[ Union, intersection, and difference of sets ] Problems on percentages and ratios $\quad]$

In the gymnasium, all students know at least one of the ancient languages - Greek or Latin, some know both languages. $85 \%$ of all students know Greek and $75 \%$ know Latin. What fraction of the students know both languages?
$100-85=15 \%$ of all the guys do not know Greek, which means they only know Latin. This means that $75-15=$ $60 \%$ speak both languages.

## Answer

$60 \%$.

35 ravens flew into the glade. Unexpectedly, the ravens took off and split into two flocks: one flock perched on the branches of an old birch tree, and the other on an alder. After some time, 5 ravens flew from the birch to the alder, and the same number of ravens flew away from the birch altogether, after which there were twice as many ravens left on the birch as on the alder. How many ravens were there in each of the two flocks initially?

## Solution

35 ravens minus the 5 that flew away leaves 30. Since after this, there were twice as many ravens on the birch as on the alder, there were 20 ravens on the birch and 10 on the alder. But before that, 5 ravens had flown from the birch to the alder, so initially, there were 5 ravens on the alder. And on the birch - the remaining 30 ravens.

## Answer

30 and 5 ravens.