Question

At his usual rowing 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 in still water be $x$ mph and the speed of the current be $y$ mph.

Speed of upstream $=(x-y)$

Speed of downstream $=(x+y)$

$\frac{12}{x-y}-\frac{12}{x+y}=6$

$\Rightarrow$ $\frac{12x+12y-12x+12y}{x^2-y^2}=6$

$\Rightarrow$ $24y=6(x^2-y^2)$

$\Rightarrow$ $x^2-y^2=4y$

$\Rightarrow$ $x^2=4y+y^2$ ------ ( 1 )

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

$\Rightarrow$ $24x+12y-24x+12y=1(4x^2-y^2)$

$\Rightarrow$ $24y=(4x^2-y^2)$

$\Rightarrow$ $x^2=\frac{24y+y^2}{4}$ ------ ( 2 )

Now, comparing ( 1 ) and ( 2 ) we get,

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

$\Rightarrow$ $16y+4y^2=24y+y^2$

$\Rightarrow$ $3y^2=8y$

$\therefore$ $y=\frac{8}{3}$

The speed of the current is $=\frac{8}{3}$ mph $=2\frac{2}{3}$ mph