Answer
Rate in still water $=4.5$ miles per hour.
Rate of the current $=1.5$ miles per hour.
Work Step by Step
Let the rate in still water $=x$ miles per hour.
and the rate of the current $=y$ miles per hour.
Effective speed with the current $=x+y$ miles per hour.
Effective speed against the current $=x-y$ miles per hour.
Use distance formual.
$\Rightarrow Speed \times Time = Distance$
First condition (with the current):-
$24$ miles in $4$ hours.
Substitute all values into the formula.
$\Rightarrow (x+y)\times 4 = 24$
Divide both sides by $4$.
$\Rightarrow \frac{1}{4}\cdot (x+y)\times 4 = \frac{1}{4}\cdot 24$
Simplify.
$\Rightarrow x+y= 6$ ......(1)
Second condition (against the current):-
$\frac{3}{4}$ of the distance
$=\frac{3}{4}\cdot 24=18\; miles$
$18$ miles in $6$ hours.
Substitute all values into the formula.
$\Rightarrow (x-y)\times 6 = 18$
Divide both sides by $6$.
$\Rightarrow \frac{1}{6}\cdot (x-y)\times 6 = \frac{1}{6}\cdot 18$
Simplify.
$\Rightarrow x-y= 3$ ......(2)
Add equation (1) and (2).
$\Rightarrow x+y+x-y=6+3$
Simplify.
$\Rightarrow 2x=9$
Divide both sides by $2$.
$\Rightarrow \frac{2x}{2}=\frac{9}{2}$
Simplify.
$\Rightarrow x=4.5$.
Substitute the value of $x$ into equation (1).
$\Rightarrow 4.5+y=6$
Subtract $4.5$ from both sides.
$\Rightarrow 4.5+y-4.5=6-4.5$
Simplify.
$\Rightarrow y=1.5$
