Answer
$\dfrac{23}{25}$
Work Step by Step
Let $x$ be the numerator. Then $x+2$ is the denominator.
The conditions of the problem translate to
\begin{array}{l}\require{cancel}
\dfrac{x-3}{x+2+5}=\dfrac{2}{3}
.\end{array}
By cross-multiplication and the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
3(x-3)=2(x+2+5)
\\
3(x-3)=2(x+7)
\\
3x-9=2x+14
\\
3x-2x=14+9
\\
x=23
.\end{array}
The numerator, $x,$ is $23$ and the denominator, $x+2,$ is $25.$ Hence, the fraction is $
\dfrac{23}{25}
.$
