1.5 Rounding and estimation — Learn

Prerequisites: place value (1.1); the four operations (1.2).

By the end of this section you can: round to the nearest 10, 100 or whole number; round to a given number of decimal places or significant figures; and use estimation to check an answer.

1. Rounding to the nearest 10, 100 or whole number

Find the place value you are rounding to. Look at the digit to its right: if it is 5 or more, round up; otherwise round down.

Example 1. Round $4\,367$ to the nearest $100$.

The hundreds digit is $3$. The digit to its right is $6 \geq 5$, so round up: $4\,367 \approx 4\,400$. (The symbol $\approx$ means "is approximately equal to".)

Example 2. Round $2.7$ to the nearest whole number. The tenths digit is $7 \geq 5$, so $2.7 \approx 3$.

2. Decimal places

To round to $n$ decimal places, keep $n$ digits after the point and look at the next one.

Example 3. Round $3.14159$ to 2 decimal places.

Keep $3.14$; the next digit is $1 < 5$, so round down: $3.14159 \approx 3.14$.

3. Significant figures

Significant figures count from the first non-zero digit. Rounding to significant figures is useful when a number is large or small.

Example 4. Round $4\,367$ to 2 significant figures.

The first two significant figures are $4$ and $3$. The next digit is $6 \geq 5$, so round up: $4\,367 \approx 4\,400$.

Example 5. Round $0.004\,367$ to 2 significant figures.

Leading zeros are not significant. The first significant figure is $4$, so the first two are $4$ and $3$, giving $0.004\,4$.

Watch out. Treating leading zeros as significant figures. $0.004\,367$ to 2 s.f. is $0.004\,4$, not $0.004\,37$.

4. Estimation

Estimating means rounding each number first, so the arithmetic is quick. It is a check, not an exact answer.

Example 6. Estimate $19.7 \times 6.2$.

Round: $20 \times 6 = 120$. The exact answer is $122.14$, so the estimate is close — the calculation is reasonable.

Try it: rounding

Predict first: what is $3.14159$ to 2 decimal places? Say it before you use the tool, then check. Now switch to significant figures and predict $3.14159$ to 2 s.f.

Explain: for a given number of decimal places or significant figures, which single digit decides whether you round up or down? Why?

5. Progress check

The stall's profit was $£39.20$. Round it to the nearest pound for a poster: $£39.20 \approx £39$.

Now estimate as a check: revenue is about $95 \times 1.20 + 6 \times 3 \approx 100 \times 1.2 + 18 = 138$; cost is about $120 \times 0.65 + 8 \times 2 = 78 + 16 = 94$; so the profit should be about $138 - 94 = £44$. The exact $£39.20$ is a little lower, but it is the same size, so the answer is reasonable.

You should now be able to

• round to the nearest 10, 100 or whole number;
• round to a given number of decimal places or significant figures;
• estimate a calculation by rounding first;
• use an estimate to judge whether an exact answer is reasonable.