

  4. 4

    Write \(\dfrac{3}{8}\) as a percentage.

    1. A3.8%
    2. B38%
    3. C0.375%
    4. D37.5%
    Show answerHide answer

    D: 37.5%

    \(3 \div 8 = 0.375\), and \(0.375 \times 100 = 37.5\%\).

  5. 5

    What is 15% of 60?

    1. A15
    2. B6
    3. C9
    4. D90
    Show answerHide answer

    C: 9

    10% is 6 and 5% is 3, so 15% is 9.

  6. 6

    What is the multiplier for a decrease of 8%?

    1. A8
    2. B0.92
    3. C1.08
    4. D0.08
    Show answerHide answer

    B: 0.92

    100% − 8% = 92%, which is 0.92.

  7. 7

    Increase £60 by 25%.

    1. A£15
    2. B£75
    3. C£240
    4. D£85
    Show answerHide answer

    B: £75

    25% of 60 is 15, and 60 + 15 = 75.

  8. 8

    After a 20% decrease, a price is £48. What was the original price?

    1. A£57.60
    2. B£60
    3. C£96
    4. D£38.40
    Show answerHide answer

    B: £60

    The sale price is 80% of the original, so the original is 48 ÷ 0.8 = £60.

  9. 9

    A price rises from 40 to 50. What is the percentage increase?

    1. A25%
    2. B20%
    3. C80%
    4. D10%
    Show answerHide answer

    A: 25%

    The change is 10, and \(\dfrac{10}{40} \times 100 = 25\%\). It is divided by the original, not the new price.

  10. 10

    Write 12.5% as a fraction in its simplest form.

    1. A\(\dfrac{1}{8}\)
    2. B\(\dfrac{1}{12}\)
    3. C\(\dfrac{1}{5}\)
    4. D\(\dfrac{5}{8}\)
    Show answerHide answer

    A: \(\dfrac{1}{8}\)

    \(12.5\% = \dfrac{12.5}{100} = \dfrac{1}{8}\).

Powers, Roots and Indices

Just this lesson
  1. 1 Work out [2 marks]

    Work out (a) \(11^2\) [1 mark] (b) \(\sqrt[3]{125}\) [1 mark]

    Show answerHide answer

    Model answer

    (a) \(11 \times 11 = 121\). (b) \(5 \times 5 \times 5 = 125\), so \(\sqrt[3]{125} = 5\).

    Mark scheme

    • (a) 121 — B1
    • (b) 5 — B1
  2. 2 Simplify [2 marks]

    Simplify (a) \(x^4 \times x^5\) [1 mark] (b) \(y^8 \div y^3\) [1 mark]

    Show answerHide answer

    Model answer

    (a) Add the indices: \(x^{4+5} = x^9\). (b) Subtract the indices: \(y^{8-3} = y^5\).

    Mark scheme

    • (a) \(x^9\) — B1
    • (b) \(y^5\) — B1
  3. 3 Work out [2 marks]

    Work out the value of \(2^{-3} \times 4\). Give your answer as a fraction.

    Show answerHide answer

    Model answer

    \(2^{-3} = \dfrac{1}{8}\), so \(\dfrac{1}{8} \times 4 = \dfrac{4}{8} = \dfrac{1}{2}\). Alternatively, \(4 = 2^2\), so \(2^{-3} \times 2^2 = 2^{-1} = \dfrac{1}{2}\).

    Mark scheme

    • \(2^{-3} = \dfrac{1}{8}\) or \(2^{-3} \times 2^2\) — M1
    • \(\dfrac{1}{2}\) — A1
  4. 4 Find [3 marks]

    (a) Find the value of \(n\) when \(2^n = \dfrac{1}{32}\). [2 marks] (b) Find the value of \(n\) when \(3^n \times 3^4 = 3\). [1 mark]

    Show answerHide answer

    Model answer

    (a) \(\dfrac{1}{32} = \dfrac{1}{2^5} = 2^{-5}\), so \(n = -5\). (b) \(n + 4 = 1\), so \(n = -3\).

    Mark scheme

    • (a) \(32 = 2^5\) or \(\dfrac{1}{2^5}\) — M1
    • (a) \(-5\) — A1
    • (b) \(-3\) — B1
  5. 5 Work out [3 marks]

    Work out the value of (a) \(125^{\frac{1}{3}}\) [1 mark] (b) \(81^{\frac{3}{4}}\) [2 marks]

    Show answerHide answer

    Model answer

    (a) The cube root of 125 is 5. (b) \(81^{\frac{1}{4}} = 3\), then \(3^3 = 27\).

    Mark scheme

    • (a) 5 — B1
    • (b) \(\sqrt[4]{81} = 3\) or \(81^{\frac{1}{4}} = 3\) — M1
    • (b) 27 — A1
  6. 6 Show that [2 marks]

    Show that \(5^7 + 5^7 + 5^7 + 5^7 + 5^7 = 5^8\).

    Show answerHide answer

    Model answer

    There are five lots of \(5^7\), so the sum is \(5 \times 5^7 = 5^1 \times 5^7 = 5^{1+7} = 5^8\).

    Mark scheme

    • \(5 \times 5^7\) — M1
    • \(5^1 \times 5^7 = 5^8\) with a clear conclusion — A1
  7. 7 Simplify [2 marks]

    Simplify \((3a^2b^3)^2\).

    Show answerHide answer

    Model answer

    Square every part: \(3^2 \times (a^2)^2 \times (b^3)^2 = 9a^4b^6\).

    Mark scheme

    • Two of \(9\), \(a^4\), \(b^6\) correct — M1
    • \(9a^4b^6\) — A1

Quick check

  1. 1

    Between which two whole numbers does \(\sqrt{40}\) lie?

    1. A6 and 7
    2. B5 and 6
    3. C4 and 5
    4. D7 and 8
    Show answerHide answer

    A: 6 and 7

    \(6^2 = 36\) and \(7^2 = 49\), and 40 is between 36 and 49.

  2. 2

    Write \(16 \times 8\) as a single power of 2.

    1. A\(4^7\)
    2. B\(2^{24}\)
    3. C\(2^{12}\)
    4. D\(2^7\)
    Show answerHide answer

    D: \(2^7\)

    \(16 = 2^4\) and \(8 = 2^3\), so \(2^4 \times 2^3 = 2^7\).

  3. 3

    What is the value of \(\sqrt{196}\)?

    1. A98
    2. B14
    3. C16
    4. D13
    Show answerHide answer

    B: 14

    14 × 14 = 196.

  4. 4

    What is the value of \(4^3\)?

    1. A12
    2. B81
    3. C64
    4. D256
    Show answerHide answer

    C: 64

    \(4 \times 4 \times 4 = 64\), not \(4 \times 3 = 12\).

  5. 5

    Simplify \(a^5 \times a^3\).

    1. A\(a^2\)
    2. B\(a^{15}\)
    3. C\(a^8\)
    4. D\(2a^8\)
    Show answerHide answer

    C: \(a^8\)

    Add the indices when multiplying: \(a^{5+3} = a^8\).

  6. 6

    Simplify \((3^2)^4\).

    1. A\(3^8\)
    2. B\(9^8\)
    3. C\(3^6\)
    4. D\(3^{16}\)
    Show answerHide answer

    A: \(3^8\)

    For a power of a power, multiply the indices: \(3^{2 \times 4} = 3^8\).

  7. 7

    What is the value of \(5^{-2}\)?

    1. A\(-10\)
    2. B\(\dfrac{1}{25}\)
    3. C\(-25\)
    4. D\(\dfrac{1}{10}\)
    Show answerHide answer

    B: \(\dfrac{1}{25}\)

    A negative index means a reciprocal: \(5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}\).

  8. 8

    What is the value of \(7^0\)?

    1. A1
    2. BUndefined
    3. C0
    4. D7
    Show answerHide answer

    A: 1

    Any non-zero number to the power 0 is 1.

  9. 9

    Simplify \(2^6 \div 2^{-2}\).

    1. A\(2^4\)
    2. B\(2^{-3}\)
    3. C\(2^8\)
    4. D\(2^{-12}\)
    Show answerHide answer

    C: \(2^8\)

    Subtract the indices: \(6 - (-2) = 8\), so the answer is \(2^8\).

  10. 10

    What is the value of \(8^{\frac{1}{3}}\)?

    1. A24
    2. B\(\dfrac{8}{3}\)
    3. C\(\dfrac{1}{8}\)
    4. D2
    Show answerHide answer

    D: 2

    A power of one third means the cube root, and \(2 \times 2 \times 2 = 8\).