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

(Problem 6)
Line 1: Line 1:
 
== Problem 6 ==
 
== 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?  
+
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 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
 
 
 
  
 +
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 ==
 
== Solutions ==

Revision as of 18:08, 8 December 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

$\textbf{(A)}$

Invalid username
Login to AoPS