Question

Amelia needs to estimate the quantity $\dfrac{a}{b}-c,$ where $a,$ $b,$ and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\dfrac{a}{b}-c?$

She rounds all three numbers up.

She rounds $a$ and $b$ up, and she rounds $c$ down.

She rounds $a$ and $c$ up, and she rounds $b$ down.

She rounds $a$ up, and she rounds $b$ and $c$ down.

She rounds $c$ up, and she rounds $a$ and $b$ down.

Solution:

To make $\dfrac{a}{b}$ larger, round the numerator $a$ up and the denominator $b$ down. To make $-c$ larger, round $c$ down.

Only choice $\text{(D)}$ does all three: it rounds $a$ up while rounding $b$ and $c$ down, so every change pushes the estimate above the exact value. In the other choices at least one change works the wrong way, so the estimate is not guaranteed to be larger.

Thus, the correct answer is D.

2015 AMC 12A Problem 5

Solution:

Problem 5 in Other Years