[ Arithmetic. Mental calculation, etc.]

What angle do the hour and minute hands form at 4:12?

#
The first method. At 12:00, each of the hands is pointing vertically upwards. Let's find the angles that each of the hands forms with this position at 4 hours and 12 minutes.

The hour hand turns $30^{\circ}$ every hour, and in 12 minutes, it will turn an additional $30^{\circ} \div 5 = 6^{\circ}$. Thus, it will form an angle of $126^{\circ}$ with the vertical (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_79aa1f75de69bad6c7b2g-30.jpg?height=453&width=484&top_left_y=-4&top_left_x=781)

The minute hand will turn one-fifth of its full rotation in 12 minutes, which is $360^{\circ} \div 5 = 72^{\circ}$. Therefore, the desired angle is $126^{\circ} - 72^{\circ} = 54^{\circ}$.

The second method. At 4:00, the minute hand is pointing upwards, and the hour hand has passed 20 minute divisions from the vertical position. In the next 12 minutes, the minute hand will turn 12 divisions, and the hour hand will turn one minute division, as 12 minutes constitute one-fifth of an hour. Thus, there are $21 - 12 = 9$ minute divisions between the hour and minute hands. Considering that one minute division is $360^{\circ} \div 60 = 6^{\circ}$, we get that the angle between the hands is $54^{\circ}$.

## Answer

$54^{\circ}$.