Estimation

Estimation is a very useful skill for scenarios where we do not need an exact answer, just one that is good enough. For example, being able to estimate a bill can prevent us from being severely overcharged, and help us identify potential errors when solving other problems.

We estimate by finding a value that is close enough to the correct answer. We start by identifying the term with the most significant impact, and then improve our approximation if necessary.

If one meal costs $5.88, which of the following is the best approximation for the cost of eight meals?

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We round the cost of the meal up to $6.00 and obtain
$8 \times \$5.88 \approx 8 \times \$6.00 = \$48.00$.

Note that since we rounded the price up, this is an over-approximation, and the actual cost is below $48.00.

If a pen costs $1.89 and an eraser costs $0.99, which of the following is the best estimate for the cost of 10 pens and 5 erasers?

(A) $\ \ \$10.00$
(B) $\ \ \$15.00$
(C) $\ \ \$20.00$
(D) $\ \ \$25.00$
(E) $\ \ \$30.00$

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We can round the price of a pen up from $1.89 to $2.00, and also round up the price of an eraser from $0.99 to $1.00. Then we get:

$$10 \times \$1.89 + 5 \times \$0.99 \approx 10 \times \$2.00 + 5 \times \$1.00 = \$25.00.$$

What is the integer that is closest to $\sqrt{108}?$

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$100 < 108 < 121$, so the integer part of $\sqrt{ 108}$ is 10.
Since $10.5 ^2 = (10 + 0.5)^2 = 100 + 10 + 0.25 = 110.25$, it follows that $\sqrt{ 108 } < 10.5$.

Thus, the integer that is closest to $\sqrt{108}$ is 10.