Denote the number of footballers by $n$, the number of basketball players by $k$ and the number of students practicing both sports by $d$. One fifth of the footballers also plays basketball. So $\frac{n}{5} = d$. One seventh of the basketball players also plays football, so $\frac{k}{7} = d$. This implies $n = 5d$ and $k = 7d$.

There are $k - d$ basketball players who do not play football and $n - d$ football players who do not play basketball. Altogether, there are $(n - d) + (k - d)$ students who only practice one of the sports. So $110 = n + k - 2d$. We get $110 = 10d$ or $d = 11$. Hence, 11 students practice both sports.