Question 739464
You just stated that the original fraction is 2/3 and you ask what is this fraction.  It's the same: 2/3.  


Did you mean that the ratio of two variables is 2/3, and then according to the description, find those two numbers?  That would be worded a little differently.


Let numerator be n
Let denominator be d.
{{{n/d = 2/3}}}.


The longer part of the description gives {{{(n+14)/(d-9)=d/n}}}.


Working with the longer equation,{{{ n(n+14)=d(d-9)}}}
{{{n^2+14n=d^2-9d}}}.


Using the 2/3 equation, {{{n=2d/3}}}, so try substituting:
{{{d^2-9d=4d^2/9+14*2d/3}}}
Multiply both sides by 9
{{{9d^2-81d=4d^2+84d}}}
.
.
{{{5d^2-165d=0}}}
{{{d^2-33d=0}}}
{{{d(d-33)=0}}}
... {{{highlight(d=33)}}}
meaning {{{highlight(n=22)}}}.