## 3. In the Cafeteria

Every day while having breakfast in the cafeteria, I noticed that snacks are always served between 7 and 8 o'clock, when both the hour and minute hands are equally distant from the number 6, and coffee is placed on the table at the moment when the minute hand catches up with the hour hand.

How much time does it take me to have breakfast in this cafeteria (or more precisely, to eat the appetizer and the main course - before the coffee)?
3. Let 7 hours $x$ minutes be the time when the appetizer is served. The angle formed by the hour hand and the segment connecting the center of the clock face with the number 6 is (in degrees

$$
\frac{360}{12}+\left(\frac{x}{60} \cdot \frac{360}{12}\right)=30\left(1+\frac{x}{60}\right)=30+\frac{x}{2}
$$

The angle formed by the minute hand with the same segment is

$$
180-x \frac{360}{60}=180-6 x
$$

Since these two angles are equal,

$$
x=300 / 13 \approx 23 \text { min } 5 \text{ s. }
$$

So, the appetizers are served at approximately 7:23:05. Let 7 hours $y$ minutes be the time when the coffee is served. The angle formed by the hour hand with the same ray directed at the number 6 is

$$
30+y / 2
$$

The angle formed by the minute hand with the same direction is

$$
6 y-180
$$

Since these two angles are equal, then

$$
y=420 / 11 \approx 38 \text { min } 11 \text { s, }
$$

i.e., the coffee is served at approximately 7:38:11.

Thus, in this cafeteria, it takes about 15 min 6 s to eat the appetizers and the main course.