Rounding, Estimation, and Bounds

Rounding

• Decimal places: 3.4567 to 2 d.p. = 3.46.
• Significant figures: count from the first nonzero digit.

- 0.004 572 to 2 s.f. = 0.0046

- 38 450 to 3 s.f. = 38 500

Estimation

Round every number to 1 significant figure, then calculate.

Example. (48.7 × 3.1) ÷ 0.52 ≈ (50 × 3) ÷ 0.5 = 300.

Upper and lower bounds

A measurement rounded to some accuracy lies within half a unit either side.

• 7.3 cm to 1 d.p.: lower bound 7.25, upper bound 7.35.
• 400 m to the nearest 10 m: from 395 m to 405 m.

Written as an error interval: 7.25 ≤ x < 7.35.

Bounds in calculations

Calculation	Maximum	Minimum
a + b	UB + UB	LB + LB
a − b	UB − LB	LB − UB
a × b	UB × UB	LB × LB
a ÷ b	UB ÷ LB	LB ÷ UB

Example. A rectangle is 6 cm by 4 cm, both to the nearest cm. Maximum area = 6.5 × 4.5 = 29.25 cm².

Example. Speed = distance ÷ time; the maximum speed uses the largest distance and the smallest time.

Quick summary

• Significant figures start at the first nonzero digit.
• Estimate with 1 s.f. numbers.
• Bounds are half a unit either side; for division, maximum = UB ÷ LB.