Upper and Lower Bounds
When a measurement has been rounded, the true value lies within a range. Bounds questions test whether you can find that range and, crucially, which bound to use in a calculation — which is where nearly all the marks are lost.
1. Finding bounds
Halve the unit of rounding, then add and subtract.
| Rounded to | Half-unit | Value | Bounds |
|---|---|---|---|
| nearest 10 | 5 | 40 | 35 ≤ x < 45 |
| nearest whole number | 0.5 | 7 | 6.5 ≤ x < 7.5 |
| 1 decimal place | 0.05 | 4.5 | 4.45 ≤ x < 4.55 |
| 2 decimal places | 0.005 | 2.36 | 2.355 ≤ x < 2.365 |
| nearest 100 | 50 | 300 | 250 ≤ x < 350 |
Divide the rounding unit by 2 — this was the rule tutors repeated, and it works for every case. For 1 d.p. the unit is 0.1, so the half-unit is 0.05.
Worked example: 4.5 correct to 1 decimal place.
- Lower bound = 4.5 − 0.05 = 4.45
- Upper bound = 4.5 + 0.05 = 4.55
The lower bound is a subtraction, not a guess. Giving 4.4 instead of 4.45 was a specific recorded error — the half-unit is 0.05, not 0.1.
With significant figures, identify the place value of the last significant digit first. For 3200 to 2 s.f., the last significant digit is in the hundreds, so the half-unit is 50: bounds are 3150 ≤ x < 3250.
Notation: the lower bound is included (≤) and the upper bound is not (<), because a value exactly at the upper bound would round up. Write it as an inequality when asked.
2. Calculating with bounds — the decision table
This is the heart of the topic. To get the largest or smallest possible result, you must pick the right bound for each value.
| Operation | Maximum | Minimum |
|---|---|---|
| Addition a + b | UB + UB | LB + LB |
| Subtraction a − b | UB − LB | LB − UB |
| Multiplication a × b | UB × UB | LB × LB |
| Division a ÷ b | UB ÷ LB | LB ÷ UB |
Addition and multiplication are intuitive; subtraction and division are not. For those two, the bounds are mixed — and using the same bound for both values was the single most common error recorded in lessons.
Why subtraction mixes: to make a difference as large as possible, start as big as you can and take away as little as you can → UB − LB.
Why division mixes: dividing by a smaller number gives a bigger answer. So for the maximum, take the largest numerator and the smallest denominator → UB ÷ LB.
Think about direction before substituting. Ask “would a bigger denominator make my answer bigger or smaller?” — the answer determines the bound.
3. Worked examples
Addition. A rectangle has sides 8 cm and 5 cm, each to the nearest cm. Find the maximum perimeter.
- Bounds: 7.5 ≤ l < 8.5 and 4.5 ≤ w < 5.5
- Max perimeter = 2(8.5 + 5.5) = 28 cm
Find the bounds of each side first, then combine. Adding the sides and then taking bounds of the total was a documented error — do it in the right order.
Subtraction. a = 12 and b = 7, both to the nearest whole number. Find the maximum of a − b.
- a: 11.5 to 12.5; b: 6.5 to 7.5
- Maximum = UB of a − LB of b = 12.5 − 6.5 = 6
- Minimum = 11.5 − 7.5 = 4
Multiplication. A rectangle 6.2 cm by 4.5 cm, each to 1 d.p. Find the maximum area.
- 6.15 to 6.25 and 4.45 to 4.55
- Max area = 6.25 × 4.55 = 28.4375 cm²
Division — the classic speed question. A car travels 100 m (to the nearest metre) in 12 s (to the nearest second). Find the maximum speed.
- Distance: 99.5 to 100.5; Time: 11.5 to 12.5
- Maximum speed = UB distance ÷ LB time = 100.5 ÷ 11.5 = 8.74 m/s (3 s.f.)
- Minimum speed = 99.5 ÷ 12.5 = 7.96 m/s
For maximum speed, use the maximum distance and the MINIMUM time — going as far as possible in as little time as possible. Using upper bounds for both was recorded repeatedly.
Density works the same way (mass ÷ volume): maximum density uses maximum mass and minimum volume.
4. Method and accuracy
- Write the bounds of every quantity before calculating
- Decide whether you want a maximum or minimum
- Use the decision table to pick the bound for each
- Calculate
- Give the answer to a sensible accuracy with units
Use the exact bound values in the calculation — 11.5, not 11.49999. The upper bound is used as a normal number even though the true value never quite reaches it.
Show which bounds you used. Writing “100.5 ÷ 11.5” earns the method mark even if the arithmetic slips.
5. Mistakes that cost marks
Using the whole rounding unit instead of half.
Guessing the lower bound rather than subtracting.
Using the same bound for both values in a subtraction or division.
Using upper bounds for both distance and time.
Combining values first, then taking bounds.
Getting the inequality signs backwards.
Misidentifying the place value for significant-figure rounding.
Rounding the bounds themselves before calculating.
Omitting units.
Frequently asked questions
How do I find upper and lower bounds? Halve the rounding unit, then add it for the upper bound and subtract it for the lower.
What are the bounds of 4.5 to 1 d.p.? 4.45 to 4.55.
Why is the upper bound not included? Because a value exactly at the upper bound would round up to the next value.
How do I find a maximum sum? Add the upper bounds.
How do I find a maximum difference? Upper bound of the first minus lower bound of the second.
How do I find a maximum from a division? Upper bound ÷ lower bound — a smaller denominator gives a bigger answer.
How do I find the maximum speed? Maximum distance ÷ minimum time.
Why do subtraction and division mix the bounds? Because making one part bigger makes the result bigger, while making the other part bigger makes it smaller.
Quick revision checklist
- I halve the rounding unit to find bounds
- I can find bounds for decimal places and significant figures
- I write bounds as an inequality with ≤ and <
- I find the bounds of each quantity before calculating
- I add upper bounds for a maximum sum
- I use UB − LB for a maximum difference
- I multiply upper bounds for a maximum product
- I use UB ÷ LB for a maximum quotient
- I can reason about direction rather than memorising
- I can handle speed and density questions
- I show which bounds I used
- I give units and sensible accuracy
These notes cover upper and lower bounds in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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