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2012 AMC 10B Problems/Problem 6: Difference between revisions

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== Problem 6 ==
== 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>
<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>
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 ==
== Solution ==


Say Z=is the amount rounded up by and down by.  
Let's define <math>z</math> as 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''.
The problem statement tells us that Xiaoli performed the following computation:


Which translates to:
<math>\left(x+z\right) - \left(y-z\right) = x+z-y+z = x-y+2z</math>


<math>(X+Z)-(Y-Z)</math>=<math>X+Z-Y+Z</math>=<math>X+2Z-Y</math>
We can see that <math>x-y+2z</math> is greater than <math>x-y</math>, and the answer is <math>\textbf{(A)} \text{Her estimate is larger than} x-y</math>


This is <math>2Z</math> bigger than the original amount of <math>X-Y</math>.
Therefore, her estimate is larger than <math>X-Y</math>
or
<math> \textbf{(A)}</math>


==See Also==
==See Also==

Revision as of 14:54, 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 the answer is $\textbf{(A)} \text{Her estimate is larger than} x-y$


See Also

2012 AMC 10B (Problems • Answer Key • Resources)
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

These problems are copyrighted © by the Mathematical Association of America.

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