jmc

Suppose we played $x$ games. Since we won $2/9$ of the games we played, we won $(2/9)x=2x/9$ games. Therefore, we lost $x-2x/9=7x/9$ games. Since we lost 15 more games than we won, we have $$\frac{7x}{9}-\frac{2x}{9}=15.$$Simplifying the left side gives $5x/9=15$, and solving this equation gives $x=\boxed{27}$ games played.