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Difference between revisions of "2012 AMC 10B Problems/Problem 6"

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

$\textbf{(A)}$ The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.