# 2012 AMC 12B 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? $\textbf{(A) } \text{Her estimate is larger than } x - y \qquad \textbf{(B) } \text{Her estimate is smaller than } x - y \qquad \textbf{(C) } \text{Her estimate equals } x - y$ $\textbf{(D) } \text{Her estimate equals } y - x \qquad \textbf{(E) } \text{Her estimate is } 0$

## Solution

The original expression $x-y$ now becomes $(x+k) - (y-k)=(x-y)+2k>x-y$, where $k$ is a positive constant, hence the answer is $\boxed{\textbf{(A)}}$.

## Solution 2

The problem never says what $x$ and $y$ are, so we can decide what they are. Let $x = 1.6$ and $y = 1.4$. We round $x$ to $2$ and $y$ to $1$. Then the new $x - y = 1$, while the original $x - y = 0.2$. Thus, the new $x - y$ is greater than the original $x - y$. The answer is $\boxed{\textbf{(A)}}$. ~Extremelysupercooldude

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