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 miles 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?

Answer 1

Hint: Find the speed in the downstream and the speed in the upstream when Rahul is rowing in his usual speed and then apply the condition given in the problem to find the equation of two variables and then find the speed in downstream and the upstream when Rahul doubles his speed and use then to apply the given condition in the problem. Use both the equations to get the desired result.
It is given that Rahul can travel 12 miles downstream at his usual rowing rate in a certain river in
6 hours less than it takes him to travel the same distance upstream. It is also given that if he
doubles his rowing speed, he will take only one hour less time the upstream 12 miles.
We have to find the speed of the current in miles per hour.
Assume that the usual speed of Rahul is $x$mph and the speed of the current is$y$mph.
Then the speed in upstream and speed in downstream is given as:
Upstream Speed$ = \left( {x - y} \right)$mph
Downstream speed$ = \left( {x + y} \right)$mph
Distance$ = 12$ miles
Time taken to travel 12 miles in upstream\[ = \dfrac{{12}}{{x - y}}\]
Time taken to travel 12 miles in downstream\[ = \dfrac{{12}}{{x + y}}\]
Then according to the question,
Time taken to travel in upstream - Time taken to travel in upstream = 6
\[\dfrac{{12}}{{x - y}} - \dfrac{{12}}{{x + y}} = 6\]
\[\dfrac{{12\left( {x + y} \right) - 12\left( {x - y} \right)}}{{\left( {x - y} \right)\left( {x + y}
\right)}} = 6\]
Perform a cross multiplication:
\[12\left( {x + y} \right) - 12\left( {x - y} \right) = 6\left( {{x^2} - {y^2}} \right)\]
Simplify the equation:
\[24y = 6\left( {{x^2} - {y^2}} \right)\]
\[4y = {x^2} - {y^2}\]
\[{x^2} = {y^2} + 4y\] … (1)
Now, it is given that Rahul doubles his speed, then his new speed becomes $2x$mph.
Then the speed in upstream and speed in downstream is given as:
Upstream Speed$ = \left( {2x - y} \right)$mph
Downstream speed$ = \left( {2x + y} \right)$mph
Distance$ = 12$ miles
Time taken to travel 12 miles in upstream\[ = \dfrac{{12}}{{2x - y}}\]
Time taken to travel 12 miles in downstream\[ = \dfrac{{12}}{{2x + y}}\]

Then according to the question,
Time taken to travel in upstream - Time taken to travel in upstream = 1
\[\dfrac{{12}}{{2x - y}} - \dfrac{{12}}{{2x + y}} = 1\]
$ \Rightarrow 12\left( {2x + y} \right) - 12\left( {2x - y} \right) = 4{x^2} - {y^2}$
$ \Rightarrow 24y = 4{x^2} - {y^2}$
$ \Rightarrow 4{x^2} = {y^2} + 24y$ … (2)
Substitute the value ${x^2}$ from the equation (1):
$4\left( {{y^2} + 4y} \right) = {y^2} + 24y$
$4{y^2} + 16y = {y^2} + 24y$
$ \Rightarrow 3{y^2} = 8y$
$ \Rightarrow 3y = 8$
$ \Rightarrow y = \dfrac{8}{3}$
So, the speed of the current is $\dfrac{8}{3}$mph.

Note: Upstream means that we are traveling in the opposite direction of the flow of the river and
the downstream means that we are traveling in the direction of the flow of the river. Therefore,
the speed in the upstream is taken as the difference of the speed of the object with the speed of
flow of the river, and the speed in the downstream is taken as the sum of the speed of the object
and the speed of the flow of the river.