Upper and lower bounds calculator (error intervals)

Q: How do you find upper and lower bounds?

Halve the unit the value was rounded to, then subtract it for the lower bound and add it for the upper bound.

Q: What is an error interval?

The range of possible true values, written lower bound ≤ x < upper bound.

Q: Is anything I type sent to a server?

No. The calculation runs in your browser and nothing is stored.

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When a measurement has been rounded, its true value lies in a range. Enter the rounded value and how it was rounded, and the calculator gives the lower bound, the upper bound and the error interval, written as:

\text{lower bound} \leq x < \text{upper bound}

For rounding, the bounds are half a unit of accuracy either side of the value.

ValueAccuracyNumber of decimal places

Error interval

7.35 ≤ x < 7.45

Lower bound 7.35 · upper bound 7.45

Working

1Rounded to 1 decimal place: the step is 0.1, so half of it is 0.05
2Lower bound = 7.4 − 0.05 = 7.35
3Upper bound = 7.4 + 0.05 = 7.45
4Error interval: 7.35 ≤ x < 7.45

Bounds in calculations? The AI tutor explains which bound to use for the biggest or smallest answer.Try it free

Worked examples

7.4 to 1 decimal place: the unit is 0.1, half of it is 0.05. Error interval: $7.35 \leq x < 7.45$ .
300 to the nearest 10: half of 10 is 5. Error interval: $295 \leq x < 305$ .
0.62 to 2 significant figures: the unit is 0.01. Error interval: $0.615 \leq x < 0.625$ .

Why < and not ≤ at the top

7.45 would round up to 7.5, not 7.4, so the upper bound itself isn't included: the sign is strictly less than. But 7.35 does round to 7.4, so the lower bound is included with $\leq$ . Exam markers look for exactly these signs.

Truncation

Truncating means chopping digits off, not rounding. If 3.7 is a value truncated to 1 decimal place, the true value could be anything from 3.7 up to (but not including) 3.8: $3.7 \leq x < 3.8$ . Truncated error intervals start at the value itself.

Bounds in calculations

To find the largest or smallest possible result of a calculation:

Calculation	Maximum	Minimum
$a + b$	upper + upper	lower + lower
$a - b$	upper − lower	lower − upper
$a \times b$	upper × upper	lower × lower
$a \div b$	upper ÷ lower	lower ÷ upper

Example: a rectangle measures 8 cm and 5 cm, each to the nearest cm. The largest possible area is $8.5 \times 5.5 = 46.75$ cm².

Common mistakes

Adding a whole unit instead of half.
Using $\leq$ at the upper bound.
For division, using upper ÷ upper for the maximum (it's upper ÷ lower).

Practise with our bounds questions, and see our guide to significant figures.

Exam tips

Write the unit of accuracy first ("to the nearest 0.1"), then halve it.
For calculations with bounds, show which bound you used for each value.
Error intervals appear on both tiers at GCSE; calculations with bounds are usually Higher tier.

Frequently asked questions

How do you find upper and lower bounds?

Halve the unit the value was rounded to, then subtract it for the lower bound and add it for the upper bound.

What is an error interval?

The range of possible true values, written lower bound ≤ x < upper bound.

What is the error interval for 7.4 to 1 decimal place?

7.35 ≤ x < 7.45.

Is anything I type sent to a server?

No. The calculation runs in your browser and nothing is stored.

Sources

GCSE mathematics: subject content and assessment objectives. Department for Education.
Cambridge IGCSE Mathematics 0580 syllabus. Cambridge International.

Bounds and error intervals questions and answers

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