At a glance
- Cambridge 0580
- Core C1.10; Extended E1.10
- Edexcel 4MA1
- Foundation 1.8C; Higher 1.8A
- Half-unit rule
- Bounds are ± half the rounding unit
- Division rule
- max = upper ÷ lower
Which bounds to use in a calculation
| To find | For a + b or a × b | For a - b | For a ÷ b |
|---|---|---|---|
| Upper bound | upper a and upper b | upper a and lower b | upper a and lower b |
| Lower bound | lower a and lower b | lower a and upper b | lower a and upper b |
Cambridge Core does not ask for bounds of calculations; Cambridge Extended (E1.10.2) and Edexcel Higher (1.8A) do.
The key ideas
Find the rounding unit, halve it, and add and subtract. 'To the nearest 10' means a unit of 10, so ± 5. 'To 1 decimal place' means 0.1, so ± 0.05. 'To 2 significant figures' depends on the size of the number: 3400 to 2 s.f. has a unit of 100, so the bounds are 3350 and 3450.
The error interval uses ≤ for the lower bound and < for the upper bound: 3350 ≤ x < 3450. Exam questions accept the upper bound written as 3450 even though x cannot equal it exactly.
When values are combined, think about what makes the answer biggest or smallest. Adding or multiplying bigger numbers gives a bigger result. Subtracting a smaller number, or dividing by a smaller number, also gives a bigger result. This is why differences and quotients mix an upper bound with a lower bound.
Final answers from bounds often need judgement. If a question asks for a value to an appropriate degree of accuracy, round both bounds and give the most accurate value on which they agree.
Worked example 1 (Edexcel Higher): areas from rounded lengths
- A rectangle measures 12 cm by 8 cm, each to the nearest cm. Calculate the smallest possible area as a percentage of the largest possible area.
- Bounds: 11.5 ≤ length < 12.5 and 7.5 ≤ width < 8.5.
- Smallest area = 11.5 × 7.5 = 86.25 cm^2. Largest area = 12.5 × 8.5 = 106.25 cm^2.
- Percentage = 86.25 ÷ 106.25 × 100 = 81.176...%, so 81.2% to 3 s.f.
Worked example 2 (Cambridge Extended): bounds of a speed
- A runner covers 100 m, measured to the nearest metre, in 12.4 s, measured to the nearest 0.1 s. Find the lower bound of the speed.
- Bounds: 99.5 ≤ d < 100.5 and 12.35 ≤ t < 12.45.
- Speed = distance ÷ time. The smallest speed uses the smallest distance and the largest time: 99.5 ÷ 12.45 = 7.9919...
- Lower bound = 7.99 m/s (3 s.f.). For comparison, the upper bound is 100.5 ÷ 12.35 = 8.14 m/s.
Worked example 3: a difference and a perimeter
- Two lengths are 8.6 cm and 3.2 cm, each to 1 decimal place. Find the upper bound of their difference.
- Bounds: 8.55 to 8.65 and 3.15 to 3.25.
- Largest difference = largest first minus smallest second = 8.65 - 3.15 = 5.5 cm.
- Perimeter: a square has side 5 cm to the nearest cm. The upper bound of its perimeter is 4 × 5.5 = 22 cm.
Common mistakes that cost marks
- Using ± 1 unit instead of ± half a unit: 7 m to the nearest metre is 6.5 to 7.5, not 6 to 8.
- Writing the upper bound as 7.49 or 7.4999. The upper bound is 7.5.
- Using both upper bounds for a difference or a quotient.
- Rounding the bounds before the final step, which can change the answer.
- Getting the unit wrong for significant figures in large numbers, such as using ± 0.5 for 3400 to 2 s.f.
- Answering with only one bound when the question asks for both, or for the error interval.
Exam technique and how a tutor helps
Write all the bounds down first, in a little list, before doing any calculation. Then write the calculation with the bounds you have chosen, such as '99.5 ÷ 12.45', so that the examiner can award the method mark even if a calculator slip follows. Keep the unrounded answer on your calculator until the very end.
Bounds can appear inside other topics: the bounds of an area of a circle, a density, or an angle from trigonometry. The method is the same every time: decide whether you want the answer big or small, then choose each input accordingly.
Students often memorise the table above without understanding it, then freeze on an unusual formula. In one-to-one lessons a tutor asks the student to reason aloud about which input makes the result bigger, which works for any formula, and uses quick verbal drills on rounding units until the half-unit step is automatic.
Self-check: can you do these?
- A mass is 2.4 kg to 1 decimal place. Write its error interval. (Answer: 2.35 ≤ m < 2.45)
- A number is 3400 to 2 significant figures. Write its bounds. (Answer: 3350 and 3450)
- a = 20 and b = 4, each to the nearest whole number. Find the lower bound of a ÷ b. (Answer: 19.5 ÷ 4.5 = 4.33)
- A length is 50 cm to the nearest 10 cm. Write its lower and upper bounds. (Answer: 45 cm and 55 cm)
- Explain why the upper bound of a - b uses the lower bound of b.
Common questions
Why is the upper bound 7.5 and not 7.49?
Because the bound is the boundary value itself. Any number just below 7.5 rounds to 7, so 7.5 is the limit. There is no single largest number below 7.5, so the bound is written as 7.5, with x < 7.5 in an error interval.
Is calculating with bounds on the Core or Foundation paper?
Cambridge Core and Edexcel Foundation ask for the bounds of a single rounded value. Using bounds in calculations is Cambridge Extended (E1.10) and Edexcel Higher (1.8A).
How do I give an answer to a suitable degree of accuracy using bounds?
Round the upper and lower bounds of the result to the same accuracy. Choose the most accurate rounding where both give the same value and use that as your answer.
What about truncated values?
A value truncated (cut off) rather than rounded has its full unit above it: a time truncated to 12 s lies from 12 to 13 s. Read the question to see which applies.
How do LiveTutor lessons work?
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Sources
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