__Boats and Streams__

__ ____Important Facts and Formula__

- The speed of water or stream that denoted by v km/hr.
- Speed of boat or boatman in calm water which we denoted by u km/hr.
**Speed downstream****=**(u + v) km / hr**Speed upstream****=**(u – v) km / hr.- In water, the direction along with stream is called Downstream.
- The direction of boat against the stream is called Upstream.

If the **speed downstream is x km / hr** and the **speed upstream is y km / hr,**** **then

**Speed in still water****=**1 / 2 ( x + y ) km / hr**Rate of Stream****=**1 / 2 ( x – y ) km / hr .

__Example:__

A boy can row upstream at 6 km/hr and downstream at 12 km/hr. Find boy’s rate in still water and the rate of current?

__Answer:__

We Know the formula of

Rate in still water that is = 1 / 2 ( x + y )km /hr

So we applied formula 1 / 2 ( 12 + 6 ) = 1 / 2 X 18 = 9 km/ hr. and

we also know the formula of

Rate of current that is = 1 / 2 ( x – y )km /hr

So we applied formula of 1 / 2 (12 – 6) = 3 km / hr.

__Example:__** **A boy can row downstream at 24 km and upstream 16 km . If he has 8 hours to cover each distance, then what is the velocity of the current?

__Answer:__

The rate of downstream = 24 / 8 km/hr ,

The rate of upstream = 16 / 8 km /hr .

So, the velocity of the current is 1 / 2 ( 24 / 8 – 16 / 8 ) km /hr = 8 / 8 = 1 km /hr

__Example:__

A man can go 40 km/hr upstream 36 km/hr downstream . Find the speed of current & speed of man in still water?

__Answer:__

So , Speed of current Y is

= U – V / 2

= 40 – 36 / 2 = 4 / 2

= 2 km/hr .

So, Speed of man in still water x is

= U + V / 2

= 40 + 36 / 2

= 38 km/hr

__Example:__

A small ship covers a certain distance downstream in 1 hour ,when it comes back in 3 /2 hours . If the speed of the stream be 4 km/hr,

what is the speed of the boat in still water ?

__Answer:__

Suppose the speed of the ship in still water be x km/hr , Then

speed of downstream = ( x + 4 ) km /hr .

Speed of upstream = ( x – 4 ) km /hr .

So , ( x + 4 ) x 1 = ( x – 4 ) x 3 / 2

2x + 8 = 3x – 12

= 3x – 2x = -12 – 8

= x = 20 km /hr .

__Example:__

If a boat goes 7km upstream in 21km and the stream is 5 kmph, then the speed of the boat?

__Answer:__

Rate of stream = d / t = 7 x 60 / 21 = 20 kmph.

Let speed in still water be x km / hr.

Then, speed upstream = (x – 5) km /hr.

So, (x – 5) = (20 – 5) = 15 km /hr.

__Example:____
__A boat can travel with a speed of 14 km / hr in still water. If the speed of the stream is 4 km/ hr, find the time taken to go 72 km downstream.

__Answer:__Speed of downstream (14 + 4 ) = 18 km/ hr

Time taken to travel 72 km downstream

= 72 / 18 = 4 hrs

So time taken to go downstream is 4 hrs.

__Example:__

A man can row upstream at 10 km /hr And downstream at 18 kmph. what is the speed of the stream ?

__Answer:__

Speed of stream = 1 / 2(a – b )

1 / 2 (18 – 10) kmph = 4 kmph

__Example:__

Kapil can row a certain distance downstream in 8 hours and upstream the distance in 10 hours , If the stream flows rate at of 4 km / hour then , Find the speed of Kapil in still water .

__Answer:__

Speed of Kapil in still water be z km /hr

downstream speed = ( u – v)

upstream speed = ( u + v)

__Shortcut trick:__

Speed of Kapil in still water is =

rate ( upstream speed + downstream speed / upstream speed – downstream speed )

4 ( 10 + 8 ) / 10 – 8 = 36 km / hr .