8.1 On the island, $\frac{2}{3}$ of all men are married and $\frac{3}{5}$ of all women are married. What fraction of the island's population is married?
8.1 On the island, $\frac{2}{3}$ of all men are married and $\frac{3}{5}$ of all women are married. What fraction of the island's population is married?

Solution: Let M be the number of men on the island, and W be the number of women. The number of families, if counted by husbands, is $\frac{2}{3} M$; on the other hand, if counted by wives, the number of families is $\frac{3}{5} W$. Therefore, $\frac{2}{3} M = \frac{3}{5} W$, from which we get $W = \frac{10}{9} M$. The total number of people who are married is $\frac{2}{3} M + \frac{3}{5} W = \frac{2}{3} M + \frac{2}{3} M = \frac{4}{3} M$. The total population of the island is $M + W = M + \frac{10}{9} M = \frac{19}{9} M. \quad$ Therefore, the fraction of the population that is married is

$$
\frac{\frac{4}{3} M}{\frac{19}{9} M} = \frac{4}{3} \cdot \frac{9}{19} = \frac{12}{19}
$$