4. Given ten distinct rational numbers, among them, the sum of any nine is an irreducible proper fraction with a denominator of 22 (a proper fraction where the numerator and denominator have no common divisor). Then the sum of these ten rational numbers $S=$ $\qquad$
4. $\frac{5}{9}$.

The irreducible proper fraction with a denominator of 22 is $\frac{i}{22}$, where $i=1,3,5,7,9,13,15,17,19,21$.

According to the problem, ten distinct rational numbers $a_{1}, a_{2}, \cdots, a_{10}$, each time taking nine of them, the sum $S-a_{i}$ $(i=1,2, \cdots, 10)$ should be one of the ten irreducible proper fractions above.
$$
\begin{array}{l}
\text { Hence } 9 S=\frac{1}{22}+\frac{3}{22}+\frac{5}{22}+\frac{7}{22}+\frac{9}{22}+\frac{13}{22}+ \\
\frac{15}{22}+\frac{17}{22}+\frac{19}{22}+\frac{21}{22} \\
=5 .
\end{array}
$$

Therefore, $S=\frac{5}{9}$.