Question

In the following exercises, translate to a system of equations and solve.

A motorboat travels 60 miles down a river in three hours but takes five hours to return upstream. Find the rate of the boat in still water and the rate of the current.

Short answer

The rate of boat in still water and rate of current are 16 and 4 respectively.

Step-by-step solution

Step 1. Given Information     

The given data is that motorboat travels 60 miles down a river in three hours but takes five hours to return upstream.

Step 2. Explanation

Let x be the rate of the boat in still water and y be the rate of current.

The boat ride down for 3 hours and the distance covered is 60 miles and the actual rate of boat is $x+y$and the distance covered can be expressed as$3(x+y)=60--\left(1\right)$

The boat return trip took 5 hours. Since the upstream downs the boat, the actual rate of the boat is$x-y$and the distance covered is$5(x-y)=60--\left(2\right)$

Step 3. Calculation  

Multiply equation (1) with 5 and equation (2) with 3. Thus, the revised equations are,

$$\begin{gathered} 5(3x+3y)=5\left(60\right) \\ 15x+15y=300--\left(3\right) \\ And \\ 3(5x-5y)=3\left(60\right) \\ 15x-15y=180--\left(4\right) \end{gathered}$$

Solve the equations (3) and (4) by subtracting equation (4) from equation (3),

$$\begin{gathered} 15x+15y-15x+15y=300-180 \\ 30y=120 \\ y=4 \end{gathered}$$

Substitute the value of y in equation (1) to get the value of x.

$$\begin{gathered} 3x+3\left(4\right)=60 \\ 3x=48 \\ x=16 \end{gathered}$$