Question

Rekha can row her boat at the speed of $5$ km/hour in still water. If it takes one hour more to row the boat $5.25$ km upstream than to return downstream. Find the speed of the stream.
A. 9 km/hour
B. 5 km/hour
C. 2 km/hour
D. None of these

Answer 1

Hint: Assume the speed of the stream as $xkm/hr$. Then the speed of the boat becomes $5-x$ in upstream and the speed of the boat becomes $5+x$ in the downstream. Then, use the formula Time=distance/speed. Solve the formulated equations to get the value of x.

Complete step-by-step answer:

Before proceeding with the question, we must know that upstream means that the boat is moving opposite to the direction of flow of the water and downstream means that the boat is moving along the direction of flow of the water. Therefore, the speed of the boat increases in downstream and speed of boat decreases in upstream. We must know the formula Time=distance/speed.

In this question, we have been given that Rekha can row her boat at the speed of $5km/hr$ in still water. If the boat takes one hour more to row the boat $5.25km/hr$ upstream than to return downstream, then we have to find out the speed of the stream.

Let us assume the speed of the stream as \[\left( x \right)km/hr\]. Therefore the speed of the stream in upstream becomes $(5-x)km/hr$ and the speed of the stream becomes $(5+x)km/hr$ in the downstream.

We know that the distance covered by the boat is $5.25km$. We also know that the formula for time is given by Time=distance/speed.

Therefore, we can write that the time taken by the boat while going up-stream is $=\dfrac{5.25}{5-x}hr$ and the time taken by the boat while going down-stream is $=\dfrac{5.25}{5+x}hr$.

In the question, it has been given that while going upstream it takes one hour more than going downstream.

Therefore, the equation can be framed as $\dfrac{5.25}{5-x}-\dfrac{5.25}{5+x}=1$.

Now, taking the L.C.M of the denominators and simplifying the equation, we get,

$\Rightarrow \dfrac{5.25\left( 5+x \right)-5.25\left( 5-x \right)}{\left( 5-x \right)\left( 5+x \right)}=1$

Taking out $5.25$ common we get,

$\Rightarrow \dfrac{5.25\left( 5+x-5+x \right)}{\left( 5+x \right)\left( 5-x \right)}=1$

Simplifying the terms inside the bracket, we get,

$\Rightarrow \dfrac{5.25\left( 2x \right)}{\left( 5+x \right)\left( 5-x \right)}=1$

By cross multiplication we get,

$\Rightarrow 10.5x=\left( 5+x \right)\left( 5-x \right)$

$\Rightarrow 10.5x=25-{{x}^{2}}$

$\Rightarrow {{x}^{2}}+10.5x-25=0$

Using the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ we can find the value of $x$. Here, we have$a=1,b=10.5,c=-25$. Therefore, substituting these in the formula, we get,

$\Rightarrow x=\dfrac{-10.5\pm \sqrt{{{\left( 10.5 \right)}^{2}}-4\times 1\times \left( -25 \right)}}{2\times 1}$

\[\Rightarrow x=\dfrac{-10.5\pm \sqrt{110.25+100}}{2}\]

$\Rightarrow x=\dfrac{-10.5\pm 14.5}{2}$

Here, we have to take $14.5$ and not the negative value as $x$ is the speed and speed cannot be negative.

$\Rightarrow x=\dfrac{-10.5+14.5}{2}$

$\therefore x=2$

Hence, we have got the speed of the stream as $2km/hr$. So, the correct answer is option C.


Note: We have to be careful while getting the values of speed, as negative values are not acceptable. Always remember that in case of upstream, the speed of boat decreases, while in downstream, the speed of boat increases. The mistake that is commonly made is by taking the downstream speed as (x-5) km/hr and the upstream speed as (x+5) km/hr.