Bounds in complex calculations
Objective. Choose extremal input bounds in multi-stage formulae.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
Bounds in complex calculations means selecting the upper or lower bound of each rounded measurement so that a multi-stage formula gives the true maximum or minimum possible result, since different operations push the extreme the opposite way. To maximise a fraction, use the largest upper bound for the numerator and the smallest lower bound for the denominator; to minimise, swap these choices. Each individual bound is found from the rounding precision, e.g. a length given as 8.3 cm to 1 d.p. has bounds 8.25 and 8.35. For example, to find the maximum of a/b with a=8.3 and b=4.2 (1 d.p.), use 8.35/4.15 = 2.0120…
Worked example
A speed is calculated as distance ÷ time, where distance = 120 m (to nearest 5 m) and time = 15 s (to nearest s). Find the upper bound for speed.
- 1Distance upper bound = 120 + 2.5 = 122.5 m
- 2Time lower bound = 15 − 0.5 = 14.5 s
- 3Upper bound of speed = largest numerator ÷ smallest denominator = 122.5 ÷ 14.5
- 4Compute 122.5 ÷ 14.5 = 8.448… m/s
Answer: 8.45 m/s (3 s.f.)
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Start practisingCommon mistake. A frequent mistake is using the upper bound for every value regardless of whether it increases or decreases the final result, such as using both upper bounds to minimise a fraction when the denominator's upper bound actually makes the fraction smaller, not larger.
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