Question

10. A boat goes 6 km upstream and 57 km downstream in S hours. In 9 hours it can go 21 km upstream and
38 km downstream. Determine the speed of stream and that of boat in still water.​

Answer 1

Given:

A boat goes 6 km upstream and 57 km downstream in 5 hours

In 9 hours the same boat can go 21 km upstream and  38 km downstream

To find:

The speed of stream and speed of the boat in still water.​

Solution:

Let's assume,

"x" km/hr → speed of the boat in still water

"y" km/hr → speed of the stream

∴ Speed of the boat in downstream = (x + y) km/hr

and

∴ Speed of the boat in upstream = (x - y) km/hr

We know,

$\boxed{\bold{Time = \frac{Distance}{Speed} }}$

First condition:

Time taken to travel upstream = $\frac{6}{x - y} \:hr$

Time taken to travel downstream = $\frac{57}{x + y} \:hr$

∴ Total time taken to travel the entire journey ⇒  $\frac{6}{x - y} + \frac{57}{x + y}= 5$ ........ (i)

Second condition:

Time taken to travel upstream = $\frac{21}{x - y} \:hr$

Time taken to travel downstream = $\frac{38}{x + y} \:hr$

∴ Total time taken to travel the entire journey ⇒  $\frac{21}{x - y} + \frac{38}{x + y}= 9$ ........ (ii)

Let $\frac{1}{x - y} = a \: and \:\frac{1}{x+y} = b$

So, equation (i) & (ii) will become,

$6a + 57b = 5$ ..... (iii)

and

$21a + 38b = 9$ ...... (iv)

On multiplying eq. (iii) by 21 and eq. (iv) by 6 and then subtracting both the equations, we get

126a + 1197b = 105

126a + 228b = 54

-        -              -

-----------------------------

       969b = 51

-----------------------------

∴ b = $\frac{51}{969} = \frac{1}{19}$

Substituting the value of b in (iii), we get

$6a + 57 \times \frac{1}{19} = 5$

$\implies 6a = 5 - 3$

$\implies 6a = 2$

$\implies a = \frac{1}{3}$

∴ $\frac{1}{x-y} = a = \frac{1}{3} \Rightarrow x - y = 3 \:........\: (v)$

∴ $\frac{1}{x+y} = b = \frac{1}{19} \Rightarrow x + y = 19 \:........\: (vi)$

Now, adding eq. (v) & (vi), we get

x - y = 3

x + y = 19

---------------

2x = 22

----------------

∴ $\bold{x = \frac{22}{2} = 11\: km/hr}$

Substituting the value of x in eq. (v), we get

$11 - y = 3$

$\implies y = 11 - 3$

$\implies \bold{y = 8\:km/hr}$

Thus,

$\boxed{\bold{The \:speed\:of \:boat\:in\:still\:water\:is \:11\:km/hr.}}\\\\\boxed{\bold{The \:speed\:of \:the\:stream\:is \:8\:km/hr.}}$

----------------------------------------------------------------------------------------

Also View:

A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream is the same time. Find the speed of the boat in still water and the speed of the stream?

https://brainly.in/question/15748

A boat can go 20 km upstream and 30 km downstream in 3 hour. It can go 20km downstream and 10 km upstream in 1 × 2/3 hours find speed of boat in still water and speed of stream?

https://brainly.in/question/18019235

The speed of the boat in still water is 15km/h.it takes 4 1/2 hours to travel 30 km down the stream and back to the starting point. Find the speed of the stream.

https://brainly.in/question/6752289