Question

The time taken to travel 30 km upstream and 44 km downstream is 14 hrs. If the distance covered upstream is doubled and the distance covered downstream is increased by 11 km, then the total time taken is 11 hrs more than earlier. Find the speed of the stream and the speed of the boat. $\left[5\right]$

Answer 1

Let, us assume that the speed of boat in still water is x $\frac{km}{hr}$ and speed of stream is y$\frac{km}{hr}$.

So, the speed of boat upstream will be (x-y)$\frac{km}{hr}$.

Similarly the speed of boat downstream will be (x+y)$\frac{km}{hr}$.

We know that,
($time=\frac{distance}{speed}$)

Using the above formula we can form the equation in two variables.

Taking the first case
$\Rightarrow \frac{30}{x-y}$ + $\frac{44}{x+y}$ = 14

Taking up the second case
$\Rightarrow \frac{60}{x-y}$ + $\frac{55}{x+y}$ = 25 $\left[2mark\right]$

Now we have the equation in two variables but the equations are not linear. So, we will assume $\frac{1}{x-y}$ as u and $\frac{1}{x+y}$ as v.

So substituting u and v in the above two equations we get.

$\Rightarrow$ 30u + 44v = 14

$\Rightarrow$ 60u + 55v = 25

We can solve the above two equations using elimination method.

$\Rightarrow$60u + 88v = 28

$\Rightarrow$60u + 55v = 25

Subtracting the above two equations we get v = $\frac{1}{11}$

Substituting v in any of above two equation we get u = $\frac{1}{3}$ $\left[1mark\right]$

Now as we have assumed

$\frac{1}{x-y}$ = u and $\frac{1}{x+y}$ = v

Putting the values of u and v

We get another linear equation in x,y

$\Rightarrow$ x - y = 3

$\Rightarrow$ x + y = 11

Solving the above two equations we get x = 7, y = 4

So, speed of boat in still water is
7 $\frac{km}{hr}$ and speed of stream is 4 $\frac{km}{hr}$. $\left[2mark\right]$