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

(Created page with "== Problem 6 == 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, an...")
 
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<math>(X+Z)-(Y-Z)</math>=<math>X+Z-Y+Z</math>=<math>X+2Z-Y</math>
 
<math>(X+Z)-(Y-Z)</math>=<math>X+Z-Y+Z</math>=<math>X+2Z-Y</math>
  
This is 2Z bigger than the original amount of <math>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>
 
Therefore, her estimate is larger than <math>X-Y</math>
 
  
 
or
 
or
 
  
 
<math> \textbf{(A)}</math>
 
<math> \textbf{(A)}</math>

Revision as of 19:07, 25 February 2012

Problem 6

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


Solutions

Say Z=is 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.

Which translates to:

$(X+Z)-(Y-Z)$=$X+Z-Y+Z$=$X+2Z-Y$

This is $2Z$ bigger than the original amount of $X-Y$.

Therefore, her estimate is larger than $X-Y$

or

$\textbf{(A)}$

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