A motorboat takes 5 hours to travel 100 miles going upstream. The return trip takes 2 hours going downstream. What is the rate of the boat in still water and what is the rate of the current?
Let S be the distance travelled by the boat, both upstream and downstream = 100 miles
Let Vc be the speed of the current
Let Vm be the speed of the motorboat (in still water)
Let Vu be the speed of the boat as it travels upstream = Vm - Vc
Let Tu be the time to travel upstream = 5 hrs
Let Vd be the speed of the boat as it travels downstream = Vm + Vc
Let Td be the time to travel downstream = 2 hrs
Travelling upstream
S = Vu*Tu = 5Vu = 5(Vm - Vc)
Travelling downstream
S = Vd*Td = 2Vd = 2(Vm + Vc)
Substituting Values,
100 = 5Vm - 5Vc
100 = 2Vm + 2Vc
Substituting for Vm = 50 - Vc into the 1st eqn above,
100 = 5(50 - Vc) - 5Vc
100 = 250 - 5Vc - 5Vc
10Vc = 150
Vc = 15 mph
Vm = 35 mph