✓ Revise Right course
To find upper and lower bounds, halve the degree of accuracy, then subtract it for the lower bound and add it for the upper. 12 questions answered.
Limits of Accuracy and Upper and Lower Bounds is part of Measures and accuracy in GCSE AQA Higher Maths.
4 min
Reading time
22
Lessons in this topic
GCSE AQA Higher
Pathway
Reviewed by Revise Right
Content review
To find upper and lower bounds, halve the degree of accuracy, then subtract it for the lower bound and add it for the upper. 12 questions answered.
AQA GCSE Maths mock draft, 26–31 Oct. Draft coverage: Maths Paper 1 (Higher tier). Reviewed for an original-practice sitting; check student availability before registering. Marking and results will follow later, after separate approval. Eligible new accounts receive a five-day Pro trial when signup email verification completes. Previous trials keep their original expiry. After expiry, new papers need Pro unless you have an explicitly confirmed free sitting offer for this event. That offer adds no Pro time or AI learning. Saved attempts and released results remain readable. No card or automatic charge.
Use this lesson to stay inside Measures and accuracy while connecting the detail back to the wider Number topic.
Read the core explanations from Limits of Accuracy and Upper and Lower Bounds before testing yourself from memory.
Every measurement you read in a maths problem has been rounded. A length of 8 cm could actually be anywhere from 7.5 cm up to (but not including) 8.5 cm — both round to 8 cm to the nearest centimetre. This raises a real question: if you use that measurement in a calculation, how wrong could your answer be?
That is exactly what limits of accuracy and upper and lower bounds let you answer. Rather than pretending a measurement is exact, you work out the worst-case and best-case values — and that tells you how much you can trust your result.
When a value has been rounded, there is a range of original values that could have produced it. The lower bound is the smallest value in that range, and the upper bound is the largest. Together they define the limits of accuracy.
The inequality is strict at the upper end ($<$ not $\leq$) because a value exactly at the upper bound would round up to the next value, not to $x$. This is a detail examiners test directly, so it is worth locking in now.
The method is the same regardless of the degree of accuracy — identify what the rounding unit is, halve it, then add and subtract. Here are two contrasting examples to build fluency.
Worked example: Finding Bounds: Rounded to the Nearest Integer
A plank of wood is measured as 12 m, rounded to the nearest metre. Find the upper and lower bounds of its length.
Answer: Lower bound = 11.5 m, Upper bound = 12.5 m. Limits: $11.5 \leq x < 12.5$
Worked example: Finding Bounds: Rounded to 1 Decimal Place
A mass is recorded as 4.6 kg, rounded to 1 decimal place. Find the upper and lower bounds.
Answer: Lower bound = 4.55 kg, Upper bound = 4.65 kg. Limits: $4.55 \leq x < 4.65$
Finding bounds for a single measurement is the foundation. The real exam challenge is combining bounds from two or more measurements in a calculation. The rule here is not obvious, and it is where most marks are dropped.
The key principle: to get the upper bound of the result, you must combine the bounds of each measurement in the way that makes the result as large as possible. To get the lower bound of the result, combine them to make the result as small as possible.
| Operation | Upper bound of result | Lower bound of result |
|---|---|---|
| $A + B$ | $\text{UB}(A) + \text{UB}(B)$ | $\text{LB}(A) + \text{LB}(B)$ |
| $A - B$ | $\text{UB}(A) - \text{LB}(B)$ | $\text{LB}(A) - \text{UB}(B)$ |
| $A \times B$ | $\text{UB}(A) \times \text{UB}(B)$ | $\text{LB}(A) \times \text{LB}(B)$ |
| $A \div B$ | $\text{UB}(A) \div \text{LB}(B)$ | $\text{LB}(A) \div \text{UB}(B)$ |
The subtraction and division rows are the ones that trip students up. For subtraction, the upper bound of the result uses the lower bound of the quantity being subtracted — because subtracting less gives a larger answer. For division, dividing by a smaller number gives a larger result, so the upper bound uses the lower bound of the divisor.
Worked example: Upper and Lower Bounds of an Area
A rectangle has length $L = 9$ cm and width $W = 4$ cm, both rounded to the nearest centimetre. Calculate the upper bound and lower bound of the area. Give your answers to 2 decimal places where necessary.
Answer: Upper bound of area = 42.75 cm², Lower bound of area = 29.75 cm²
Worked example: Upper and Lower Bounds of a Division (Speed Calculation)
Mohammed runs a distance of $d = 100$ m, rounded to the nearest metre, in a time of $t = 12$ s, rounded to the nearest second. Calculate the upper bound of his speed. Give your answer to 3 significant figures.
Answer: Upper bound of speed = 8.74 m/s (3 s.f.)
Sometimes an exam question asks you to use bounds to make a decision or give a sensible answer. This is where you apply and interpret the limits of accuracy rather than just calculate them.
A typical question might ask: "Can you be certain that the total length exceeds 20 m?" To answer this, you check the lower bound of the total. If the lower bound is greater than 20 m, then yes — even in the worst case, the total exceeds 20 m. If the lower bound is less than or equal to 20 m, you cannot be certain.
Worked example: Interpreting Bounds to Make a Decision
Two planks have lengths $A = 3.2$ m and $B = 2.9$ m, both rounded to 1 decimal place. Aisha needs a combined length of at least 6 m. Can she be certain the planks are long enough?
Answer: Aisha can be certain the total is ≥ 6.00 m. However, she cannot be certain it strictly exceeds 6 m since the lower bound is exactly 6.00 m.
These problems cover the full range of bound types you will meet in an Higher-tier exam. Work through each one before checking the solution.
You now have the method. This section focuses on the specific errors that cost marks in GCSE Higher exam questions on this topic. Remember to include a teaching or transition between blocks to reinforce learning.
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-21.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
To find upper and lower bounds, halve the degree of accuracy, then subtract it for the lower bound and add it for the upper bound. A plank measured as 12 m to the nearest metre has a lower bound of 11.5 m and an upper bound of 12.5 m.
Limits of accuracy are the lower and upper bounds of a rounded value, the smallest and largest values that could round to it. If x is rounded to a degree of accuracy d, the lower bound is x - d/2 and the upper bound is x + d/2.
The right degree of accuracy for a bounds answer is the one the question asks for, such as 3 significant figures, and rounding should happen only at the final step. For example, 100.5 ÷ 11.5 = 8.739... is given as 8.74 m/s to 3 significant figures.
For a value x rounded to a degree of accuracy d, the lower bound is x - d/2 and the upper bound is x + d/2. The true value satisfies lower bound ≤ true value < upper bound.
The limits of accuracy rules say the upper bound of A + B is UB(A) + UB(B), of A - B is UB(A) - LB(B), of A × B is UB(A) × UB(B), and of A ÷ B is UB(A) ÷ LB(B). The lower bounds are LB(A) + LB(B), LB(A) - UB(B), LB(A) × LB(B) and LB(A) ÷ UB(B).
The upper bound of A - B is UB(A) - LB(B), because subtracting less gives a larger answer, and the lower bound is LB(A) - UB(B). For division, the upper bound of A ÷ B is UB(A) ÷ LB(B), because dividing by a smaller number gives a larger result.
The upper bound is not included because a value exactly at the upper bound would round up to the next value. For example, a mass of exactly 4.65 kg would round to 4.7 kg, so the limits for 4.6 kg run from 4.55 kg up to, but not including, 4.65 kg.
To decide whether a result is certain, check the worst case using the bounds. Two planks of 3.2 m and 2.9 m, both to 1 d.p., have a lower bound total of 3.15 + 2.85 = 6.00 m, so they are certainly at least 6 m long, but not certainly more than 6 m.
A limit of accuracy is one end of the range of original values that could have produced a rounded measurement: the lower bound or the upper bound. A length of 8 cm to the nearest centimetre could be anywhere from 7.5 cm up to, but not including, 8.5 cm.
The bounds of 4.6 kg to 1 decimal place are 4.55 kg and 4.65 kg, so 4.55 ≤ x < 4.65. The degree of accuracy is 0.1 kg, so 0.05 kg is subtracted and added, and 4.65 is not included because it would round to 4.7.
The lower bound of a subtraction A - B is LB(A) - UB(B): the smallest possible first value minus the largest possible second value. Subtracting more gives a smaller answer, which is why the upper bound of B is used.
The upper bound of a sum A + B is UB(A) + UB(B), and the lower bound is LB(A) + LB(B). For planks of 3.2 m and 2.9 m to 1 decimal place, the lower bound of the total is 3.15 + 2.85 = 6.00 m.