At his usual running rate, Rahul can travel 12 miles downstream in a certain river in 6 hours less
than it takes him to travel the same distance upstream. But if he could double his usual rowing
rate for his 24 miles round trip the downstream 12 miles would then take only one hour less than
the upstream 12 miles.$2\frac{2}{3}$  is the speed of the current in miles per hour?

Let the speed in still water be x mph and the speed of the currently be y mph.

Speed of upstream =(x−y)

Speed of downstream =(x+y)

According to the question,

$$\begin{gathered} \frac{12}{x-y}-\frac{12}{x+y}=6 \\ Or, \frac{12x+12y-12x+12y}{x^2-y^2}=6 \\ Or, 24y=6(x^2-y^2) \\ or, x^2-y^2=4y \\ or, x^2=4y+y^2..........\left(i\right) \end{gathered}$$

Again,

$\frac{12}{(2x-y)}-\frac{12}{(2x+y)}=1$

or, $$\begin{gathered} 24x+12y-24x+12y=(4 x^2-y^2) \\ or, 24y=(4 x^2-y^2) \\ or, x^2=\frac{24y+y^2}{24} ........\left(ii\right) \end{gathered}$$

Comparing equations (i) and (ii)

$4y+y^2=\frac{24y+y^2}{24}$

$$\begin{gathered} Or, 16y+4 y^2= 24y+ y^2 \\ Or, 3y^2=8y \\ Or, y=\frac{8}{3}=2\frac{2}{3} mph \end{gathered}$$

The speed of the current is $2\frac{2}{3} MPH$