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Exam questions · Maths · Standard Form and Accuracy

Error Intervals and Bounds

  • 6 exam questions
  • 15 marks
  • 9 quick checks
  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 \(L\).

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    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 \(x\).

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    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 \(n\).

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    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 \(a + b\).

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    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.

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    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 \(a - b\).

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    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\(12 \leq L < 13\)
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    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\(3.65 \leq x < 3.75\)
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    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\(5 \leq n \leq 6\)
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    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\(39 \leq x < 41\)
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    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
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    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\(0.35 \leq x < 0.45\)
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    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
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    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 \(b\)
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    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\(^2\)
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    A: \(46.75\) cm\(^2\)

    \(8.5 \times 5.5 = 46.75\).