A cyclist bikes x distance at 10miles per hour and returns over the same path at 8miles per hour. what is the cyclists average rate for the round trip in miles per hour

Average speed = total distance travelled / time taken

in this case total distance = 2x miles

time taken for first half of the journey = x/10 hours and for second part its x / 8 hours

average speed = 2x / (x/10 + x/8) = 2 x / (4x + 5x / 40)

= 2x * 40 / 9x = 80 / 9 mph 8.89 mph