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2012 AMC 12B Problem 6

Problem 6 of 25EasierAlgebra

In order to estimate the value of x−yx-y where xx and yy are real numbers with x>y>0,x \gt y \gt 0, Xiaoli rounded xx up by a small amount, rounded yy down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?

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Solution

Let d>0d \gt 0 be the small amount. Xiaoli computes (x+d)−(y−d)(x+d)-(y-d) =(x−y)+2d.=(x-y)+2d. Since 2d>0,2d \gt 0, her estimate exceeds the true value x−y.x-y. Thus, the correct answer is A.

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Concepts: estimation · inequality

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.