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

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== Problem==
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== Problem ==
 
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?
 
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>.
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<math>\textbf{(A) } \text{Her estimate is larger than } x - y \qquad \textbf{(B) } \text{Her estimate is smaller than } x - y \qquad \textbf{(C) } \text{Her estimate equals } x - y</math>
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<math>\textbf{(D) } \text{Her estimate equals } y - x \qquad \textbf{(E) } \text{Her estimate is } 0</math>
  
'''(B)''' Her estimate is smaller than <math>x-y</math>.
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== Solution ==
 
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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 <math>\boxed{\textbf{(A)}}</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==
 
 
 
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)'''.
 
  
 
== See Also ==
 
== See Also ==
 
 
{{AMC12 box|year=2012|ab=B|num-b=5|num-a=7}}
 
{{AMC12 box|year=2012|ab=B|num-b=5|num-a=7}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 00:25, 19 October 2020

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?

$\textbf{(A) } \text{Her estimate is larger than } x - y \qquad \textbf{(B) } \text{Her estimate is smaller than } x - y \qquad \textbf{(C) } \text{Her estimate equals } x - y$ $\textbf{(D) } \text{Her estimate equals } y - x \qquad \textbf{(E) } \text{Her estimate is } 0$

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 $\boxed{\textbf{(A)}}$.

See Also

2012 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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