On an Analog clock, what angle is made by the minute and hour hand at exactly 4:09?

This problem is so much easier on a digital clock ... but ...

There are 360° in a circle.

Between any two numbers on the clock, there will be 360° ÷ 12  =  30°.  (This means that the hour hand moves 30° when going from 4:00 to 5:00.)

I'm going to measure all the degree from 12:00 going in a clockwise direction.

First, let's just consider the hour hand.

At exactly 4:00, the hour hand will be 4 x 30° = 120° from 12:00.

At 4:09, the hour hand will go another 9/60th of the distance from 4:00 to 5:00.  (9/60) x 30° = 4.5°.

So the hour hard will be 120° + 4.5° = 124.5° from 12:00.

Now, let's consider the minute hand. After 9 minutes, it will be 9/60th of the way around the dial, so it will be (9/60) x 360° = 54° from 12:00.

The distance between the two hands is:  124.5° - 54°  =  70.5°.