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CAMBRIDGE O LEVEL · 2025–2027

Estimation.

Choose sensible accuracy and use estimates to check whether an answer makes sense.

Syllabus 1.9Visual notesWorked examples
BY THE END

Round to decimal places and significant figures · estimate calculations · choose accuracy in context

01

Round to a stated level of accuracy

A measurement such as 8.473 m may contain more detail than we need. Rounding chooses the nearest value at a stated level of detail, such as the nearest tenth of a metre.

First identify the digit you intend to keep. The next digit decides which neighbouring rounded value is nearer. Decimal places count from the decimal point; significant figures start at the first non-zero digit. These describe accuracy in different ways, especially for small numbers.

Locate the last place you want to keep. Look at the next digit: 0–4 leaves it unchanged; 5–9 increases it by 1. Remove the remaining digits, using zeros to preserve place value.

Decimal places (d.p.)

Count places after the decimal point, including zeros.

0.04786 → 0.05 (2 d.p.)

For whole-number rounding: 5764 → 6000 to the nearest thousand.

Significant figures (s.f.)

Start at the first non-zero digit. Zeros between significant digits count.

0.04786 → 0.048 (2 s.f.)

Leading zeros locate the digits; they are not significant.

5764 → 6000

The hundreds digit is 7, so 5000 rounds up to 6000.

Pause and explainWhy is the first significant figure of 0.00472 the digit 4?

The leading zeros only locate the decimal value. The 4 is the first digit that contributes to the measured amount.

02

Estimate before you calculate

An estimate deliberately replaces awkward numbers with nearby simpler ones. Its purpose is to predict the approximate size of an answer or check whether a calculator result is plausible.

Round each input to the accuracy requested, often one significant figure, and write the simplified calculation before evaluating it. This shows how the estimate was made. Do not confuse this with calculating exactly first and then rounding the final answer.

Replace awkward numbers by convenient approximations, often to 1 significant figure, then calculate mentally. Use ≈ to show an approximation.

WORKED EXAMPLE

Estimate (97.9 × 0.765) ÷ 4.13 using 1 significant figure.

97.9 → 1000.765 → 0.84.13 → 4
(97.9 × 0.765)4.13 ≈ (100 × 0.8)4 = 20

The estimate checks the size of a calculator result. It does not replace an accurate answer when one is requested.

WORKED EXAMPLE

Estimate the cost of 19 items at $4.85 each.

19 × 4.85 ≈ 20 × 5 = $100

The exact cost is $92.15, so a result near $100 is plausible.

Pause and explainWhy estimate 19.8 × 5.1 using 20 × 5?

Those nearby values make the product easy to evaluate as about 100, providing a useful check on the original calculation.

03

Make the accuracy fit the context

The appropriate final accuracy depends on what the answer represents. A price usually needs cents, while a number of buses must be a whole number that provides enough seats.

Keep full working precision until the end, then choose rounding that makes sense in the situation. Sometimes this means rounding up rather than to the nearest integer. Always ask whether your final number can actually perform the job described by the question.

Money and measurements

$18 ÷ 7 = $2.571428… per item, or $2.57 to the nearest cent. Keep extra digits during intermediate work.

Whole objects

53 people in vehicles holding 8 each need 53 ÷ 8 = 6.625 vehicles, so book 7 vehicles. Here the context requires rounding up.

Pause and explainIf 3.2 buses are needed to carry a group, how many buses must be booked?

Four. Three buses would not provide enough capacity, so the practical answer rounds up.

04

Your turn. Is the size sensible?

State the rounding accuracy before doing any work. If the task is an estimate, show the rounded inputs; if it is a final-answer rounding task, keep the original inputs during calculation.

Check whether trailing zeros communicate the requested accuracy. Then interpret the result in context: nearest, whole number, enough capacity, or a sensible money amount may require different final decisions.

1

Round 0.006307 to 2 significant figures and to 4 decimal places.

Reveal worked answer
2 s.f.: 0.0063. 4 d.p.: 0.0063. The written values happen to agree here, but the counting methods differ.
2

Round 9.996 to 2 decimal places.

Reveal worked answer
10.00. The third decimal digit is 6, causing carrying through the 9s. Keep two decimal places.
3

Estimate (48.7 × 0.193) ÷ 9.82 using 1 significant figure.

Reveal worked answer
(50 × 0.2) ÷ 10 = 1.
4

A 4 m ribbon is cut into 0.35 m pieces. How many complete pieces are possible?

Reveal worked answer
4 ÷ 0.35 = 11.42857… Only 11 complete pieces can be cut, so round down for this context.