Difference between revisions of "2015 AMC 10A Problems/Problem 15"

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==Solution==
 
==Solution==
  
You can create the equation  <math>\frac{x+1}{y+1}=(1.1)(\frac{x}{y})</math>
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You can create the equation   
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<math>\frac{x+1}{y+1}=(1.1)(\frac{x}{y})</math>
 +
 
 +
<math>\frac{x+1}{y+1}=\frac{1.1x}{y}</math>
 +
 
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<math>(x+1)(y)=(1.1x)(y+1)</math>
 +
 
 +
<math>xy+y=1.1xy+1.1x</math>
 +
 
 +
<math>y=.1xy+1.1x</math>
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 +
<math>10y=xy+11x</math>

Revision as of 17:46, 4 February 2015

Problem

Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\%$?

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{infinitely many}$

Solution

You can create the equation $\frac{x+1}{y+1}=(1.1)(\frac{x}{y})$

$\frac{x+1}{y+1}=\frac{1.1x}{y}$

$(x+1)(y)=(1.1x)(y+1)$

$xy+y=1.1xy+1.1x$

$y=.1xy+1.1x$

$10y=xy+11x$

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