At what time, between 4 and 5 o'clock, will the minute and hour hands be together?

To calculate the time when the minute and hour hands are together between 4 and 5 o'clock, we need to consider the relative speeds of the two hands.

The minute hand moves at a rate of 6 degrees per minute, while the hour hand moves at a rate of 0.5 degrees per minute (since it covers 30 degrees in 60 minutes).

Let's denote the time when the minute and hour hands are together as "x" minutes past 4 o'clock.

The minute hand will cover x minutes at a rate of 6 degrees per minute, resulting in an angular displacement of 6x degrees.

The hour hand will cover x minutes at a rate of 0.5 degrees per minute, resulting in an angular displacement of 0.5x degrees.

To find the time when the minute and hour hands are together, we need to solve the equation:

6x = 0.5x + 120

x = (120/5.5) = (240/11) minutes = 21 (9/11) minutes

Therefore, the minute and hour hands are together at 4 hours, 21 minutes, and 9/11 minutes past 4 o'clock.