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Question

A boat on a river travels 24 miles downstream in only 3 hours. It takes the same boat 6 hours to travel 24 miles upstream. What are the speed of the boat and the speed of the current?

 

boat=   mph

 

current=  mph

Answer 1

Hi Corina, This is an application of the D(istance) = R(ate) * T(ime) formula. We need to find the rate of the boat with the current and the rate of the boat against the current.

 

The distance traveled in both the directions is 24 miles.

 

Equation 1, down river:

 

D = R_d * T, where R_d is the rate of the boat plus the rate of the river.

 

24 = R_d * 3

 

R_d = 8, distance, 24, divided by time, 3.

 

Equation 2, up river:

 

D = R_u * T, where R_u is the rate of the river minus the rate of the boat.

 

24 = R_u * 6

 

R_u = 4, distance, 24, divided by time, 6.

 

Rate down stream:

 

The rate down stream is equal to the rate of the current plus the rate of the boat.

 

R_d = R_r + R_b

 

Substitute for R_d:

 

8 = R_r + R_b

 

Solve for the rate of the river:

 

R_r = 8 - R_b

 

Rate up stream:

 

The rate up stream is equal to the rate of the current minus the rate of the boat.

 

R_u = R_r - R_b

 

Substitute for R_u

 

4 = R_r - R_b

 

Solve for the rate of the river:

 

R_r = 4 + R_b

 

The rate of the river down stream must equal the rate of the river up stream, which gives:

 

8 - R_b = 4 + R_b

 

Solve for R_b:

 

2R_b = 4

 

Divide both sides by 2:

 

R_b = 2

 

The rate of the boat is 2 miles per hour.

 

Return to rate down stream equation gives:

 

8 = R_r + R_b

 

Substitute for R_b

 

8 = R_r + 2

 

Solve for R_r:

 

R_r = 6

 

The rate of the river is 6 miles per hour.

 

Questions?