Starting at home, luis traveled uphill to the hardware store for 30 minutes at just 8 mph. he then traveled back home along the same path downhill at a speed of 24 mph. what is his average speed for the entire trip from home to the hardware store and back?

In order to solve this problem, we need to know how far Luis travelled and how long it took him.  First, let's see how far Luis travelled.

We can figure this out by doubling how far he went up hill.  Since he travelled for 30 minutes at 8 miles per hour, he went 4 miles up the hill.  Since he also went back home, Luis went a total of 8 miles.

Now, we figure out how long it took him.  The problem tells us it took him 30 minutes to get up the hill.  We can add that to however long it took him to get down the hill.  Since he went 4 miles down the hill, that is 1/6th of 24, so it will take him 1/6th of an hour, or 10 minutes.  It took him a total of 40 minutes to go up and down the hill.

Now, we have the distance he went, 8 miles, and how long it took him, 40 minutes.  40 minutes could also be written as 2/3rd hour.  In order to convert this to miles per hour, we multiply both the distance and the time by 3/2.  This is equal to 12 miles per hour, so the answer is 12 mph.