Problem 9. At the conference, there were several scientists, some of whom speak English, some French, and some German. The organizers of the conference noticed that among those who speak English, exactly $1 / 5$ speak French, and exactly $1 / 3$ speak German; among those who speak French, exactly $1 / 8$ speak English, and exactly $1 / 2$ speak German; finally, among those who speak German, exactly $1 / 6$ speak English. What fraction of those who speak German also speak French?
Answer: $2 / 5$.

Solution. Let $x$ be the number of scientists who speak German. Then $x / 6$ scientists speak both German and English; thus, $x / 2$ scientists speak English, and $x / 10$ scientists speak both English and French; therefore, $4 x / 5$ scientists speak French, and $2 x / 5$ scientists speak both French and German.