2015 AMC 10A Problems/Problem 15
Problem
Consider the set of all fractions 
, where 
and 
are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by 
, the value of the fraction is increased by 
?
Solution
You can create the equation 
Cross multiplying and combining like terms gives 
.
This can be factored into 
.
Either 
has to be positive and 
has to be negative or vice versa, but if as negative, then would be negative, so that halves our search space.
Similarly, 
, so 
and 
, so 
.
This leaves the factor pairs: 
.
But we can't stop here because and must be relatively prime.
gives 
and 
. 
and 
are not relatively prime so this doesn't work.
gives 
and 
. This doesn't work.
gives 
and 
. This does work.
We found one valid solution so the answer is 
.
