Answer
$4$ miles per hour.
Work Step by Step
Average speed of the motorboat in still water is $20$ miles per hour.
Let the speed of the current be $x$.
Speed of the motorboat against the current $20-x$.
Speed of the motorboat with the current $20+x$.
Formula for the time is $=\frac{Distance}{Speed}$.
The given condition is
$\Rightarrow \frac{3}{20+x}=\frac{2}{20-x}$
The LCD is $(20+x)(20-x)$.
Multiply the equation by LCD to clear fractions.
$\Rightarrow (20+x)(20-x)\left (\frac{3}{20+x}\right )=(20+x)(20-x)\left (\frac{2}{20-x}\right )$
Cancel common factors.
$\Rightarrow 3(20-x)=2(20+x)$
Use distributive property.
$\Rightarrow 60-3x=40+2x$
Add $3x-40$ to both sides.
$\Rightarrow 60-3x+3x-40=40+2x+3x-40$
Add like terms.
$\Rightarrow 20=5x$
Divide both sides by $5$.
$\Rightarrow \frac{20}{5}=\frac{5x}{5}$
Simplify.
$\Rightarrow 4=x$
Hence, the current's rate is $4$ miles per hour.
Note: Check if the solution is correct. The equation is defined for all real values of $x$ except the zeros of the denominators, which are $-20$ and $20$. Since our solution is different than those values, it is correct.
