# 2012 AMC 10B Problems/Problem 6

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

## See Also

 2012 AMC 10B (Problems • Answer Key • Resources) Preceded byProblem 5 Followed byProblem 7 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions

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

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