Bounds — error intervals and bounds calculations
Every measurement has an inherent error. WJEC tests bounds at Foundation, Intermediate and Higher.
Error interval from a rounded measurement
If a value v has been rounded to a given precision p, the original could be anywhere in [v − p/2, v + p/2):
- Lower bound (LB) = v − p/2.
- Upper bound (UB) = v + p/2.
The upper bound uses < not ≤ because v + p/2 itself rounds up.
✦Worked example— Examples
- 4.5 to 1 d.p.: precision 0.1, so LB 4.45, UB 4.55. Range [4.45, 4.55).
- 350 to nearest 10: precision 10, so LB 345, UB 355.
- 6.3 to 2 s.f.: precision 0.1, so LB 6.25, UB 6.35.
Bounds in calculations
When values are combined, the rules depend on the operation. Let A have bounds [A_LB, A_UB] and B have [B_LB, B_UB], all positive.
| Operation | Upper bound | Lower bound |
|---|---|---|
| A + B | A_UB + B_UB | A_LB + B_LB |
| A − B | A_UB − B_LB | A_LB − B_UB |
| A × B | A_UB × B_UB | A_LB × B_LB |
| A ÷ B | A_UB ÷ B_LB | A_LB ÷ B_UB |
The subtraction and division rules invert because making the denominator smaller (or the subtractor smaller) makes the result larger.
✦Worked example
A car travels d = 250 m (to nearest 10 m) in t = 8 s (to nearest 0.1 s). Find the upper bound for speed.
- d_UB = 255 m, d_LB = 245 m.
- t_UB = 8.05 s, t_LB = 7.95 s.
- Speed UB = d_UB / t_LB = 255 / 7.95 = 32.075… m/s.
- Speed LB = d_LB / t_UB = 245 / 8.05 = 30.43… m/s.
"Truncated to" wording
If a value is truncated (chopped) at a given precision, the lower bound is the stated value but the upper bound stops at v + p (not v + p/2).
- 6.3 truncated → range [6.3, 6.4).
WJEC mostly uses "rounded to"; check the wording.
WJEC exam tip
When asked for a value "to a suitable degree of accuracy" after a bounds calculation, give the result to the precision where UB and LB AGREE. If UB = 32.07 and LB = 30.43, only the leading digit (3 of 30/32) is reliable, so a sensible quoted answer is "approximately 30 m/s".
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