To calculate the average speed of a man who travels a certain distance at a rate of 20 miles an hour and returns at the rate of 30 miles an hour, we can use the following formula:

Average speed = 2 * (Distance traveled) / (Total time taken)

We know that the distance traveled in both directions is the same. Let the distance traveled be d miles.

The total time taken is equal to the time taken to travel d miles at a rate of 20 miles an hour plus the time taken to travel d miles at a rate of 30 miles an hour.

Total time taken = d / 20 + d / 30 = 5d / 60 hours

Substituting the given values in the formula for average speed, we get:

Average speed = 2 * d / (5d / 60) = 24 miles per hour

Therefore, the average speed of the man is 24 miles per hour.

So the answer is 24