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

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== Problem== | == Problem== | ||

− | In order to estimate the value of x-y where x and y are real numbers with x>y>0 | + | 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? |

+ | '''(A)''' Her estimate is larger than <math>x-y</math>. | ||

+ | '''(B)''' Her estimate is smaller than <math>x-y</math>. | ||

+ | |||

+ | '''(C)''' Her estimate equals <math>x-y</math>. | ||

+ | |||

+ | '''(D)''' Her estimate equals <math>y-x</math>. | ||

+ | |||

+ | '''(E)''' Her estimate is <math>0</math>. | ||

==Solution== | ==Solution== | ||

− | + | 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 23:49, 27 February 2012

## Problem

In order to estimate the value of where and are real numbers with , Xiaoli rounded up by a small amount, rounded 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 .

**(B)** Her estimate is smaller than .

**(C)** Her estimate equals .

**(D)** Her estimate equals .

**(E)** Her estimate is .

## Solution

The original expression now becomes , where is a positive constant, hence the answer is **(A)**.