Question

A boat goes 30km upstream and 20 km downstream in 7 hours.in 6 hours it can go 18km upstream and 30 km downstream.determine the speed of the stream and that of the boat in still water

Answer 1

let the speed of boat be X km/h
let the speed of stream be y km/h
speed of upstream=X-Y
speed of downstream=X+Y
A.T.Q,
30/X-Y + 20/X+Y = 7
18/X-Y + 30/X+Y = 6
Let 1/X-Y be a and 1/X+Y be b 
So, 30a + 20b = 7               -1
18a + 30b =6                     -2
Mutiplying 1 by 3 and 2 by 5
so , 90a + 60b = 21
90a + 150b = 30
so , b= 1/10 =1/X+Y
a = 1/6 = 1/X-Y
SO X+Y = 10
X-Y = 6
SO X=8 km/h
Y=2 km/h

Answer 2

Given:

The upstream distance in 7 hours = 30km

The downstream distance in 7 hours = 20km

The upstream distance in 6 hours = 18km

The downstream distance in 6 hours = 30km

To find:

The speed of the stream and that of the boat in still water

Solution:

Let the speed of the boat in still water be x km per hour. and the speed of the stream be y km per hour.

We know that during the upstream journey, the velocity of the boat is x-y and during the downstream journey the velocity is x+y.

 

Case I :

The boat goes 30 km upstream and 20 km downstream in 7 hours.

Since, Time = distance/speed

⇒  $\frac{30}{x-y} + \frac{20}{x+y} = 7$

Similarly,

Case II:

In 6 hours, it can go 18 km upstream and 30 km downstream.

⇒$\frac{18}{x-y} +\frac{30}{x+y} = 6$

Let us assume that 1/(x - y) = M

1/(x + y) = N

Substituting in the above equations,

30M + 20N = 7

or M = (7 - 20N)/30     - (1)

Also, 18M + 30N = 6    - (2)

Solving equations 1 and 2,

18[(7 - 20N)/30] + 30N = 6  

⇒ (126 - 360N)/30 + 30N = 6

or 126 - 360N + 900N = 180  

or N = 54/540

= 1/10

Substituting the value of N in equation (1) we get :

M = (7 - 20 × 1/10)/30

or M = (7 - 2)/30 = 5/30

= 1/6

Now, M = 1/(x - y) = 1/6

⇒ x - y = 6      - (3)

Also, N = 1/(x + y) = 1/10

or x + y = 10      -(4)

Solving equations (3) and (4) we get,

y = 2

Substituting the value of y in equation 3

x = 6 + 2

or  x= 8

Hence, the speed of boat in still water is 8 km/hr and the speed of the stream is 2 km/hr.