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 7 300 000 in standard form. [2 marks]

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    Model answer

    The decimal point moves 6 places left, so \(7.3 \times 10^6\).

    Mark scheme

    • 7.3 seen — M1
    • \(7.3 \times 10^6\) — A1
  2. 2 Write [2 marks]

    Write \(4.1 \times 10^{-2}\) as an ordinary number. [2 marks]

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    Model answer

    Move the decimal point 2 places right: 0.041.

    Mark scheme

    • Moves the decimal point 2 places — M1
    • 0.041 — A1
  3. 3 Explain [2 marks]

    Which is bigger, \(6 \times 10^{-3}\) or \(2 \times 10^{-2}\)? Give a reason. [2 marks]

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    Model answer

    \(2 \times 10^{-2} = 0.02\) and \(6 \times 10^{-3} = 0.006\), so \(2 \times 10^{-2}\) is bigger, because \(-2\) is greater than \(-3\).

    Mark scheme

    • \(2 \times 10^{-2}\) — B1
    • A correct reason — Q1
  4. 4 Write [3 marks]

    (a) Write 0.00086 in standard form. [2 marks] (b) Write \(2.5 \times 10^4\) as an ordinary number. [1 mark]

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    Model answer

    (a) The point moves 4 places right, so \(8.6 \times 10^{-4}\). (b) 25 000.

    Mark scheme

    • (a) 8.6 seen — M1
    • (a) \(8.6 \times 10^{-4}\) — A1
    • (b) 25 000 — B1
  5. 5 Write [2 marks]

    Write \(0.8 \times 10^5\) in standard form. [2 marks]

    Show answerHide answer

    Model answer

    \(0.8 = 8 \times 10^{-1}\), so \(0.8 \times 10^5 = 8 \times 10^4\).

    Mark scheme

    • \(80\,000\) or \(8 \times 10^{-1}\) seen — M1
    • \(8 \times 10^4\) — A1
  6. 6 Work out [3 marks]

    A virus is 0.00000012 m long. [3 marks] (a) Write this length in standard form. [2 marks] (b) A bacterium is \(3 \times 10^{-6}\) m long. Which is longer, the virus or the bacterium? [1 mark]

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    Model answer

    (a) The point moves 7 places right, so \(1.2 \times 10^{-7}\) m. (b) \(3 \times 10^{-6}\) is bigger than \(1.2 \times 10^{-7}\), so the bacterium is longer.

    Mark scheme

    • (a) 1.2 seen — M1
    • (a) \(1.2 \times 10^{-7}\) — A1
    • (b) The bacterium — 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 \((3 \times 10^2) \times (2 \times 10^5)\). Give your answer in standard form. [2 marks]

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    Model answer

    \(3 \times 2 = 6\) and \(10^2 \times 10^5 = 10^7\), so \(6 \times 10^7\).

    Mark scheme

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

    Work out \((8 \times 10^9) \div (4 \times 10^6)\). Give your answer in standard form. [2 marks]

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    Model answer

    \(8 \div 4 = 2\) and \(10^9 \div 10^6 = 10^3\), so \(2 \times 10^3\).

    Mark scheme

    • \(8 \div 4\) or \(10^3\) — M1
    • \(2 \times 10^3\) — A1
  3. 3 Work out [3 marks]

    Work out \((5 \times 10^4) \times (6 \times 10^3)\). Give your answer in standard form. [3 marks]

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    Model answer

    \(5 \times 6 = 30\) and \(10^4 \times 10^3 = 10^7\), giving \(30 \times 10^7 = 3 \times 10^8\).

    Mark scheme

    • \(30 \times 10^7\) — M1
    • Adjusts to standard form — M1
    • \(3 \times 10^8\) — A1
  4. 4 Work out [3 marks]

    Work out \(4 \times 10^5 + 6 \times 10^4\). Give your answer in standard form. [3 marks]

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    Model answer

    \(400\,000 + 60\,000 = 460\,000 = 4.6 \times 10^5\).

    Mark scheme

    • Same power or ordinary numbers — M1
    • 460 000 or \(4.6 \times 10^5\) seen — M1
    • \(4.6 \times 10^5\) — A1
  5. 5 Work out [3 marks]

    A grain of sand is \(2 \times 10^{-3}\) m wide. \(5 \times 10^6\) grains are put in a line, side by side. Work out the length of the line. Give your answer in standard form. [3 marks]

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    Model answer

    \((5 \times 10^6) \times (2 \times 10^{-3}) = 10 \times 10^3 = 1 \times 10^4\) m.

    Mark scheme

    • \(5 \times 2 = 10\) or \(10^6 \times 10^{-3} = 10^3\) — M1
    • \(10 \times 10^3\) — M1
    • \(1 \times 10^4\) m — A1
  6. 6 Work out [3 marks]

    Work out \((9 \times 10^{-3}) \div (3 \times 10^{-5})\). Give your answer in standard form. [3 marks]

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    Model answer

    \(9 \div 3 = 3\) and \(10^{-3} \div 10^{-5} = 10^{-3 - (-5)} = 10^2\), so the answer is \(3 \times 10^2\).

    Mark scheme

    • \(9 \div 3 = 3\) — M1
    • \(10^2\) seen — M1
    • \(3 \times 10^2\) — 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]

    Write 0.0764 (a) correct to 2 decimal places, (b) correct to 2 significant figures. [2 marks]

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    Model answer

    (a) The next digit is 6, so 0.08. (b) The first two significant figures are 7 and 6, and the next digit 4 rounds down: 0.076.

    Mark scheme

    • (a) 0.08 — B1
    • (b) 0.076 — B1
  2. 2 Write [2 marks]

    Write 58 700 correct to 1 significant figure. [2 marks]

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    Model answer

    The first figure is 5, and the next digit 8 rounds it up to 6, with zeros to hold the place: 60 000.

    Mark scheme

    • 6 seen — M1
    • 60 000 — A1
  3. 3 Estimate [3 marks]

    Work out an estimate for \(\dfrac{6.1 \times 19.7}{0.41}\). [3 marks]

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    Model answer

    \(\dfrac{6 \times 20}{0.4} = \dfrac{120}{0.4} = 300\).

    Mark scheme

    • Rounds to 6, 20 and 0.4 — M1
    • \(\dfrac{6 \times 20}{0.4}\) — M1
    • 300 — A1
  4. 4 Estimate [3 marks]

    Work out an estimate for \(\dfrac{298 \times 0.52}{5.1}\). [3 marks]

    Show answerHide answer

    Model answer

    \(\dfrac{300 \times 0.5}{5} = \dfrac{150}{5} = 30\).

    Mark scheme

    • Rounds to 300, 0.5 and 5 — M1
    • \(\dfrac{300 \times 0.5}{5}\) — M1
    • 30 — A1
  5. 5 Write [2 marks]

    Write 3.996 correct to 2 decimal places. [2 marks]

    Show answerHide answer

    Model answer

    The next digit is 6, so round the 9 up, which carries: 4.00.

    Mark scheme

    • 4 or 4.0 seen — M1
    • 4.00 — A1
  6. 6 Estimate [3 marks]

    (a) Work out an estimate for \(5.2 \times 8.3\). [2 marks] (b) Is your estimate bigger or smaller than the exact value? Give a reason. [1 mark]

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    Model answer

    (a) \(5 \times 8 = 40\). (b) Both numbers were rounded down, so the estimate is smaller than the exact value.

    Mark scheme

    • (a) 5 and 8 seen — M1
    • (a) 40 — A1
    • (b) Smaller, because both were rounded down — Q1

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]

    \(x = 40\) correct to the nearest 10. Write down the error interval for \(x\). [2 marks]

    Show answerHide answer

    Model answer

    Half of 10 is 5, so \(35 \leq x < 45\).

    Mark scheme

    • 35 and 45 seen — M1
    • \(35 \leq x < 45\) — A1
  2. 2 Write down [2 marks]

    \(x = 0.4\) correct to 1 significant figure. Write down the error interval for \(x\). [2 marks]

    Show answerHide answer

    Model answer

    The unit is 0.1, so half is 0.05, and \(0.35 \leq x < 0.45\).

    Mark scheme

    • 0.35 and 0.45 seen — M1
    • \(0.35 \leq x < 0.45\) — A1
  3. 3 Write down [2 marks]

    The number \(n\) is 5 after it has been truncated to an integer. Write down the error interval for \(n\). [2 marks]

    Show answerHide answer

    Model answer

    \(5 \leq n < 6\).

    Mark scheme

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

    \(a = 9\) cm and \(b = 4\) cm, each correct to the nearest centimetre. Work out the upper bound of \(a - b\). [3 marks]

    Show answerHide answer

    Model answer

    The upper bound of \(a\) is 9.5 and the lower bound of \(b\) is 3.5, so the upper bound of \(a - b\) is \(9.5 - 3.5 = 6\).

    Mark scheme

    • 9.5 or 3.5 seen — M1
    • \(9.5 - 3.5\) — M1
    • 6 — A1
  5. 5 Work out [3 marks]

    The side of a square is 20 cm, correct to the nearest 10 cm. [3 marks] (a) Write down the lower bound of the side. [1 mark] (b) Work out the lower bound of the area of the square. [2 marks]

    Show answerHide answer

    Model answer

    (a) 15 cm. (b) \(15 \times 15 = 225\) cm\(^2\).

    Mark scheme

    • (a) 15 — B1
    • (b) \(15 \times 15\) — M1
    • (b) 225 cm\(^2\) — A1
  6. 6 Work out [3 marks]

    \(a = 2.4\) and \(b = 1.6\), each correct to 1 decimal place. Work out the upper bound of \(a + b\). [3 marks]

    Show answerHide answer

    Model answer

    \(2.45 + 1.65 = 4.1\).

    Mark scheme

    • 2.45 or 1.65 seen — M1
    • \(2.45 + 1.65\) — M1
    • 4.1 — 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 \(\dfrac{7}{20}\) as a decimal. [1 mark] (b) Write \(\dfrac{2}{3}\) as a recurring decimal, using dot notation. [1 mark]

    Show answerHide answer

    Model answer

    (a) \(7 \div 20 = 0.35\). (b) \(0.\dot{6}\).

    Mark scheme

    • (a) 0.35 — B1
    • (b) \(0.\dot{6}\) — B1
  2. 2 Write [2 marks]

    Write \(\dfrac{5}{9}\) as a recurring decimal, using dot notation. [2 marks]

    Show answerHide answer

    Model answer

    \(5 \div 9 = 0.5555\ldots = 0.\dot{5}\).

    Mark scheme

    • 0.555 seen — M1
    • \(0.\dot{5}\) — A1
  3. 3 Show that [3 marks]

    Write \(0.\dot{1}\dot{8}\) as a fraction in its simplest form. [3 marks]

    Show answerHide answer

    Model answer

    Let \(x = 0.1818\ldots\). Then \(100x = 18.1818\ldots\), so \(99x = 18\) and \(x = \dfrac{18}{99} = \dfrac{2}{11}\).

    Mark scheme

    • \(100x = 18.1818\ldots\) — M1
    • \(99x = 18\) — M1
    • \(\dfrac{2}{11}\) — A1
  4. 4 Show that [3 marks]

    Write \(0.1\dot{6}\) as a fraction in its simplest form. [3 marks]

    Show answerHide answer

    Model answer

    Let \(x = 0.1666\ldots\). Then \(100x = 16.666\ldots\) and \(10x = 1.666\ldots\). Subtracting, \(90x = 15\), so \(x = \dfrac{15}{90} = \dfrac{1}{6}\).

    Mark scheme

    • \(100x\) and \(10x\) seen — M1
    • \(90x = 15\) — M1
    • \(\dfrac{1}{6}\) — A1
  5. 5 Explain [2 marks]

    Here are four numbers: \(\dfrac{22}{7}\), \(\sqrt{25}\), \(\sqrt{11}\) and \(0.\dot{4}\). Which number is irrational? Give a reason. [2 marks]

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    Model answer

    \(\sqrt{11}\). The others can be written as fractions: \(\dfrac{22}{7}\), \(5\) and \(\dfrac{4}{9}\). \(\sqrt{11}\) is not a perfect square, so it cannot be written as a fraction.

    Mark scheme

    • \(\sqrt{11}\) — B1
    • A correct reason — Q1
  6. 6 Show that [3 marks]

    Show that \(0.\dot{5}\dot{4} = \dfrac{6}{11}\). [3 marks]

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    Model answer

    Let \(x = 0.5454\ldots\). Then \(100x = 54.5454\ldots\), so \(99x = 54\) and \(x = \dfrac{54}{99} = \dfrac{6}{11}\).

    Mark scheme

    • \(100x = 54.5454\ldots\) — M1
    • \(99x = 54\) — M1
    • \(\dfrac{54}{99} = \dfrac{6}{11}\) — Q1

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 \(x = \dfrac{15}{90} = \dfrac{1}{6}\).