why is the answer 21,845 and not 218,450?

Question

How can you tell whether

$$\frac{(250,000)(5.47)}{6.26}$$

is closer to 21,845 or 218,450 without calculating it exactly?

Thank you.

Answer 1

If we just estimate by orders of magnitude, using the fact that

$$5.47 \approx 6.26$$

(at least up to factors of $10$ this is true), then

$$\frac{250,000 \cdot 5.47}{6.26} = 250,000 \cdot \frac{5.47}{6.26} \approx 250,000 \cdot 1$$

so the answer should be on the scale of hundred-thousands. Alternatively, as powers of $10$, the answer should rougly be

$$\frac{10^5 \cdot 10^1}{10^1} = 10^{5 + 1 - 1} = 10^5$$

again suggesting an answer on the order of a hundred thousand.