Difference between revisions of "2014 AMC 12B Problems/Problem 7"

(Created page with "==Problem== For how many positive integers <math>n</math> is <math>\frac{n}{30-n}</math> also a positive integer? <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\...")
 
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==Solution==
 
==Solution==
We know that <math>n \le 30</math> or else <math>30-n</math> will be negative, resulting in a negative fraction.  We also know that <math>n \ge 15</math> or else the fraction's denominator will exceed its numerator making the fraction unable to equal a positive integer value.  Substituting all values <math>n</math> from <math>15</math> to <math>30</math> gives us integer values for <math>n=15, 20, 24, 25, 27, 28, 30</math>.  Counting them up, we have <math>\boxed{\textbf{(D)}\ 7}</math> possible values for <math>n</math>.   
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We know that <math>n \le 30</math> or else <math>30-n</math> will be negative, resulting in a negative fraction.  We also know that <math>n \ge 15</math> or else the fraction's denominator will exceed its numerator making the fraction unable to equal a positive integer value.  Substituting all values <math>n</math> from <math>15</math> to <math>30</math> gives us integer values for <math>n=15, 20, 24, 25, 27, 28, 29</math>.  Counting them up, we have <math>\boxed{\textbf{(D)}\ 7}</math> possible values for <math>n</math>.   
  
 
(Solution by kevin38017)
 
(Solution by kevin38017)

Revision as of 17:47, 20 February 2014

Problem

For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}}\ 7\qquad\textbf{(E)}\ 8$ (Error compiling LaTeX. Unknown error_msg)

Solution

We know that $n \le 30$ or else $30-n$ will be negative, resulting in a negative fraction. We also know that $n \ge 15$ or else the fraction's denominator will exceed its numerator making the fraction unable to equal a positive integer value. Substituting all values $n$ from $15$ to $30$ gives us integer values for $n=15, 20, 24, 25, 27, 28, 29$. Counting them up, we have $\boxed{\textbf{(D)}\ 7}$ possible values for $n$.

(Solution by kevin38017)