Q 118 - Intermediate Algebra

Question

A motor boat traveled $18$ miles down a river in two hours but going back upstream, it took $4.5$ hours due to the current. Find the rate of the motor boat in still water and the rate of the current. (Round to the nearest hundredth).

Step-by-Step Solution

Answer

The rate of the boat in still water is $6.5$ mph.

The rate of the boat of the current is $2.5$ mph.

Step 1. Given information.

Consider the given question,

Distance traveled by a motor boat is $18$ miles.

Time taken by the boat to travel downstream is $2$ hours.

Time taken by the boat to travel upstream is $4.5$ 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 $18$ miles downstream in $2$hours due to the current.

So, $$\begin{gathered} 2\left(x+y\right)=18 \\ x+y=9 ....... \left(\mathrm{i}\right) \end{gathered}$$

When the boat travels $18$ miles upstream in $4.5$ hours, as the current will slow the motor boat.

So, $$\begin{gathered} 4.5\left(x-y\right)=18 \\ x-y=\frac{18}{4.5} \\ x-y=4 ...... \left(\mathrm{ii}\right) \end{gathered}$$

Step 3. Solve the equations.

Add equations (i) and (ii),

$$\begin{gathered} \left(x+y\right)+\left(x-y\right)=9+4 \\ 2x=13 \\ x=6.5 \end{gathered}$$

Substituting $x=6.5$ in equation (ii),

$$\begin{gathered} 6.5-y=4 \\ y=2.5 \end{gathered}$$

Step 4. Check the answers.

Substitute the values in equation (i),

$$\begin{gathered} 2\left(6.5+2.5\right)=18 \\ 2\left(9\right)=18 \\ 18=18 \end{gathered}$$

This is true.

Substitute the values in equation (ii),

$$\begin{gathered} 6.5-2.5=4 \\ 4=4 \end{gathered}$$

This is true.

The rate of the boat in still water is $6.5$mph.

The rate of the boat of the current is  $2.5$ mph.