Let $n$ be the number of cups (the number of people in the family), and $x$ be the amount of milk consumed (in cups). Then the amount of coffee consumed is $n-x$. Katya drank one cup of coffee with milk, which consisted of one quarter of all the milk $(x / 4)$ and one sixth of all the coffee $((n-x) / 6)$. We get

$$
\begin{gathered}
\frac{x}{4}+\frac{(n-x)}{6}=1 \\
3 x+2(n-x)=12 \\
x+2 n=12
\end{gathered}
$$

Since $n$ is an integer, it follows from the last equation that $x$ is an integer, and even ($x=12 - 2 n$). Moreover, $x \leq n$, since the amount of milk consumed is certainly no more than the total amount of the drink.

By a small enumeration, we find that the last equation has three solutions:

$$
n=6, x=0 ; \quad n=5, x=2 ; \quad n=4, x=4 .
$$

The first and last solutions correspond to the case where everyone drank just milk or just coffee, while the second solution corresponds to the case where they drank coffee with milk.

## Answer

5 people.
Booin D.A.

The whole family drank a full cup of coffee with milk, and Katya drank a quarter of all the milk and a sixth of all the coffee. How many people are in the family?