There are 60 minute graduations around the 360 degree clock face, so the minute
graduations are 6 degrees apart.  

Therefore, when the minute hand is 7 minute graduations behind the hour hand,
the angle between the hands is 7 times 6 degrees or 42 degrees.

The numerals are 30 degrees apart.  Therefore at 5:00 the hands are 5x30 = 150
degrees apart.

The minute hand travels 360 degrees per hour, which is 6 degrees per minute.

The hour hand travels 360 degrees per 12 hours, which is 30 degrees per hour, or
0.5 degrees per minute.

So the angle between the hands is either increasing at 6.0-0.5 = 5.5 degrees per
minute or decreasing at 5.5 degrees per minute.

At 5:00 the angle between the hands is 150 degrees, and is decreasing at 5.5
degrees per minute.  So the angle between the hands must decrease from 150
degrees to 42 degrees, which is a decrease of 150-42=108 degrees. So since the
angle between the hands is decreasing at the rate of 5.5 degrees per minute,
that will take it 108/5.5 minutes to decrease to 42 degrees.  108/5.5 = 1080/55
= 216/11 = 19 7/11 minutes, so the minute hand will be seven minutes behind the
hour hand at 5:19 7/11.

Edwin