Upper and Lower Bounds and Error Intervals — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the range of values a rounded measurement could really have taken. By the end of this guide you should be able to find the upper and lower bounds of a rounded number and write the error interval using inequalities.

You should also be able to find bounds for a value rounded to decimal places or significant figures, calculate with bounds to find the largest and smallest possible results of a sum, difference, product or quotient, and give an answer to a suitable degree of accuracy.

The organising idea is that a rounded number is not a value but a range. A length recorded as 24 cm to the nearest centimetre was never exactly 24 — it was anything from 23.5 up to but not including 24.5, and the rounding threw that detail away. Every calculation using it therefore has a range of possible answers too. Once you accept that a measurement is a range, the whole topic becomes a question of which end of each range to use to make the answer as large or as small as possible.

Key terms and definitions

Lower bound — the smallest value the measurement could have been.

Upper bound — the value it must be below. It is the limit, not an attainable value.

Error interval — the range of possible values, written with inequalities.

Degree of accuracy — how the value was rounded: to the nearest unit, to decimal places, or to significant figures.

Half a unit — half of the rounding degree, which is what is added and subtracted to find the bounds.

Truncated — cut off rather than rounded, which gives different bounds.

Core concepts

Finding the bounds

A value rounded to a given degree of accuracy could be up to half a unit either side.

For 24 cm to the nearest cm, half a unit is 0.5, so the lower bound is 23.5 and the upper bound is 24.5.

The lower bound is attainable: 23.5 rounds up to 24, so the measurement really could have been 23.5. The upper bound is not: 24.5 rounds up to 25, so the measurement must be below it.

That asymmetry is the reason the error interval uses two different inequality signs, and it is the single detail this topic is built to test.

Writing the error interval

Use ≤ for the lower bound and < for the upper:

23.5 ≤ x < 24.5

Reading it aloud helps: x is at least 23.5 and less than 24.5.

Using ≤ at both ends is the commonest error, and it claims the measurement could have been 24.5, which would have rounded to 25 instead.

Matching the half-unit to the accuracy

The half-unit depends entirely on how the value was rounded.

To the nearest whole number, half a unit is 0.5. To the nearest 10, it is 5. To the nearest 0.1 or 1 decimal place, it is 0.05. To 2 decimal places, it is 0.005.

So 3.7 to 1 d.p. gives 3.65 ≤ x < 3.75, and 250 to the nearest 10 gives 245 ≤ x < 255.

For significant figures, work out what the last significant digit is worth. In 4500 to 2 s.f., the last significant digit is the 5, worth 100, so half a unit is 50, giving 4450 ≤ x < 4550.

Getting this first step right matters more than anything else here, because every later calculation inherits it.

Truncation is different

A truncated value has simply been cut off rather than rounded, so it cannot be above the stated figure.

A length truncated to 24 cm lies in 24 ≤ x < 25, since anything from 24 up to just under 25 truncates to 24.

Note the lower bound is the stated value itself, not half a unit below it. Questions do use the word truncated deliberately, so watch for it.

Calculating with bounds: addition and multiplication

To make a sum or a product as large as possible, use the upper bound of everything. To make it as small as possible, use the lower bound of everything.

For a rectangle measuring 8 cm by 5 cm, each to the nearest cm, the bounds are 7.5 to 8.5 and 4.5 to 5.5.

The largest possible area is 8.5 × 5.5 = 46.75 cm², and the smallest is 7.5 × 4.5 = 33.75 cm².

That is the straightforward case, because making each part bigger makes the answer bigger.

Calculating with bounds: subtraction and division

Here the rule reverses for the second quantity, and this is where marks are lost.

For the largest difference, take the upper bound of the first and the lower bound of the second — subtracting as little as possible leaves as much as possible.

For the smallest difference, take the lower bound of the first and the upper bound of the second.

Division behaves the same way: the largest quotient comes from the upper bound on top and the lower bound underneath, since dividing by a smaller number gives a bigger answer.

So for a distance of 100 m to the nearest metre in a time of 20 s to the nearest second, the greatest possible speed is 100.5 ÷ 19.5 = 5.15 m/s, and the least is 99.5 ÷ 20.5 = 4.85 m/s.

Using the upper bound for both would give 100.5 ÷ 20.5, which is neither the largest nor the smallest — it is simply wrong.

A rule that covers every case

Rather than memorising four separate rules, ask what makes the answer bigger.

Dividing by something smaller makes the answer bigger, so use the lower bound underneath. Subtracting something smaller leaves more, so use the lower bound second.

Working from that question handles any formula, including ones combining several operations.

Giving an answer to a suitable degree of accuracy

Some questions ask for the answer to an appropriate accuracy, which means comparing the two bounds and quoting the digits they agree on.

If the upper and lower bounds of a speed are 5.1538 and 5.1282, the two agree on 5.1 and diverge after that, so the sensible answer is 5.1 m/s, and the reason is that the bounds agree to that many figures.

Quoting more digits than the bounds support is the error; quoting fewer wastes information. The answer needs the reason as well as the value.

Worked examples

Example 1: Bounds and an error interval

A mass is given as 6.8 kg to 1 decimal place. Write the error interval.

Rounding to 1 decimal place means the half-unit is 0.05.

The lower bound is 6.8 − 0.05 = 6.75, and the upper bound is 6.8 + 0.05 = 6.85.

The error interval is 6.75 ≤ m < 6.85.

Both signs matter: 6.75 rounds up to 6.8 so it is possible, while 6.85 rounds up to 6.9 so it is not.

Example 2: A maximum area

A rectangular field measures 45 m by 32 m, each to the nearest metre. Find the greatest possible area.

The bounds are 44.5 ≤ length < 45.5 and 31.5 ≤ width < 32.5.

For the greatest area, multiply the two upper bounds: 45.5 × 32.5 = 1478.75 m².

Strictly this value is not attainable, since neither upper bound is reachable, but it is the standard expected answer and is described as the upper bound of the area.

Example 3: Division, and a suitable accuracy

A car travels 250 m, measured to the nearest 10 m, in 12 s, measured to the nearest second. Find the upper and lower bounds for its speed, and give the speed to a suitable degree of accuracy.

The distance bounds are 245 ≤ d < 255, since half of 10 is 5. The time bounds are 11.5 ≤ t < 12.5.

The greatest speed uses the largest distance and the smallest time: 255 ÷ 11.5 = 22.17 m/s.

The least speed uses the smallest distance and the largest time: 245 ÷ 12.5 = 19.6 m/s.

The two bounds are 22.17 and 19.6, which do not even agree on the first digit. The only safe statement is that the speed is about 20 m/s, and a question asking for a suitable accuracy would expect the reason: the bounds differ too much to justify any further precision.

Common mistakes and how to avoid them

Using ≤ at both ends of the error interval. The upper bound is not attainable, so it takes <.

Using 0.5 regardless of the accuracy. The half-unit matches the rounding: 5 for the nearest 10, 0.05 for 1 d.p.

Treating truncation as rounding. A truncated value has the stated figure as its lower bound.

Using upper bounds throughout for a division or subtraction. Dividing or subtracting needs the lower bound for the second quantity to maximise the answer.

Forgetting to work out what the last significant figure is worth. In 4500 to 2 s.f., it is worth 100, so the half-unit is 50.

Quoting an answer to more figures than the bounds support. Compare the two bounds and keep only the digits they agree on.

Rounding the bounds themselves. Keep them exact, since they are the limits of the range.

Exam technique for "Upper and Lower Bounds"

Write both bounds for every quantity before doing any calculation, and label them clearly as upper and lower. That list is usually worth a mark and prevents the wrong bound being picked later.

State the half-unit you are using and where it came from, especially for significant figures.

For a maximum or minimum, ask what makes the answer bigger rather than reciting a rule — it handles subtraction and division correctly without extra memorising.

Use the exact bound values in the calculation rather than rounded versions.

When asked for a suitable degree of accuracy, quote the digits the two bounds agree on and give that agreement as your reason.

Check the inequality signs in the final error interval one last time.

Quick revision summary

A rounded number is a range, not a value. Every calculation using it therefore has a range of possible answers.

The bounds lie half a unit either side, where the half-unit matches the accuracy: 0.5 for the nearest whole number, 5 for the nearest 10, 0.05 for 1 d.p., 0.005 for 2 d.p.

For significant figures, work out what the last significant digit is worth: 4500 to 2 s.f. gives 4450 ≤ x < 4550.

The error interval uses ≤ for the lower bound and < for the upper, because the lower bound is attainable and the upper is not: 23.5 ≤ x < 24.5.

Truncation differs: 24 truncated gives 24 ≤ x < 25.

To maximise a sum or product use all upper bounds. To maximise a difference or quotient, use the upper bound first and the lower bound second — subtracting or dividing by less gives more.

For a suitable degree of accuracy, quote only the digits on which the upper and lower bounds agree, and say that is why.