Answer
$\dfrac{6}{10} \text{ or } \dfrac{3}{5}$
Work Step by Step
Let $x$ be the denominator. Then the numerator is $x-4.$
The conditions of the problem translate to
\begin{array}{l}\require{cancel}
\dfrac{x-4+2}{x+2}=\dfrac{2}{3}
\\\\
\dfrac{x-2}{x+2}=\dfrac{2}{3}
.\end{array}
By cross-multiplication and by using the properties of equality, then
\begin{array}{l}\require{cancel}
3(x-2)=2(x+2)
\\\\
3x-6=2x+4
\\\\
3x-2x=4+6
\\\\
x=10
.\end{array}
Hence, the numerator, $x-4,$ is $6$ and the denominator, $x,$ is $10.$ Hence, the fraction is $
\dfrac{6}{10} \text{ or } $\dfrac{3}{5}
.$
