Q 119 - Intermediate Algebra

Question

A river cruise boat sailed $80$ miles down the Mississippi River for four hours. It took five hours to return. Find the rate of the cruise boat in still water and the rate of the current.

Step-by-Step Solution

Answer

The rate of the cruise boat in still water is $18$ mph.

The rate of the current is $2$ mph.

Step 1. Given information.

Consider the given question,

Distance traveled by the cruise boat is $80$ miles.

Time taken by the cruise boat to travel downstream is $4$ hours.

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

Assume $x$ to be the rate of the cruise 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 $80$ miles downstream in $4$ hours,  then the current will help the cruise boat.

So, $$\begin{gathered} 4\left(x+y\right)=80 \\ x+y=20 ...... \left(\mathrm{i}\right) \end{gathered}$$

When the travels $80$ miles upstream in $5$ hours, then the current will slow the cruise boat.

So,  $$\begin{gathered} 5\left(x-y\right)=80 \\ x-y=16 ...... \left(\mathrm{ii}\right) \end{gathered}$$

Step 3. Solve the equations.

Add equations (i) and (ii),

$$\begin{gathered} \left(x+y\right)+\left(x-y\right)=20+16 \\ 2x=36 \\ x=18 \end{gathered}$$

Substituting $x=18$ in equation (i),

$$\begin{gathered} 18+y=20 \\ y=2 \end{gathered}$$

Step 4. Check the answers.

Substitute the values in equation (i),

$$\begin{gathered} 4\left(18+2\right)=80 \\ 4\left(20\right)=80 \\ 80=80 \end{gathered}$$

This is true.

Substitute the values in equation (ii),

$$\begin{gathered} 18-2=16 \\ 16=16 \end{gathered}$$

This is true.

The rate of the cruise boat in still water is $18$ mph.

The rate of the current is $2$ mph.