UPPER AND LOWER BOUNDS IGCSE MATHS: THE FULL WORKED GUIDE

Upper and Lower Bounds IGCSE Maths: The Full Worked Guide

450 cm to the nearest 10 cm doesn't actually mean 450 cm. It means anywhere from 445 cm to 455 cm, and that gap is worth an entire mark on its own before you've done any actual calculating.

Bounds themselves aren't the hard part. Add or subtract half the degree of accuracy and you're done. What actually costs marks is picking the wrong bound once two rounded measurements get multiplied or divided together, and forgetting that a rounded number was never exact to begin with.

The One Rule Behind Every Bounds Question

The upper bound is the greatest value a rounded or measured quantity could actually have been before rounding. The lower bound is the smallest. Both come from one rule: take half the degree of accuracy the number was rounded to, then add it for the upper bound or subtract it for the lower.

Cambridge International (CAIE) tests this under "Estimation and Accuracy" in Extended IGCSE Mathematics (0580), and it's a consistent source of dropped marks on real scripts. The method itself is short. The notation has to be exact.

Finding a Bound: Add or Subtract Half the Degree of Accuracy

If a length is given as 450 cm to the nearest 10 cm, the degree of accuracy is 10, so half of it is 5.

Upper bound = 450 + 5 = 455. Lower bound = 450 − 5 = 445.

Only "half the degree of accuracy" ever changes from question to question. Every rounded value in an exam question is really hiding a range, and bounds notation is simply how you write that range down properly.

Why "Half the Degree of Accuracy" Is the Part Worth a Mark on Its Own

The degree of accuracy is the unit a value was rounded to: nearest 10, nearest whole number, or a stated number of significant figures. Half of that unit is the furthest the true value could sit from the rounded one before it would round to something else instead.

Get the degree of accuracy wrong and both bounds are wrong, even with flawless arithmetic afterwards. That's exactly why examiners award a mark specifically for identifying it correctly, separate from the mark for the calculation itself.

Bounds for Numbers Rounded to Significant Figures

Apply the same rule, but work out first what one unit of that significant figure actually represents. A mass given as 250 kg to 2 significant figures has been rounded to the nearest 10 kg, because the "5" sits in the tens column, so the bounds are 245 kg and 255 kg. Not 249.5 and 250.5, which is the mistake that shows up most often when students confuse "2 s.f." with "rounded to a whole number."

Where This Gets Genuinely Harder: Calculated Quantities

Bounds stop being simple addition and subtraction once a question asks for the bound of a calculated quantity, an area, a speed, a density, built from two or more rounded measurements. You have to pick the correct bound for each measurement before calculating anything, and that choice depends on whether it's being multiplied, divided into, or divided by.

Which Bound Goes Where: A Quick Reference

To find the...

Use upper bounds for

Use lower bounds for

Maximum of a sum or product

all values added or multiplied

n/a

Minimum of a sum or product

n/a

all values added or multiplied

Maximum of A ÷ B

A

B

Minimum of A ÷ B

B

A

Worked Example: The Upper Bound of a Field's Area

A rectangular field is measured as 15 m by 9 m, both to the nearest metre. Find the upper bound of the area.

Step 1: The degree of accuracy for both is 1 m, so half of it is 0.5 m. Length bounds: 14.5 m to 15.5 m. Width bounds: 8.5 m to 9.5 m.

Step 2: Area is length times width, a product, so the upper bound uses the upper bound of both measurements: 15.5 × 9.5 = 147.25 m².

Step 3: For contrast, the lower bound uses both lower bounds: 14.5 × 8.5 = 123.25 m². Quoting the rounded 15 × 9 = 135 m² as if it were exact, instead of giving the full range from 123.25 m² to 147.25 m², scores zero even when every step of the multiplication is flawless.

Notation That Actually Earns the Accuracy Mark

Label which value is the upper bound and which is the lower, or use inequality notation, for example 445 ≤ x < 455. CAIE mark schemes check for this labelling specifically, and an unlabelled pair of otherwise correct numbers can still lose the accuracy mark.

For a value rounded to the nearest 10, use ≤ for the lower bound and a strict < for the upper, since 455 itself would round to 460, not 450.

For more worked bounds and rounding questions mapped to the CAIE 0580 syllabus, see our IGCSE Extended Mathematics number hub.

Frequently Asked Questions

What is the difference between upper bound and lower bound? The upper bound is the largest value a rounded number could have been before rounding; the lower bound is the smallest. Both come from adding or subtracting half the degree of accuracy. For 450 to the nearest 10, that's 455 (upper) and 445 (lower).

How do you find the upper and lower bound of a number rounded to 1 decimal place? The degree of accuracy is 0.1, so half of it is 0.05. A length given as 6.4 cm to 1 d.p. has bounds of 6.35 cm (lower) and 6.45 cm (upper).

Do you use upper or lower bounds to find the maximum value of a division? For A ÷ B, the maximum uses the upper bound of A divided by the lower bound of B, since dividing by the smallest possible number gives the largest answer. The minimum uses the lower bound of A divided by the upper bound of B.

Why do exam boards test upper and lower bounds instead of just exact numbers? Because no real-world measurement is exact. Every ruler, scale, and stopwatch has a limit of accuracy. CAIE tests bounds specifically to check students understand that a rounded measurement represents a range of possible true values, not a precise fact.

Where to Spend Your Last Ten Minutes on This Topic

If you only have time to fix one thing, fix the division table above. Almost every student can add and subtract half a degree of accuracy correctly; far fewer can say with confidence which bound goes on top and which goes on the bottom once a division is involved.

Take three past paper questions that combine two rounded measurements into one calculation, work out both bounds properly, and check your labelling matches what a mark scheme would actually accept. If you're not sure whether your inequality notation would survive CAIE's marking, the AI Tutor can check it against the same criteria examiners use.