2012 AMC 10B Problems/Problem 6

Problem

In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x > y > 0$ , Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?

$\textbf{(A) } \text{Her estimate is larger than } x-y \qquad \textbf{(B) } \text{Her estimate is smaller than } x-y \qquad \textbf{(C) } \text{Her estimate equals } y-x \\ \qquad \textbf{(D) } \text{Her estimate equals } x-y \qquad \textbf{(E) } \text{Her estimate is } 0$

Solution

Let's define $z$ as the amount rounded up by and down by.

The problem statement tells us that Xiaoli performed the following computation:

$\left(x+z\right) - \left(y-z\right) = x+z-y+z = x-y+2z$

We can see that $x-y+2z$ is greater than $x-y$ , and so the answer is $\boxed{\textbf{(A) } \text{Her estimate is larger than } x-y}$ .

2012 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2012 AMC 10B Problems/Problem 6

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