Question

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 $24$ mile round trip the downstream $12$ miles would then take only one hour less than the upstream $12$ miles. What is the speed of the current in miles per hour?

• A. $1\frac{1}{3}$
• B. $1\frac{2}{3}$
• C. $2\frac{1}{3}$
• D. $2\frac{2}{3}$

Answer 1

The correct option is D $2\frac{2}{3}$

Let the speed of the boat in still water be $x$ mph and speed of the current be $y$ mph.

Then, speed upstream $=(x-y)$ mph and stream downstream $=(x+y)$ mph
Given, $\frac{12}{x-y}-\frac{12}{x+y}=6$
$\Rightarrow \frac{12(x+y)-12(x-y)}{x^2-y^2}=6$
$\Rightarrow 24y=6(x^2-y^2)\Rightarrow x^2-y^2=4y$
$\Rightarrow x^2=4y+y^2$ ...(i)
and $\frac{12}{(2x-y)}-\frac{12}{(2x+y)}=1$
$\Rightarrow \frac{12(2x+y)-12(2x-y)}{4x^2-y^2}=1$
$\Rightarrow 24y-4x^2-y^2$

$\Rightarrow 4x^2-y^2=24y$
$\Rightarrow x^2=\frac{24y+y^2}{4}$ ......(ii)
From (i) and (ii), $4y+y^2=\frac{24y+y^2}{4}$
$\Rightarrow 16y+4y^2=24y+y^2$
$\Rightarrow 3y^2=8y$

$\Rightarrow y=\frac{8}{3}$
$\therefore$ speed of the current $=$ $\frac{8}{3}$ mph $=2\frac{2}{3}$ mph