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Maths · Standard Form and Accuracy

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Error Intervals and Bounds

Writing error intervals for rounded and truncated numbers, and finding upper and lower bounds of calculations.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Write an error interval for a number that has been rounded or truncated.
  2. 2Use inequality notation with the correct signs.
  3. 3Find the upper and lower bounds of a rounded measurement.
  4. 4Calculate the bounds of a sum, difference, product or quotient (Higher tier).

How wrong could it be

A rounded number is not exact, so the true value could be a little more or a little less. An error interval shows every value that would round to the given number, and it is written with inequality signs. The lower end is included and the upper end is not, because a value that is exactly halfway rounds up. Error intervals are Foundation content. Working out the bounds of a calculation, for example the biggest possible area of a rectangle, is Higher tier content on every board, and it needs a careful choice of upper or lower bounds for each number.

Error intervals

An error interval is written as \(\text{lower bound} \leq x < \text{upper bound}\).

  • Rounded to the nearest whole number

    12 cm means \(11.5 \leq L < 12.5\).

  • Rounded to 1 decimal place

    3.7 means \(3.65 \leq x < 3.75\).

  • Half the unit

    The bounds are half a unit of the rounding below and above the number.

  • Truncated

    A number truncated to a whole number, such as 5, means \(5 \leq n < 6\), since digits are cut off, not rounded.

Writing an error interval

The mass of a parcel is 4.8 kg, rounded to 1 decimal place. Write down the error interval for the mass \(m\).

Show the solutionHide the solution
  1. 1 Unit of rounding 0.1 kg, so half is 0.05 kg.
  2. 2 Lower bound \(4.8 - 0.05 = 4.75\).
  3. 3 Upper bound \(4.8 + 0.05 = 4.85\).
  4. 4 Write it \(4.75 \leq m < 4.85\).

Answer\(4.75 \leq m < 4.85\)

Bounds of a calculation (Higher tier)

To get the biggest or smallest possible answer, choose the bound of each number that pushes the answer the right way.

  • Sum

    The upper bound of \(a + b\) uses the upper bounds of \(a\) and \(b\).

  • Difference

    The upper bound of \(a - b\) uses the upper bound of \(a\) and the lower bound of \(b\).

  • Product

    The upper bound of \(a \times b\) uses both upper bounds.

  • Quotient

    The upper bound of \(a \div b\) uses the upper bound of \(a\) and the lower bound of \(b\).

Bounds of an area (Higher tier)

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

Show the solutionHide the solution
  1. 1 Bounds for the length \(7.5 \leq l < 8.5\).
  2. 2 Bounds for the width \(4.5 \leq w < 5.5\).
  3. 3 Upper bound \(8.5 \times 5.5 = 46.75\).
  4. 4 Lower bound \(7.5 \times 4.5 = 33.75\).

AnswerUpper bound 46.75 cm\(^2\), lower bound 33.75 cm\(^2\)

Test yourself

  1. 1

    What is the error interval for 12 cm, rounded to the nearest cm?

    Show answerHide answer

    \(11.5 \leq L < 12.5\).

  2. 2

    Which end of an error interval uses the less than or equal to sign?

    Show answerHide answer

    The lower end.

  3. 3

    What is the error interval for 5, truncated to a whole number?

    Show answerHide answer

    \(5 \leq n < 6\).

  4. 4

    Which bounds give the upper bound of \(a - b\)?

    Show answerHide answer

    Upper bound of \(a\) and lower bound of \(b\) (Higher tier).

  5. 5

    What is half of 0.1?

    Show answerHide answer

    0.05.

Exam technique: accuracy

Think about which direction each number pushes the answer.

  • Write both bounds for each number

    Then pick the right one.

  • Use the correct signs

    \(\leq\) at the bottom and \(<\) at the top.

  • Do not just round

    The upper bound is not 8.5 rounded.

  • Say which kind of rounding

    Truncation uses the unit, not half.

Summary and exam focus

  • An error interval is \(\text{lower} \leq x < \text{upper}\), found by going half a unit down and up.
  • Truncation cuts digits off, so the interval is one whole unit.
  • For the upper bound of a sum or product, use the upper bounds of the numbers.
  • For a difference or quotient, use the upper bound of the first number and the lower bound of the second (Higher tier).

Exam focus

\(x = 3.7\) correct to 1 decimal place. Write down the error interval for \(x\). (2 marks) (2 marks)

Half of 0.1 is 0.05, so \(3.65 \leq x < 3.75\). Both signs matter for the second mark: the lower end has \(\leq\) and the upper end has \(<\).

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Error interval
The range of values that round to a given number.
Lower bound
The smallest value in an error interval.
Upper bound
The value that the interval goes up to but does not include.
Truncation
Cutting off digits without rounding.
Limit of accuracy
The bounds of a measurement or rounded number.
Inequality
A statement using signs such as \(<\) and \(\leq\).
Rounding
Replacing a number with a simpler one that is close to it.
Half a unit
Half the size of the rounding unit.
Measurement
A quantity found by measuring, which is never exact.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Write down 2 marks Easier

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

Mark scheme — 2 marks available

  • 35 and 45 seen — M1
  • \(35 \leq x < 45\) — A1

Model answer

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

2. Exam question Write down 2 marks Easier

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

Mark scheme — 2 marks available

  • 0.35 and 0.45 seen — M1
  • \(0.35 \leq x < 0.45\) — A1

Model answer

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

3. Exam question Write down 2 marks Core

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

Mark scheme — 2 marks available

  • 5 and 6 seen — M1
  • \(5 \leq n < 6\) — A1

Model answer

\(5 \leq n < 6\).

4. Exam question Work out 3 marks Stretch

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

Mark scheme — 3 marks available

  • 9.5 or 3.5 seen — M1
  • \(9.5 - 3.5\) — M1
  • 6 — A1

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 \(9.5 - 3.5 = 6\).

5. Exam question Work out 3 marks Stretch

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]

Mark scheme — 3 marks available

  • (a) 15 — B1
  • (b) \(15 \times 15\) — M1
  • (b) 225 cm\(^2\) — A1

Model answer

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

6. Exam question Work out 3 marks Stretch

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

Mark scheme — 3 marks available

  • 2.45 or 1.65 seen — M1
  • \(2.45 + 1.65\) — M1
  • 4.1 — A1

Model answer

\(2.45 + 1.65 = 4.1\).

7. Multiple choice 1 mark Easier

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

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

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

8. Multiple choice 1 mark Easier

\(x = 3.7\) 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 \(3.65 \leq x < 3.75\) Correct

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

9. Multiple choice 1 mark Core

\(n = 5\) 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 \(5 \leq n < 6\) Correct
  4. D \(5 \leq n \leq 6\)

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

10. Multiple choice 1 mark Easier

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

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

Why: Half of 10 is 5.

11. Multiple choice 1 mark Core

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

  1. A It would round up to the next value Correct
  2. B It is too big to measure
  3. C It rounds down
  4. D It is a typing convention

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

12. Multiple choice 1 mark Core

\(x = 0.4\) 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 \(0.35 \leq x < 0.45\) Correct

Why: The unit is 0.1, so half is 0.05.

13. Multiple choice 1 mark Core

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

  1. A The upper bound of \(a\) plus the lower bound of \(b\)
  2. B The lower bound of \(a\) plus the upper bound of \(b\)
  3. C The upper bound of \(a\) plus the upper bound of \(b\) Correct
  4. D The lower bounds added

Why: The biggest total comes from the biggest values.

14. Multiple choice 1 mark Stretch

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

  1. A The upper bound of \(a\) minus the upper bound of \(b\)
  2. B The upper bound of \(a\) minus the lower bound of \(b\) Correct
  3. C The lower bound of \(a\) minus the lower bound of \(b\)
  4. D The lower bound of \(a\) minus the upper bound of \(b\)

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

15. Multiple choice 1 mark Stretch

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\(^2\) Correct
  2. B \(40\) cm\(^2\)
  3. C \(45\) cm\(^2\)
  4. D \(33.75\) cm\(^2\)

Why: \(8.5 \times 5.5 = 46.75\).