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

2012 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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All AMC 12 Problems and Solutions

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2012 AMC 12B Problems/Problem 6

Contents

Problem

Solution

Solution 2

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