Q 117 - Intermediate Algebra

Question

A motor boat 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.

Step-by-Step Solution

Answer

The rate of the boat in still water is $16$ mph.

The rate of the current is $4$ mph.

Step 1. Given information.

Consider the given equation,

Distance traveled by a motor boat is $60$ miles.

Time taken by the boat to travel downstream is $3$ hours.

Time taken by the boat to travel upstream is $5$ hours.

Assume $x$ to be the rate of the motor boat in still water and $y$ to be the rate of the current.

Step 2. Form the equations.

Consider the given question,

When the boat travels $60 miles$ downstream in $3$ hours, then the current will help the motor boat.

So, $$\begin{gathered} 3\left(x+y\right)=60 \\ x+y=20 ...... \left(\mathrm{i}\right) \end{gathered}$$

When the boat travels $60 miles$ upstream in $5$ hours, then the current will slow the motor boat.

So, $$\begin{gathered} 5\left(x-y\right)=60 \\ x-y=12 ...... \left(\mathrm{ii}\right) \end{gathered}$$

Step 3. Solve the equations.

Add equations (i) and (ii),

$$\begin{gathered} \left(x+y\right)+\left(x-y\right)=20+12 \\ 2x=32 \\ x=16 \end{gathered}$$

Substituting $x=16$ in equation (ii),

$$\begin{gathered} 16-y=12 \\ y=16-12 \\ y=4 \end{gathered}$$

Step 4. Check the answers.

Substitute the values in equation (i),

$$\begin{gathered} 16+4=20 \\ 20=20 \end{gathered}$$

This is true.

Substitute the values in equation (ii),

$$\begin{gathered} 16-4=12 \\ 12=12 \end{gathered}$$

This is true.

The rate of the boat in still water is $16$ mph.

The rate of the current is $4$ mph.