2012 AMC 12B Problems/Problem 6

Revision as of 23:50, 27 February 2012 by Yongyi781 (talk | contribs) (Solution)

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?

(A) Her estimate is larger than $x-y$.

(B) Her estimate is smaller than $x-y$.

(C) Her estimate equals $x-y$.

(D) Her estimate equals $y-x$.

(E) 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 (A).

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