it takes a motorboat 4 hours to travel 56 miles down a river, it takes 3 hours longer to make the return trip. Find the speed of the current.

Question

Solve by defining variables and creating equations, as needed.

Answer 1

v - speed of the boat;

u - speed of the current of river.

 

speed of the boat down the river is (v+u), time down the river is 4 hours;

speed of the boat up the river is (v-u), time up the river is (4+3) hours.

 

Distance  56 miles both ways.

We use formula

                         distance = speed * time

two times, down the river and up the river.

 

Down the river

                         56 = (v+u)4

                         56 =4v + 4u.

 

Up the rivier

                         56 = (v-u)7

                         56 = 7v-7u.

 

The system is

4v + 4u =56,

7v - 7u  =56.

 

We can simplify the system by dividing both sides of the first equation by 4

and both sides of the second equation by 7.

 

v + u =14,

v -  u = 8.

 

When we solve the system,

we find the boat speed is 11 mph and the river speed is 3 mph.

 

Answer 2

Let y = speed of current and x = speed of boat in still water

 

Distance = (Rate)(Time), so Time = (Distance)/(Rate)

 

speed upstream (against current) = x-y

speed downstream (with current) = x+y

 

Time downstream = 56/(x+y) = 4     So, x+y = 14.

 

Time upstream  = Time downstream + 3 = 4 + 3 = 7

 

So, 56/(x-y) = 7.  Therefore, x - y = 8

 

We have:  x + y = 14  

                x - y = 8

 

Adding these equations, we get 2x = 22    x = 11

                                                                 y = 3

 

Speed of current = y = 3 mph.