Estimation, rounding and bounds

Rounding

To a given number of decimal places (d.p.): look at the digit after the required place. If it is 5 or more, round up; if it is 4 or less, round down. Example: 3.4762 rounded to 2 d.p. = 3.48 (the third decimal is 6, so round up the 7).

To a given number of significant figures (s.f.): start counting from the first non-zero digit. Example: 0.004826 to 2 s.f. = 0.0048 (significant figures are 4 and 8; the 2 rounds down). Example: 43,724 to 3 s.f. = 43,700 (significant figures are 4, 3, 7; round 2 down).

Estimation

To estimate the answer to a calculation, round each number to 1 significant figure then calculate mentally.

Example: Estimate (48.3 × 9.7) ÷ 0.51. Round: (50 × 10) ÷ 0.5 = 500 ÷ 0.5 = 1000.

CCEA Paper 1 frequently asks you to estimate and then state whether the estimate is an overestimate or underestimate (consider how each rounding affected the result).

Error intervals and bounds

When a measurement is given to a degree of accuracy (e.g. "to the nearest cm"), the true value lies within an error interval.

If x = 7.4 (given to 1 d.p.):

• Lower bound = 7.35 (the smallest value that rounds up to 7.4)
• Upper bound = 7.45 (the largest value that rounds down to 7.4)
• Error interval: 7.35 ≤ x < 7.45

Note: the upper bound uses a strict inequality (<, not ≤) because a value of exactly 7.45 would round up to 7.5, not 7.4.

Bounds in calculations

When combining rounded measurements:

• Maximum value of a product (a × b): upper bound of a × upper bound of b.
• Minimum value of a product: lower bound of a × lower bound of b.
• Maximum value of a quotient (a ÷ b): upper bound of a ÷ lower bound of b.
• Minimum value of a quotient: lower bound of a ÷ upper bound of b.

Example: A rectangle has length 8.3 cm (to 1 d.p.) and width 4.7 cm (to 1 d.p.).

• Maximum area = 8.35 × 4.75 = 39.6625 cm²
• Minimum area = 8.25 × 4.65 = 38.3625 cm²