2012 AMC 10B Problems/Problem 6

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Problem 6

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


Solutions

Say Z=is the amount rounded up by and down by.

Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values.

Which translates to:

$(X+Z)-(Y-Z)$=$X+Z-Y+Z$=$X+2Z-Y$

This is $2Z$ bigger than the original amount of $X-Y$.

Therefore, her estimate is larger than $X-Y$

or

$\textbf{(A)}$

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