Starting at home, luis traveled uphill to the gift store for 50 minutes at just 6 mph. he then traveled back home along the same path downhill at a speed of 12 mph.
what is his average speed for the entire trip from home to the gift store and back?

Average speed for the entire trip, both ways, is

                 (Total distance) divided by (total time) .

We don't know the distance from his house to the gift store,
and we don't know how long it took him to get back.
We'll need to calculate these.

-- On the trip TO the store, it took him 50 minutes, at 6 mph.
-- 50 minutes is 5/6 of an hour.
-- Traveling at 6 mph for 5/6 of an hour, he covered 5 miles.
-- The gift store is 5 miles from his house.
-- The total trip both ways was 10 miles.

-- On the way BACK home from the store, he moved at 12 mph.
-- Going 5 miles at 12 mph, it takes  (5/12 hour) = 25 minutes.

Now we have everything we need.

Distance:
         Going:       5 miles
         Returning:  5 miles
         Total         10 miles

Time:
         Going:        50 minutes
         Returning:   25 minutes
         Total:          75 minutes  =  1.25 hours  

         Average speed for the whole trip =

                       (total distance) / (total time)

                   =      (10 miles)  /  (1.25 hours)

                   =       (10 / 1.25)  miles/hours 

                   =          8 miles per hour

D