1. A person entering a barber shop at 6 in the morning noticed that the clock hands formed an angle of $90^{\circ}$, and upon leaving, just before 7 o'clock, noticed that the clock hands formed an angle of $75^{\circ}$. How long did the person stay in the barber shop?
Solution. Let the hour hand have moved $x$ degrees; then the minute hand has moved $12x$ degrees on one side, and on the other side

$$
90^{\circ}+x+75^{\circ}
$$

Thus,

$$
165^{\circ}+x=12 x \Rightarrow x=15^{\circ}
$$

i.e., the minute hand, with its movement, has made an angle of

$$
165^{\circ}+15^{\circ}=180^{\circ}
$$

Since every $6^{\circ}$ makes one minute, we get that the person stayed 30 minutes in the barber shop.

![](https://cdn.mathpix.com/cropped/2024_06_05_edcf2076eef478aa4fe7g-08.jpg?height=384&width=367&top_left_y=243&top_left_x=1095)