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Difference between revisions of "2012 AMC 12B Problems/Problem 6"

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== Problem==
 
== Problem==
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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. Which statement is always true?
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In order to estimate the value of <math>x-y</math> where <math>x</math> and <math>y</math> are real numbers with <math>x>y>0</math>, Xiaoli rounded <math>x</math> up by a small amount, rounded <math>y</math> down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?
  
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'''(A)''' Her estimate is larger than <math>x-y</math>.
  
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'''(B)''' Her estimate is smaller than <math>x-y</math>.
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'''(C)''' Her estimate equals <math>x-y</math>.
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'''(D)''' Her estimate equals <math>y-x</math>.
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'''(E)''' Her estimate is <math>0</math>.
  
 
==Solution==
 
==Solution==
  
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Since the original equation x-y now becomes (x+k) - (y-k), where k is a constant. so simplified, the equation becomes: x-y+2k, which is greater than x-y, hence the answer is: A.
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The original expression <math>x-y</math> now becomes <math>(x+k) - (y-k)=(x-y)+2k>(x-y)</math>, where <math>k</math> is a positive constant, hence the answer is '''(A)'''.

Revision as of 00:49, 28 February 2012

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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