What is the angle between the hour and minute hand when it is 4: 20?

The arc between the 4 and the 5 is (360/12) = 30 degrees.

The hour hand takes a full hour to creep from the 4 to the 5, so it moves
at the rate of ( 30 degrees / 60 minutes ) = 1/2 degree per minute.

In 20 minutes, the hour hand moves (20 x 1/2) = 10 degrees.

At 4:20 , the minute hand is right square on the 4, and the hour hand
is 10 degrees ahead, on its way to the 5 .

The angle between them is 10 degrees.