8.1 Estimation by rounding

Estimate the sum: $\text{2,357}+\text{6,106}$ .

Notice that 2,357 is near $\underset{\text{digits}}{\underset{\text{two nonzero}}{\underbrace{\mathrm{2,400,}}}}$ and that 6,106 is near $\underset{\text{digits}}{\underset{\text{two nonzero}}{\underbrace{\mathrm{6,100.}}}}$

The sum can be estimated by $\text{2,400}+\text{6,100}=\text{8,500}$ . (It is quick and easy to add 24 and 61.)

Thus, $\text{2,357}+\text{6,106}$ is about 8,400. In fact , $\text{2,357}+\text{6,106}=\text{8,463}$ .

Estimate the difference: $\mathrm{5,203}-\mathrm{3,015}$ .

Notice that 5,203 is near $\underset{\text{digits}}{\underset{\text{two nonzero}}{\underbrace{\mathrm{5,200,}}}}$ and that 3,015 is near $\underset{\text{digit}}{\underset{\text{one nonzero}}{\underbrace{\mathrm{3,000.}}}}$

The difference can be estimated by $\mathrm{5,200}-\mathrm{3,000}=\text{2,200}$ .

Thus, $\mathrm{5,203}-\mathrm{3,015}$ is about 2,200. In fact , $\mathrm{5,203}-\mathrm{3,015}=\text{2,188}$ .

We could make a less accurate estimation by observing that 5,203 is near 5,000. The number 5,000 has only one nonzero digit rather than two (as does 5,200). This fact makes the estimation quicker (but a little less accurate). We then estimate the difference by $\mathrm{5,000}-\mathrm{3,000}=\text{2,000}$ , and conclude that $\mathrm{5,203}-\mathrm{3,015}$ is about 2,000. This is why we say "answers may vary."