Answer
The rate of the boat is $=14$ miles per hour.
The rate of the current is $=2$ miles per hour.
Work Step by Step
Step 1:- Assume unknown quantities as variables.
Let the rate of the boat is $=x$.
Let the rate of the current is $=y$.
Step 2:- Write the system of equations.
Speed of the boat with the current $=x+y$.
Speed of the boat against the current $=x-y$.
The given values are
Paddleboat travels $48$ miles with the current in $3$ hours.
Paddleboat travels $48$ miles against the current in $4$ hours.
Formula for the distance is
$\Rightarrow Time\times Speed = Distance$
In the equation form
$\Rightarrow 3(x+y)=48$ ...... (1)
$\Rightarrow 4(x-y)=48$ ...... (2)
Step 3:- Solve the system of equations.
Divide equation (1) by $3$ and equation (2) by $4$.
$\Rightarrow x+y=16$ ...... (3)
$\Rightarrow x-y=12$ ...... (4)
Add equation (3) and (4).
$\Rightarrow x+y+x-y=16+12$
Simplify.
$\Rightarrow 2x=28$
Divide both sides by $2$.
$\Rightarrow \frac{2x}{2}=\frac{28}{2}$
Simplify.
$\Rightarrow x=14$
Plug the value of $x$ into equation (3).
$\Rightarrow 14+y=16$
Isolate $y$.
$\Rightarrow y=16-14$
Simplify.
$\Rightarrow y=2$.
Step 4:- Check the answers.
Substitute the values of $x$ and $y$ into equation (2).
$\Rightarrow 4(14-2)=48$
$\Rightarrow 4(12)=48$
$\Rightarrow 48=48$. True.
