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

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== Problem 6 ==
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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>
 
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>
 
  
== Solutions ==
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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 \\ \qquad \textbf{(D) } \text{Her estimate equals } x-y \qquad \textbf{(E) } \text{Her estimate is } 0</math>
  
Say Z=is the amount rounded up by and down by.
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== Solution ==
  
''Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values''.
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Let's define <math>z</math> as the amount rounded up by and down by.
  
Which translates to:
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The problem statement tells us that Xiaoli performed the following computation:
  
<math>(X+Z)-(Y-Z)</math>=<math>X+Z-Y+Z</math>=<math>X+2Z-Y</math>
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<math>\left(x+z\right) - \left(y-z\right) = x+z-y+z = x-y+2z</math>
  
This is <math>2Z</math> bigger than the original amount of <math>X-Y</math>.
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We can see that <math>x-y+2z</math> is greater than <math>x-y</math>, and so the answer is <math>\boxed{\textbf{(A) } \text{Her estimate is larger than } x-y}</math>.
  
Therefore, her estimate is larger than <math>X-Y</math>
 
  
or
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==See Also==
  
<math> \textbf{(A)}</math>
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{{AMC10 box|year=2012|ab=B|num-b=5|num-a=7}}
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{{MAA Notice}}

Latest revision as of 15:07, 18 January 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?


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

Solution

Let's define $z$ as the amount rounded up by and down by.

The problem statement tells us that Xiaoli performed the following computation:

$\left(x+z\right) - \left(y-z\right) = x+z-y+z = x-y+2z$

We can see that $x-y+2z$ is greater than $x-y$, and so the answer is $\boxed{\textbf{(A) } \text{Her estimate is larger than } x-y}$.


See Also

2012 AMC 10B (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 10 Problems and Solutions

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