OpenRevise

Exam questions · Maths

Standard Form and Accuracy

  • 30 exam questions
  • 75 marks
  • 45 quick checks

Standard Form

Just this lesson
  1. 1 Write [2 marks]

    Write 45 000 in standard form.

    Show answerHide answer

    Model answer

    The decimal point moves 4 places left, so \(45\,000 = 4.5 \times 10^4\).

    Mark scheme

    • 4.5 seen — M1
    • \(4.5 \times 10^4\) — A1
  2. 2 Write [2 marks]

    Write \(3.2 \times 10^{-3}\) as an ordinary number.

    Show answerHide answer

    Model answer

    A power of \(-3\) means the decimal point moves 3 places to the right, giving \(0.0032\).

    Mark scheme

    • Moves the decimal point 3 places — M1
    • 0.0032 — A1
  3. 3 Write [2 marks]

    Write these numbers in order of size. Start with the smallest. \(3.4 \times 10^5\) \(2.9 \times 10^6\) \(8.7 \times 10^4\)

    Show answerHide answer

    Model answer

    Compare the powers: 4 is the smallest, then 5, then 6. The order is \(8.7 \times 10^4\), \(3.4 \times 10^5\), \(2.9 \times 10^6\).

    Mark scheme

    • Compares the powers of 10 — M1
    • \(8.7 \times 10^4\), \(3.4 \times 10^5\), \(2.9 \times 10^6\) — A1
  4. 4 Write [3 marks]

    (a) The distance from the Earth to the Sun is 150 million kilometres. Write this distance in standard form. (1 mark) (b) A cell has a width of 0.0000052 m. Write this width in standard form. (2 marks)

    Show answerHide answer

    Model answer

    (a) \(150\,000\,000 = 1.5 \times 10^8\) km. (b) The point moves 6 places right to give 5.2, so the width is \(5.2 \times 10^{-6}\) m.

    Mark scheme

    • (a) \(1.5 \times 10^8\) — B1
    • (b) 5.2 seen — M1
    • (b) \(5.2 \times 10^{-6}\) — A1
  5. 5 Explain [2 marks]

    Ravi writes the number 45 000 as \(45 \times 10^3\). Explain why this is not in standard form, and write the number correctly.

    Show answerHide answer

    Model answer

    In standard form the first number must be at least 1 and less than 10. 45 is not, so the number is \(4.5 \times 10^4\).

    Mark scheme

    • 45 is not between 1 and 10 — C1
    • \(4.5 \times 10^4\) — B1
  6. 6 Work out [3 marks]

    \(N = 7.2 \times 10^5\). Write each of these in standard form. (a) \(10N\) (1 mark) (b) \(N \div 1000\) (1 mark) (c) Write \(N\) as an ordinary number. (1 mark)

    Show answerHide answer

    Model answer

    (a) \(7.2 \times 10^6\). (b) \(7.2 \times 10^2\). (c) \(720\,000\).

    Mark scheme

    • (a) \(7.2 \times 10^6\) — B1
    • (b) \(7.2 \times 10^2\) — B1
    • (c) 720 000 — B1

Quick check

  1. 1

    What is 45 000 in standard form?

    1. A\(45 \times 10^3\)
    2. B\(4.5 \times 10^4\)
    3. C\(4.5 \times 10^3\)
    4. D\(0.45 \times 10^5\)
    Show answerHide answer

    B: \(4.5 \times 10^4\)

    The decimal point moves 4 places left to give 4.5.

  2. 2

    What is \(3.2 \times 10^{-3}\) as an ordinary number?

    1. A0.0032
    2. B0.032
    3. C3200
    4. D0.00032
    Show answerHide answer

    A: 0.0032

    A power of \(-3\) moves the decimal point 3 places to the right of the 3, giving 0.0032.

  3. 3

    Which of these is in standard form?

    1. A\(62 \times 10^4\)
    2. B\(0.62 \times 10^6\)
    3. C\(6.2 \times 5^{10}\)
    4. D\(6.2 \times 10^5\)
    Show answerHide answer

    D: \(6.2 \times 10^5\)

    The first part must be at least 1 and less than 10, and the base must be 10.

  4. 4

    What is 0.00045 in standard form?

    1. A\(4.5 \times 10^{4}\)
    2. B\(4.5 \times 10^{-3}\)
    3. C\(4.5 \times 10^{-4}\)
    4. D\(45 \times 10^{-5}\)
    Show answerHide answer

    C: \(4.5 \times 10^{-4}\)

    The decimal point moves 4 places right, so the power is \(-4\).

  5. 5

    What is \(45 \times 10^3\) in standard form?

    1. A\(4.5 \times 10^3\)
    2. B\(4.5 \times 10^4\)
    3. C\(4.5 \times 10^2\)
    4. D\(450 \times 10^2\)
    Show answerHide answer

    B: \(4.5 \times 10^4\)

    \(45 = 4.5 \times 10\), so \(45 \times 10^3 = 4.5 \times 10^4\).

  6. 6

    Which is bigger, \(2 \times 10^7\) or \(9 \times 10^6\)?

    1. A\(2 \times 10^7\)
    2. B\(9 \times 10^6\)
    3. CThey are equal
    4. DIt cannot be worked out
    Show answerHide answer

    A: \(2 \times 10^7\)

    \(2 \times 10^7 = 20\,000\,000\) and \(9 \times 10^6 = 9\,000\,000\).

  7. 7

    The distance to the Sun is about 150 million km. What is this in standard form?

    1. A\(1.5 \times 10^6\) km
    2. B\(15 \times 10^7\) km
    3. C\(1.5 \times 10^9\) km
    4. D\(1.5 \times 10^8\) km
    Show answerHide answer

    D: \(1.5 \times 10^8\) km

    150 million is 150 000 000, so the point moves 8 places to give 1.5.

  8. 8

    What is \(3.2 \times 10^5\) as an ordinary number?

    1. A32 000
    2. B3 200 000
    3. C320 000
    4. D0.000032
    Show answerHide answer

    C: 320 000

    Move the decimal point 5 places right.

  9. 9

    Which is smaller, \(5 \times 10^{-3}\) or \(8 \times 10^{-5}\)?

    1. A\(5 \times 10^{-3}\)
    2. B\(8 \times 10^{-5}\)
    3. CThey are equal
    4. DIt depends on the first part
    Show answerHide answer

    B: \(8 \times 10^{-5}\)

    \(5 \times 10^{-3} = 0.005\) and \(8 \times 10^{-5} = 0.00008\).

Calculating with Standard Form

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

    Work out \((2 \times 10^3) \times (3 \times 10^4)\). Give your answer in standard form.

    Show answerHide answer

    Model answer

    \(2 \times 3 = 6\) and \(10^3 \times 10^4 = 10^7\), so the answer is \(6 \times 10^7\).

    Mark scheme

    • \(2 \times 3\) or \(10^{3 + 4}\) — M1
    • \(6 \times 10^7\) — A1
  2. 2 Work out [2 marks]

    Work out \((6 \times 10^8) \div (3 \times 10^3)\). Give your answer in standard form.

    Show answerHide answer

    Model answer

    \(6 \div 3 = 2\) and \(10^8 \div 10^3 = 10^5\), so the answer is \(2 \times 10^5\).

    Mark scheme

    • \(6 \div 3\) or \(10^{8 - 3}\) — M1
    • \(2 \times 10^5\) — A1
  3. 3 Work out [3 marks]

    Work out \((4 \times 10^5) \times (5 \times 10^3)\). Give your answer in standard form.

    Show answerHide answer

    Model answer

    \(4 \times 5 = 20\) and \(10^5 \times 10^3 = 10^8\), giving \(20 \times 10^8\). In standard form this is \(2 \times 10^9\).

    Mark scheme

    • \(20 \times 10^8\) — M1
    • Adjusts to standard form — M1
    • \(2 \times 10^9\) — A1
  4. 4 Work out [3 marks]

    Work out \(3 \times 10^4 + 5 \times 10^3\). Give your answer in standard form.

    Show answerHide answer

    Model answer

    \(3 \times 10^4 = 30\,000\) and \(5 \times 10^3 = 5000\), so the total is \(35\,000 = 3.5 \times 10^4\).

    Mark scheme

    • Both numbers written with the same power, or as ordinary numbers — M1
    • 35 000 or \(3.5 \times 10^4\) seen — M1
    • \(3.5 \times 10^4\) — A1
  5. 5 Work out [3 marks]

    A bacterium is \(2 \times 10^{-6}\) m long. \(5 \times 10^8\) of these bacteria are put end to end in a line. Work out the length of the line. Give your answer in standard form.

    Show answerHide answer

    Model answer

    \((5 \times 10^8) \times (2 \times 10^{-6}) = 10 \times 10^2 = 1 \times 10^3\) m.

    Mark scheme

    • \(5 \times 2 = 10\) or \(10^8 \times 10^{-6} = 10^2\) — M1
    • \(10 \times 10^2\) — M1
    • \(1 \times 10^3\) m — A1
  6. 6 Work out [3 marks]

    Work out \((3.6 \times 10^7) \div (1.2 \times 10^{-2})\). Give your answer in standard form.

    Show answerHide answer

    Model answer

    \(3.6 \div 1.2 = 3\), and \(10^7 \div 10^{-2} = 10^{7 - (-2)} = 10^9\). The answer is \(3 \times 10^9\).

    Mark scheme

    • \(3.6 \div 1.2 = 3\) — M1
    • \(10^9\) seen — M1
    • \(3 \times 10^9\) — A1

Quick check

  1. 1

    What is \((2 \times 10^3) \times (3 \times 10^4)\)?

    1. A\(6 \times 10^{12}\)
    2. B\(5 \times 10^7\)
    3. C\(6 \times 10^7\)
    4. D\(6 \times 10^1\)
    Show answerHide answer

    C: \(6 \times 10^7\)

    Multiply \(2 \times 3 = 6\) and add the powers, \(3 + 4 = 7\).

  2. 2

    What is \((6 \times 10^8) \div (3 \times 10^3)\)?

    1. A\(2 \times 10^{11}\)
    2. B\(2 \times 10^5\)
    3. C\(3 \times 10^5\)
    4. D\(2 \times 10^{8}\)
    Show answerHide answer

    B: \(2 \times 10^5\)

    Divide \(6 \div 3 = 2\) and subtract the powers, \(8 - 3 = 5\).

  3. 3

    What is \((4 \times 10^5) \times (5 \times 10^3)\) in standard form?

    1. A\(2 \times 10^9\)
    2. B\(20 \times 10^8\)
    3. C\(2 \times 10^8\)
    4. D\(9 \times 10^8\)
    Show answerHide answer

    A: \(2 \times 10^9\)

    \(4 \times 5 = 20\) and \(10^5 \times 10^3 = 10^8\), so \(20 \times 10^8 = 2 \times 10^9\).

  4. 4

    What is \(3 \times 10^4 + 5 \times 10^3\) in standard form?

    1. A\(8 \times 10^4\)
    2. B\(8 \times 10^7\)
    3. C\(3.5 \times 10^3\)
    4. D\(3.5 \times 10^4\)
    Show answerHide answer

    D: \(3.5 \times 10^4\)

    \(30\,000 + 5000 = 35\,000 = 3.5 \times 10^4\).

  5. 5

    What is \(6.2 \times 10^5 - 3 \times 10^4\) in standard form?

    1. A\(3.2 \times 10^1\)
    2. B\(5.9 \times 10^4\)
    3. C\(5.9 \times 10^5\)
    4. D\(3.2 \times 10^5\)
    Show answerHide answer

    C: \(5.9 \times 10^5\)

    \(620\,000 - 30\,000 = 590\,000 = 5.9 \times 10^5\).

  6. 6

    What is \(10^7 \div 10^{-2}\)?

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

    B: \(10^9\)

    Subtract the powers: \(7 - (-2) = 9\).

  7. 7

    What is \((4 \times 10^{-3}) \times (2 \times 10^{-2})\)?

    1. A\(8 \times 10^{-5}\)
    2. B\(8 \times 10^{-6}\)
    3. C\(6 \times 10^{-5}\)
    4. D\(8 \times 10^{5}\)
    Show answerHide answer

    A: \(8 \times 10^{-5}\)

    Multiply \(4 \times 2 = 8\) and add the powers, \(-3 + (-2) = -5\).

  8. 8

    What is \((3.6 \times 10^7) \div (1.2 \times 10^{-2})\)?

    1. A\(3 \times 10^5\)
    2. B\(3 \times 10^{-9}\)
    3. C\(4.8 \times 10^5\)
    4. D\(3 \times 10^9\)
    Show answerHide answer

    D: \(3 \times 10^9\)

    \(3.6 \div 1.2 = 3\) and \(10^7 \div 10^{-2} = 10^9\).

  9. 9

    \(5 \times 10^8\) bacteria, each \(2 \times 10^{-6}\) m long, are placed end to end. How long is the line?

    1. A\(1 \times 10^2\) m
    2. B\(10 \times 10^2\) m
    3. C\(1 \times 10^3\) m
    4. D\(1 \times 10^{14}\) m
    Show answerHide answer

    C: \(1 \times 10^3\) m

    \(5 \times 2 = 10\) and \(10^8 \times 10^{-6} = 10^2\), so \(10 \times 10^2 = 1 \times 10^3\).

Rounding and Estimating

Just this lesson
  1. 1 Write [2 marks]

    (a) Write 4.678 correct to 1 decimal place. (1 mark) (b) Write 4.678 correct to the nearest whole number. (1 mark)

    Show answerHide answer

    Model answer

    (a) The next digit is 7, so \(4.7\). (b) The next digit is 6, so 5.

    Mark scheme

    • (a) 4.7 — B1
    • (b) 5 — B1
  2. 2 Write [2 marks]

    Write 0.004567 correct to 2 significant figures.

    Show answerHide answer

    Model answer

    The first significant figure is 4, the second is 5, and the next digit 6 rounds the 5 up. The answer is 0.0046.

    Mark scheme

    • 0.004 or 0.005 seen, or a correct method — M1
    • 0.0046 — A1
  3. 3 Estimate [3 marks]

    Work out an estimate for \(\dfrac{4.97 \times 20.1}{0.49}\).

    Show answerHide answer

    Model answer

    Round each number to 1 significant figure: \(\dfrac{5 \times 20}{0.5} = \dfrac{100}{0.5} = 200\).

    Mark scheme

    • At least two numbers rounded to 1 s.f., such as 5 and 20 — M1
    • \(\dfrac{5 \times 20}{0.5}\) — M1
    • 200 — A1
  4. 4 Estimate [3 marks]

    Work out an estimate for \(\dfrac{39.8 \times 5.1}{0.21}\).

    Show answerHide answer

    Model answer

    \(\dfrac{40 \times 5}{0.2} = \dfrac{200}{0.2} = 1000\).

    Mark scheme

    • Rounds to 40, 5 and 0.2 — M1
    • \(\dfrac{40 \times 5}{0.2}\) — M1
    • 1000 — A1
  5. 5 Write [2 marks]

    Write 6482 (a) correct to 2 significant figures, (b) correct to the nearest hundred.

    Show answerHide answer

    Model answer

    (a) The first two figures are 6 and 4, and the next digit 8 rounds the 4 up: 6500. (b) The hundreds digit is 4, and the next digit 8 rounds it up: 6500.

    Mark scheme

    • (a) 6500 — B1
    • (b) 6500 — B1
  6. 6 Estimate [3 marks]

    (a) Work out an estimate for \(\dfrac{4.9 \times 9.8}{0.52}\). (2 marks) (b) Is your estimate bigger or smaller than the exact value? Give a reason. (1 mark)

    Show answerHide answer

    Model answer

    (a) \(\dfrac{5 \times 10}{0.5} = 100\). (b) Both numbers on the top were rounded up, and the bottom was rounded down, which makes the fraction bigger. So the estimate is bigger than the exact value.

    Mark scheme

    • (a) \(\dfrac{5 \times 10}{0.5}\) — M1
    • (a) 100 — A1
    • (b) Bigger, with a reason about the rounding — C1

Quick check

  1. 1

    What is 4.678 to 1 decimal place?

    1. A4.6
    2. B4.68
    3. C5
    4. D4.7
    Show answerHide answer

    D: 4.7

    The next digit is 7, so round up.

  2. 2

    What is 6482 to 2 significant figures?

    1. A65
    2. B6400
    3. C6500
    4. D6480
    Show answerHide answer

    C: 6500

    The first two figures are 6 and 4, and the next digit 8 rounds the 4 up to 5, with zeros to hold the place.

  3. 3

    What is 0.004567 to 2 significant figures?

    1. A0.0045
    2. B0.0046
    3. C0.00457
    4. D0.0047
    Show answerHide answer

    B: 0.0046

    The first significant figure is the 4, and the next digit 6 rounds the 5 up.

  4. 4

    What is 83.7 to the nearest 10?

    1. A80
    2. B90
    3. C84
    4. D83
    Show answerHide answer

    A: 80

    83.7 is closer to 80 than to 90.

  5. 5

    What is the best estimate for \(\dfrac{4.97 \times 20.1}{0.49}\)?

    1. A20
    2. B2000
    3. C100
    4. D200
    Show answerHide answer

    D: 200

    \(\dfrac{5 \times 20}{0.5} = \dfrac{100}{0.5} = 200\).

  6. 6

    To estimate a calculation, to how many significant figures do you round each number?

    1. A2
    2. B3
    3. C1
    4. D0
    Show answerHide answer

    C: 1

    One significant figure keeps the arithmetic easy.

  7. 7

    The bottom of a fraction is rounded up. What happens to the estimate?

    1. AIt is larger than the true value
    2. BIt is smaller than the true value
    3. CIt is exactly right
    4. DIt cannot be said
    Show answerHide answer

    B: It is smaller than the true value

    A bigger number on the bottom makes the fraction smaller.

  8. 8

    What is the best estimate for \(\dfrac{39.8 \times 5.1}{0.21}\)?

    1. A1000
    2. B100
    3. C10 000
    4. D400
    Show answerHide answer

    A: 1000

    \(\dfrac{40 \times 5}{0.2} = \dfrac{200}{0.2} = 1000\).

  9. 9

    What is 0.0305 to 2 significant figures?

    1. A0.030
    2. B0.03
    3. C0.0305
    4. D0.031
    Show answerHide answer

    D: 0.031

    The significant figures are 3 and 0. The next digit is 5, so round the 0 up to 1.

Error Intervals and Bounds

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

    The length of a pencil is 12 cm, correct to the nearest centimetre. Write down the error interval for the length \(L\).

    Show answerHide answer

    Model answer

    Half a centimetre below and above 12: \(11.5 \leq L < 12.5\).

    Mark scheme

    • 11.5 and 12.5 seen — M1
    • \(11.5 \leq L < 12.5\) — A1
  2. 2 Write down [2 marks]

    \(x = 3.7\) correct to 1 decimal place. Write down the error interval for \(x\).

    Show answerHide answer

    Model answer

    Half of 0.1 is 0.05, so \(3.65 \leq x < 3.75\).

    Mark scheme

    • 3.65 and 3.75 seen — M1
    • \(3.65 \leq x < 3.75\) — A1
  3. 3 Write down [2 marks]

    The number \(n\) is 5 after it has been truncated to a whole number. Write down the error interval for \(n\).

    Show answerHide answer

    Model answer

    Truncating cuts off the digits, so \(n\) was at least 5 and less than 6: \(5 \leq n < 6\).

    Mark scheme

    • 5 and 6 seen — M1
    • \(5 \leq n < 6\) — A1
  4. 4 Work out [3 marks]

    \(a = 6.4\) and \(b = 2.5\), each correct to 1 decimal place. Work out the upper bound of \(a + b\).

    Show answerHide answer

    Model answer

    The upper bounds are \(6.45\) and \(2.55\), so the upper bound of the sum is \(6.45 + 2.55 = 9\).

    Mark scheme

    • 6.45 or 2.55 seen — M1
    • \(6.45 + 2.55\) — M1
    • 9 — A1
  5. 5 Work out [3 marks]

    A rectangle has a length of 8 cm and a width of 5 cm, each correct to the nearest centimetre. Work out the upper bound of the area of the rectangle.

    Show answerHide answer

    Model answer

    The upper bounds are 8.5 cm and 5.5 cm, so the upper bound of the area is \(8.5 \times 5.5 = 46.75\) cm\(^2\).

    Mark scheme

    • 8.5 or 5.5 seen — M1
    • \(8.5 \times 5.5\) — M1
    • 46.75 cm\(^2\) — A1
  6. 6 Work out [3 marks]

    \(a = 6.4\) and \(b = 2.5\), each correct to 1 decimal place. Work out the lower bound of \(a - b\).

    Show answerHide answer

    Model answer

    For the smallest difference, use the lower bound of \(a\) and the upper bound of \(b\): \(6.35 - 2.55 = 3.8\).

    Mark scheme

    • 6.35 or 2.55 seen — M1
    • \(6.35 - 2.55\) — M1
    • 3.8 — A1

Quick check

  1. 1

    A length is 12 cm to the nearest cm. What is the error interval?

    1. A\(11.5 \leq L < 12.5\)
    2. B\(11 \leq L < 13\)
    3. C\(11.5 < L \leq 12.5\)
    4. D\(12 \leq L < 13\)
    Show answerHide answer

    A: \(11.5 \leq L < 12.5\)

    Half a unit below and above 12, with the lower end included.

  2. 2

    \(x = 3.7\) correct to 1 decimal place. What is the error interval?

    1. A\(3.6 \leq x < 3.8\)
    2. B\(3.7 \leq x < 3.8\)
    3. C\(3.65 < x \leq 3.75\)
    4. D\(3.65 \leq x < 3.75\)
    Show answerHide answer

    D: \(3.65 \leq x < 3.75\)

    Half of 0.1 is 0.05, so go from 3.65 up to 3.75.

  3. 3

    \(n = 5\) after being truncated to a whole number. What is the error interval?

    1. A\(4.5 \leq n < 5.5\)
    2. B\(4 \leq n < 5\)
    3. C\(5 \leq n < 6\)
    4. D\(5 \leq n \leq 6\)
    Show answerHide answer

    C: \(5 \leq n < 6\)

    Truncating cuts off the digits, so the number was at least 5 and less than 6.

  4. 4

    \(x = 40\) to the nearest 10. What is the error interval?

    1. A\(40 \leq x < 50\)
    2. B\(35 \leq x < 45\)
    3. C\(30 \leq x < 50\)
    4. D\(39 \leq x < 41\)
    Show answerHide answer

    B: \(35 \leq x < 45\)

    Half of 10 is 5.

  5. 5

    Why is the upper bound of an error interval not included?

    1. AIt would round up to the next value
    2. BIt is too big to measure
    3. CIt rounds down
    4. DIt is a typing convention
    Show answerHide answer

    A: It would round up to the next value

    A value exactly half way rounds up, so it belongs to the next number.

  6. 6

    \(x = 0.4\) correct to 1 significant figure. What is the error interval?

    1. A\(0.3 \leq x < 0.5\)
    2. B\(0.4 \leq x < 0.5\)
    3. C\(0.39 \leq x < 0.41\)
    4. D\(0.35 \leq x < 0.45\)
    Show answerHide answer

    D: \(0.35 \leq x < 0.45\)

    The unit is 0.1, so half is 0.05.

  7. 7

    What is the upper bound of \(a + b\)?

    1. AThe upper bound of \(a\) plus the lower bound of \(b\)
    2. BThe lower bound of \(a\) plus the upper bound of \(b\)
    3. CThe upper bound of \(a\) plus the upper bound of \(b\)
    4. DThe lower bounds added
    Show answerHide answer

    C: The upper bound of \(a\) plus the upper bound of \(b\)

    The biggest total comes from the biggest values.

  8. 8

    What is the upper bound of \(a - b\)?

    1. AThe upper bound of \(a\) minus the upper bound of \(b\)
    2. BThe upper bound of \(a\) minus the lower bound of \(b\)
    3. CThe lower bound of \(a\) minus the lower bound of \(b\)
    4. DThe lower bound of \(a\) minus the upper bound of \(b\)
    Show answerHide answer

    B: The upper bound of \(a\) minus the lower bound of \(b\)

    To make the difference big, take the biggest \(a\) and subtract the smallest \(b\).

  9. 9

    A rectangle is 8 cm by 5 cm, each to the nearest cm. What is the upper bound of its area?

    1. A\(46.75\) cm\(^2\)
    2. B\(40\) cm\(^2\)
    3. C\(45\) cm\(^2\)
    4. D\(33.75\) cm\(^2\)
    Show answerHide answer

    A: \(46.75\) cm\(^2\)

    \(8.5 \times 5.5 = 46.75\).

Recurring Decimals and Rational Numbers

Just this lesson
  1. 1 Write [2 marks]

    (a) Write \(\dfrac{3}{8}\) as a decimal. (1 mark) (b) Write \(\dfrac{5}{6}\) as a decimal, using dot notation. (1 mark)

    Show answerHide answer

    Model answer

    (a) \(3 \div 8 = 0.375\). (b) \(5 \div 6 = 0.8333\ldots = 0.8\dot{3}\).

    Mark scheme

    • (a) 0.375 — B1
    • (b) \(0.8\dot{3}\) — B1
  2. 2 Write [2 marks]

    Write \(\dfrac{3}{11}\) as a recurring decimal, using dot notation.

    Show answerHide answer

    Model answer

    \(3 \div 11 = 0.272727\ldots = 0.\dot{2}\dot{7}\).

    Mark scheme

    • 0.2727 seen — M1
    • \(0.\dot{2}\dot{7}\) — A1
  3. 3 Show that [3 marks]

    Write \(0.\dot{4}\dot{5}\) as a fraction in its simplest form.

    Show answerHide answer

    Model answer

    Let \(x = 0.454545\ldots\). Then \(100x = 45.4545\ldots\), so \(99x = 45\) and \(x = \dfrac{45}{99} = \dfrac{5}{11}\).

    Mark scheme

    • \(100x = 45.4545\ldots\) — M1
    • \(99x = 45\) — M1
    • \(\dfrac{5}{11}\) — A1
  4. 4 Show that [3 marks]

    Write \(0.\dot{3}\dot{6}\) as a fraction in its simplest form.

    Show answerHide answer

    Model answer

    Let \(x = 0.3636\ldots\). Then \(100x = 36.3636\ldots\), so \(99x = 36\) and \(x = \dfrac{36}{99} = \dfrac{4}{11}\).

    Mark scheme

    • \(100x = 36.3636\ldots\) — M1
    • \(99x = 36\) — M1
    • \(\dfrac{4}{11}\) — A1
  5. 5 Write down [2 marks]

    Here are four numbers: \(\sqrt{16}\), \(\pi\), \(0.\dot{3}\) and \(\sqrt{7}\). Write down the numbers that are irrational.

    Show answerHide answer

    Model answer

    \(\sqrt{16} = 4\) and \(0.\dot{3} = \dfrac{1}{3}\) are rational. \(\pi\) and \(\sqrt{7}\) are irrational.

    Mark scheme

    • \(\pi\) or \(\sqrt{7}\) — B1
    • \(\pi\) and \(\sqrt{7}\) only — B1
  6. 6 Prove [3 marks]

    Prove that \(0.\dot{7}\dot{2} = \dfrac{8}{11}\).

    Show answerHide answer

    Model answer

    Let \(x = 0.7272\ldots\). Then \(100x = 72.7272\ldots\), so \(99x = 72\). Therefore \(x = \dfrac{72}{99} = \dfrac{8}{11}\).

    Mark scheme

    • \(100x = 72.7272\ldots\) — M1
    • \(99x = 72\) — M1
    • \(\dfrac{72}{99} = \dfrac{8}{11}\) with the conclusion — C1

Quick check

  1. 1

    What is \(\dfrac{3}{8}\) as a decimal?

    1. A0.38
    2. B0.375
    3. C0.83
    4. D0.\dot{3}
    Show answerHide answer

    B: 0.375

    \(3 \div 8 = 0.375\), which stops.

  2. 2

    What is \(\dfrac{1}{3}\) as a recurring decimal?

    1. A\(0.\dot{3}\)
    2. B0.3
    3. C0.33
    4. D\(0.\dot{1}\dot{3}\)
    Show answerHide answer

    A: \(0.\dot{3}\)

    The 3 repeats for ever.

  3. 3

    What is \(\dfrac{3}{11}\) as a recurring decimal?

    1. A\(0.\dot{3}\)
    2. B\(0.\dot{2}\dot{3}\)
    3. C\(0.2\dot{7}\)
    4. D\(0.\dot{2}\dot{7}\)
    Show answerHide answer

    D: \(0.\dot{2}\dot{7}\)

    \(3 \div 11 = 0.272727\ldots\), where 27 repeats.

  4. 4

    Which fraction gives a terminating decimal?

    1. A\(\dfrac{1}{3}\)
    2. B\(\dfrac{2}{7}\)
    3. C\(\dfrac{7}{20}\)
    4. D\(\dfrac{5}{6}\)
    Show answerHide answer

    C: \(\dfrac{7}{20}\)

    \(20 = 2^2 \times 5\), so the decimal 0.35 stops.

  5. 5

    Is \(\pi\) rational or irrational?

    1. ARational
    2. BIrrational
    3. CBoth
    4. DNeither
    Show answerHide answer

    B: Irrational

    \(\pi\) cannot be written as a fraction.

  6. 6

    Is \(\sqrt{16}\) rational or irrational?

    1. ARational, because it equals 4
    2. BIrrational, because it is a root
    3. CIrrational, because it is not a fraction
    4. DNeither
    Show answerHide answer

    A: Rational, because it equals 4

    \(\sqrt{16} = 4 = \dfrac{4}{1}\).

  7. 7

    Write \(0.\dot{4}\) as a fraction.

    1. A\(\dfrac{4}{10}\)
    2. B\(\dfrac{4}{99}\)
    3. C\(\dfrac{1}{4}\)
    4. D\(\dfrac{4}{9}\)
    Show answerHide answer

    D: \(\dfrac{4}{9}\)

    \(x = 0.\dot{4}\), \(10x = 4.\dot{4}\), so \(9x = 4\).

  8. 8

    Write \(0.\dot{4}\dot{5}\) as a fraction in its simplest form.

    1. A\(\dfrac{45}{100}\)
    2. B\(\dfrac{9}{20}\)
    3. C\(\dfrac{5}{11}\)
    4. D\(\dfrac{4}{9}\)
    Show answerHide answer

    C: \(\dfrac{5}{11}\)

    \(100x - x = 45\), so \(x = \dfrac{45}{99} = \dfrac{5}{11}\).

  9. 9

    Write \(0.1\dot{6}\) as a fraction in its simplest form.

    1. A\(\dfrac{16}{99}\)
    2. B\(\dfrac{1}{6}\)
    3. C\(\dfrac{16}{100}\)
    4. D\(\dfrac{1}{16}\)
    Show answerHide answer

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

    \(100x - 10x = 15\), so \(x = \dfrac{15}{90} = \dfrac{1}{6}\).