Mathematics 4024 · Syllabus tracker
Limits Of Accuracy
Upper and lower bounds of rounded values, error intervals and bounds of calculations.
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Quick revision
Limits Of Accuracy on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Upper and lower bounds of rounded values and of calculations that use them.
1Bounds of a rounded value
- A measurement rounded to some accuracy could have been anything within half a unit either side.
- Lower bound = value − half the unit; upper bound = value + half the unit.
- The upper bound itself would round up, so the error interval is lower ⩽ x < upper.
| Given as | Unit | Lower bound | Upper bound |
|---|---|---|---|
| 7.4 cm (1 d.p.) | 0.1 | 7.35 | 7.45 |
| 3400 (nearest 100) | 100 | 3350 | 3450 |
| 12 kg (nearest kg) | 1 | 11.5 | 12.5 |
| 0.36 (2 s.f.) | 0.01 | 0.355 | 0.365 |
Worked example
A length is 7.4 cm to the nearest mm. Write its error interval.
- Nearest mm = nearest 0.1 cm.
- Half of 0.1 = 0.05.
7.35 ⩽ l < 7.45
2Bounds of calculations
- To make an answer as large as possible, choose the bounds that push it up; for as small as possible, the opposite.
- Adding and multiplying: upper with upper gives the upper bound.
- Subtracting and dividing: one of each. The biggest difference is upper − lower; the biggest quotient is upper ÷ lower.
| Calculation | Upper bound | Lower bound |
|---|---|---|
| a + b | Ua + Ub | La + Lb |
| a × b | Ua × Ub | La × Lb |
| a − b | Ua − Lb | La − Ub |
| a ÷ b | Ua ÷ Lb | La ÷ Ub |
Worked example
A car travels 120 m (nearest 10 m) in 15.2 s (1 d.p.). Find the lower bound of its speed.
- Lower distance = 115 m; upper time = 15.25 s.
- Lower speed = 115 ÷ 15.25 = 7.540…
7.54 m/s (3 s.f.)
Exam tip: Write the bounds you are using before calculating — they earn method marks.
3Giving an answer to a suitable accuracy
- Work out both the upper and lower bound of the result.
- Round both to fewer and fewer figures until they agree: that is the accuracy you can trust.
- Example: bounds 7.54 and 7.81 agree only to 1 s.f. → answer 8.
Worked example
The bounds of a length are 4.4637 m and 4.4762 m. Give the length to a suitable accuracy.
- To 3 s.f.: 4.46 and 4.48 — different.
- To 2 s.f.: 4.5 and 4.5 — the same.
4.5 m
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Upper bound of 7.4 (1 d.p.) is 7.49 | It is 7.45 — the boundary |
| Error interval 11.5 ⩽ x ⩽ 12.5 | Upper end is strict: 11.5 ⩽ x < 12.5 |
| Upper bound of a ÷ b = Ua ÷ Ub | Divide by the smallest: Ua ÷ Lb |
| Upper bound of a − b = Ua − Ub | Subtract the smallest: Ua − Lb |
★In the exam
- Find the unit first ('nearest 5' → half unit 2.5; '2 s.f.' on 360 → nearest 10).
- Keep full calculator values for bounds; round only the final answer.
- For a mixed calculation (like speed = d/t or density = m/V) decide each bound separately.