Approximation

Approximating a calculation before doing it precisely helps you spot mistakes (e.g. a misplaced decimal point) and gives you a quick sense-check of whether your final exact answer is plausible.

To approximate a calculation, round each number to 1 significant figure (or another convenient value) before calculating. E.g. to approximate 38.7 × 21.4, round to 40 × 20 = 800.

The same idea works for division and multi-step calculations: round each part first. E.g. to approximate $\frac{198\times 5.1}{9.8}$, round to $\frac{200\times 5}{10}=\frac{1000}{10}=100$. This is much faster than the exact calculation and gives a good sanity-check figure.

Once you've calculated an exact answer, compare it to your approximation. If they're wildly different, you've likely made an error (e.g. a misplaced decimal point, or pressed the wrong calculator button) and should recheck your working.

(a) Approximate: $\frac{300\times 5}{6}=\frac{1500}{6}=250$.

(b) The approximation (250) is very different from the student's answer (22.59) — roughly 10 times smaller — so the student's answer is not reasonable. A likely cause is a decimal point error, e.g. the student may have divided by 61 instead of 6.1, or made a similar factor-of-10 slip.

1. 400 × 5 = 2,000.
2. $\frac{800}{20}=$ 40.
3. Approximate: 60 × 70 = 4,200. This is close to the exact answer of 4,150, so the answer is reasonable.
4. $\frac{50\times 3}{10}=\frac{150}{10}=$ 15.
5. It gives a quick way to sanity-check the exact answer, helping you spot obvious errors (e.g. a misplaced decimal point or a calculator input mistake) before relying on the exact answer.