Rounding to a chosen accuracy

Rounding is everywhere on the maths paper. The two systems you must master are decimal places (d.p.) and significant figures (s.f.), plus knowing when each is appropriate.

Decimal places

Count digits AFTER the decimal point. To round to n d.p.:

1. Look at the (n+1)-th decimal digit.
2. If it's 5 or more, round the n-th digit up; otherwise leave it.
3. Drop everything beyond the n-th decimal place.

Examples:

• 3.846 to 2 d.p. → 3.85 (the next digit is 6, round up).
• 7.124 to 2 d.p. → 7.12 (the next digit is 4, round down).
• 0.0998 to 3 d.p. → 0.100 (the 8 carries the 9 to 10, which carries again).

⚠ Don't drop trailing zeros that the rounding produced. 4.298 to 2 d.p. is 4.30, not 4.3.

Significant figures

The first significant figure is the first non-zero digit, reading left-to-right. Subsequent digits are also significant.

Examples:

• 0.00457 has three s.f.: 4, 5, 7.
• 540,000 to 2 s.f. is 540,000 (the trailing zeros may or may not be significant — to be precise we'd write 5.4 × 10⁵).
• 0.000 4920 has 4 s.f. (4, 9, 2, 0).

To round to n s.f.:

1. Find the first non-zero digit.
2. Count n significant figures from there.
3. Look at the next digit and round accordingly.
4. Replace any digits between the rounded position and the decimal point with zeros (so the place value is preserved).

Examples:

• 38,712 to 2 s.f. → 39,000.
• 0.004726 to 2 s.f. → 0.0047.
• 6.499 to 2 s.f. → 6.5.

When to use d.p. vs s.f.

• Money is usually quoted to 2 d.p. (e.g. £14.30).
• Calculator answers in real-world contexts are usually quoted to 3 s.f. unless told otherwise.
• Lengths and measurements match the precision of the measuring device.
• Probabilities are sometimes given as fractions, decimals to 3 d.p., or percentages.

If the question doesn't say, 3 s.f. is the safe default for non-money calculator answers.

"Appropriate" accuracy

If a question says "give your answer to an appropriate degree of accuracy", check the precision of the data given.

• Inputs given to 1 d.p. → answer at 1 d.p.
• Inputs given to 3 s.f. → answer at 3 s.f.

Carrying intermediate values

Don't round in the middle of a calculation — only round at the very end.

Worked example: a circle has radius 4.7 cm. Find its circumference to 3 s.f.

• C = 2 × π × 4.7.
• Use π ≈ 3.14159… (or the calculator's π key).
• 2 × π × 4.7 = 29.530…
• To 3 s.f.: 29.5 cm.

If you'd rounded π to 3.14 mid-calculation, you'd get 29.516, which still rounds to 29.5 — but in tricky problems mid-calc rounding can flip the last digit, costing the A1.