Let the missing number be $x$. Given $1 \frac{1}{2}+2 \frac{2}{7}=3 \frac{1}{2}+x$ $$\begin{aligned} & x=1 \frac{1}{2}+2 \frac{2}{7}-3 \frac{1}{2} \\ & x=\frac{3}{2}+\frac{16}{7}-\frac{7}{2} \\ & x=\frac{21+32-49}{14} \\ & x=\frac{4}{14}=\frac{2}{7} \end{aligned}$$ Hence, the number is $\frac{2}{7}$.
Explanation as extracted from the printed page; notation may be imperfect.