Here
$\frac{4 quarts of Alcohol}{4 quarts of water}$

we need to add water to above solution to make it equal to = $\frac{3 parts of Alcohol}{5 parts of water}$
Let w= quarts of water required

Total quarts of water = (4 + w)

Therefore we can re write as -

$\frac{4 quarts of Alcohol}{(4 quarts of water + w)}$ = $\frac{3 parts of Alcohol}{5 parts of water}$(since we need to add w to make the ratio equal to 3/5)

or $\frac{4}{(4+w)}$ =$\frac{3}{5}$

or w =$\frac{8}{3}$