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

## 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 the answer is $\textbf{(A)} \text{Her estimate is larger than} x-y$

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 