Question 135
The time it took the man to travel uphill is:

time to travel uphill = distance/30 mph
time to travel downhill = distance/downhill speed

Average speed is the total distance (which is twice the distance to go one way), divided by total time:

{{{average_speed = (2*distance/(distance/30+distance/downhill_speed)) }}}

we know that average_speed is 45:

{{{45 = (2*distance/(distance/30+distance/downhill_speed)) }}}

Simplifying

{{{45=(2*distance/((distance*downhill_speed+distance*30)/(30*downhill_speed))) }}}

distance cancels out (and that's why it was not given in the problem):

{{{45=(2/((downhill_speed+30)/(30*downhill_speed))) }}}
{{{45=(2*30*downhill_speed)/(downhill_speed+30) }}}
{{{45*(downhill_speed+30) = 60*downhill_speed}}}

45*downhill_speed+30*45 = 60*downhill_speed
15*downhill_speed = 30*45
downhill_speed = 30*3 = 90