Arithmetic/Estimation

Estimation

Estimation involves working out a rough answer to a calculation. The most common way to estimate a solution to a calculation is to round the numbers up or down to numbers which are easier to calculate with.

An example: Estimate the answer to 99 × 9. In this case we can easily see that 9 is almost 10, and multiplying by 10 is really easy, so we can replace our calculation with 99 × 10 which is far more easy to calculate. As we rounded a number up we know that our estimate is going to be larger than the real answer.

Why don't we just call this a guess? The difference is that a guess is just that, a completely wild guess. An estimate is based on some extra information. So whilst you might guess that
99 × 9 is something around 1000 by just pulling a number out of the air, we estimate that 99 × 9 is close to 99 × 10, then we work out 99 × 10 exactly, which gives 990 as an estimate of
99 × 9.

When we estimate, we want our estimate to be close enough to the actual value so as to be useful. (This also applies to approximation.) In our example above, the answer deviates from the actual value by about 10%, which might not be acceptable. Estimation often involves some guesswork, so we might not actually know how accurate our estimate is. (And yes, that figure 10% is itself an estimate.)

A better way to estimate 99 × 9 is to say that 99 is close to 100, then we work out 100 × 9 exactly, which gives 900 as an estimate of 99 × 9. This estimate is 1.0101...% off. The reason this is a better estimate is that 99 deviates from 100 much less than 9 deviates from 10 in percentage terms (although the absolute difference is one in each case).