Bounds and limits of accuracy
Upper and lower bounds appear on almost every OCR J560 Paper 3 (calculator). They frequently require students to combine bounds in calculations — a challenging higher-tier topic.
What are bounds?
When a measurement is rounded, the true value lies within an interval. The lower bound is the smallest value that rounds to the given value; the upper bound is the largest.
For a value rounded to the nearest unit:
- Lower bound = value − half a unit.
- Upper bound = value + half a unit.
Example: length = 6.4 cm (to 1 d.p.)
- Rounded to nearest 0.1.
- Lower bound = 6.4 − 0.05 = 6.35 cm.
- Upper bound = 6.4 + 0.05 = 6.45 cm.
The error interval is written: 6.35 ≤ L < 6.45.
(Note: upper bound is a strict inequality — 6.45 would round to 6.5, not 6.4, so we use <.)
Combining bounds in calculations
When combining rounded measurements, use the appropriate bounds to find the maximum or minimum result:
| Calculation | Maximum result | Minimum result |
|---|---|---|
| A + B | UB(A) + UB(B) | LB(A) + LB(B) |
| A − B | UB(A) − LB(B) | LB(A) − UB(B) |
| A × B | UB(A) × UB(B) | LB(A) × LB(B) |
| A ÷ B | UB(A) ÷ LB(B) | LB(A) ÷ UB(B) |
Key insight for subtraction and division: to maximise A−B, you want A as large as possible AND B as small as possible. To minimise A÷B, divide the smallest A by the largest B.
✦Worked example
A rectangle has length L = 8.3 m (to 1 d.p.) and width W = 4.7 m (to 1 d.p.).
Bounds:
- L: 8.25 ≤ L < 8.35; W: 4.65 ≤ W < 4.75.
Upper bound of area = 8.35 × 4.75 = 39.6625 m². Lower bound of area = 8.25 × 4.65 = 38.3625 m².
Significant figures and d.p. (reminder)
- Rounded to nearest 10: bounds ± 5.
- Rounded to nearest 1: bounds ± 0.5.
- Rounded to 1 d.p.: bounds ± 0.05.
- Rounded to 2 d.p.: bounds ± 0.005.
- Rounded to 3 s.f. (e.g. 4.67): unit in last s.f. is 0.01 → bounds ± 0.005.
Common OCR exam mistakes
- Using the wrong bound in a combined calculation — particularly for A − B and A ÷ B.
- Using ≤ instead of < for the upper bound in error interval notation — the value exactly at the upper bound rounds up.
- Forgetting that "rounded to 3 significant figures" may have a smaller unit than "rounded to 3 decimal places" — always identify what the last significant digit is.
AI-generated · claude-opus-4-7 · v3-ocr-maths