title: "Bounds GCSE Maths: Upper, Lower & Error Intervals" date: "2026-06-19" excerpt: "Upper and lower bounds explained for GCSE — how to find them, write error intervals, the truncation trap, and how to use bounds in calculations, with worked examples." metaDescription: "Bounds in GCSE maths made simple — find upper and lower bounds, write error intervals, and use bounds in calculations, with worked examples and exam tips."

Why bounds in GCSE maths trip so many students up

Bounds in GCSE maths is one of those topics that feels slippery — not because the arithmetic is hard, but because it asks you to think about a number you can't quite see. When someone tells you a length is "24 cm to the nearest centimetre", the real length isn't 24. It's somewhere near 24. Upper and lower bounds are simply the two ends of that "somewhere near".

Once you understand that one idea, the whole topic falls into place: writing error intervals, dealing with truncation, and the dreaded Grade 7–9 calculation questions where you have to combine bounds. These questions appear on every exam board — AQA, Edexcel and OCR — and they're worth easy marks if you have a reliable method.

This guide covers what upper and lower bounds actually are, how to find them from any rounded value, the truncation rule that catches people out, and exactly how to use bounds in a calculation — with worked examples for each.

What are upper and lower bounds?

Every rounded measurement hides a range of possible true values. The lower bound is the smallest value that would still round to the figure you were given. The upper bound is the largest.

Take a length given as 24 cm to the nearest centimetre. Anything from 23.5 cm up to (but not including) 24.5 cm rounds to 24. So:

• Lower bound = 23.5 cm
• Upper bound = 24.5 cm

We write this as an error interval using inequalities:

$$23.5 \leq x < 24.5$$

Notice the two different signs. The lower bound uses ≤ because 23.5 does round up to 24. The upper bound uses < because 24.5 would round up to 25, not down to 24 — so it's the limit you approach but never reach. Getting these two signs the right way round is one of the most common places students drop a mark, so it's worth slowing down and checking them every single time you write an error interval — examiners award the mark for the inequality, not just the two numbers.

How to find bounds from a rounded value

There's a single method that works every time, whatever the degree of rounding:

Halve the unit of rounding, then add and subtract it from the given value.

The "unit of rounding" is the size of the gap you're rounding to. To the nearest cm, the unit is 1. To the nearest 10, the unit is 10. To 1 decimal place, the unit is 0.1.

Worked example. A mass is given as 4.6 kg to 1 decimal place. Find the error interval.

1. The unit of rounding is 0.1 (one decimal place).
2. Half the unit is 0.05.
3. Lower bound = 4.6 − 0.05 = 4.55
4. Upper bound = 4.6 + 0.05 = 4.65

So the error interval is:

$$4.55 \leq x < 4.65$$

The same method handles trickier cases. To the nearest 10, a value of 150 has a half-unit of 5, giving $145 \leq x < 155$. Don't fall into the trap of always using 0.5 — you halve whatever the unit is, not always one.

Truncation: the bounds rule that catches everyone out

Exam boards love to swap "rounded" for truncated, because the rule changes — and students who are on autopilot get it wrong.

Truncation means chopping off the extra digits rather than rounding them. If 7.86 is truncated to 1 decimal place, it becomes 7.8, not 7.9 — the rest is simply discarded.

This changes the bounds. A truncated number is always the lowest the true value could be, because nothing was rounded up. So for a value of 7.8 truncated to 1 decimal place:

• Lower bound = 7.8 (the truncated value itself)
• Upper bound = 7.9

$$7.8 \leq x < 7.9$$

The shortcut: for truncation, the given number is the lower bound. You only add the unit to find the upper bound — you never subtract. If you remember nothing else about truncation, remember that.

Using bounds in calculations — the Grade 7–9 question

The hardest bounds questions give you two or more rounded quantities and ask for the maximum or minimum of a calculation. The skill is choosing which bound to use for each number. Here are the rules:

• Adding: Maximum = UB + UB. Minimum = LB + LB.
• Subtracting: Maximum = UB − LB. Minimum = LB − UB.
• Multiplying: Maximum = UB × UB. Minimum = LB × LB.
• Dividing: Maximum = UB ÷ LB. Minimum = LB ÷ UB.

Subtraction and division are the ones to watch — you use opposite bounds top and bottom. To make a division as big as possible, you want the biggest top and the smallest bottom.

Worked example. A car travels 150 m (to the nearest 10 m) in 20 s (to the nearest second). Find the maximum possible average speed.

1. Bounds for distance: $145 \leq d < 155$
2. Bounds for time: $19.5 \leq t < 20.5$
3. Speed = distance ÷ time, so the maximum speed uses the largest distance and the smallest time:

$$\text{max speed} = \frac{155}{19.5} = 7.95 \text{ m/s (to 3 s.f.)}$$

The minimum speed would use the smallest distance over the largest time: $145 \div 20.5 = 7.07$ m/s.

There's a neat follow-up examiners use: "Give the speed to a suitable degree of accuracy." If the upper and lower bounds of your answer agree when rounded — say both round to 7.9 m/s to 2 significant figures — you can confidently state the answer to that accuracy, because every possible true value falls in that range. This shows you understand why bounds matter, not just how to calculate them.

Common mistakes examiners flag

• Mixing up the inequality signs — the lower bound takes ≤, the upper bound takes <. Practise writing the full error interval every time so it becomes automatic.
• Always halving to 0.5 — you halve the unit of rounding. Nearest 10 means half-unit 5; nearest 100 means 50.
• Treating truncation like rounding — with truncation the given number is the lower bound; you only ever add to reach the upper bound.
• Using the same bounds top and bottom in a division or subtraction — these need opposite bounds. Maximum division = biggest ÷ smallest.
• Rounding too early — keep full bound values through the calculation and only round the final answer.

Summary

Bounds in GCSE maths come down to one idea: a rounded number stands in for a whole range of true values, and the upper and lower bounds are the two ends of that range. Find them by halving the unit of rounding (or, for truncation, by treating the given value as the lower bound), write the error interval with the correct signs, and choose opposite bounds for subtraction and division.

The fastest way to make this automatic is targeted practice on bounds questions until the method is second nature. If you want to know which topics like this are costing you the most marks, the Bow Tie Maths app builds a personal Topic Radar from your practice answers — so you always know exactly where to focus next.