## Boats and Streams

"Boats and streams" is an application of concepts of speed, time and distance. Speed of river either aides a swimmer/boat, while traveling with the direction of river or it apposes when traveling against the direction of river.

Still Water: If the speed of water of a river is zero, then water is considered to be still water.

Stream Water: If the water of a river is moving at a certain speed, then it is called as stream water.

Speed of Boat: Speed of boat means speed of boat in still water

Upstream: If a boat or a swimmer moves in the opposite direction of the stream, then it is called upstream.

Downstream: If a boat or a swimmer moves in the same direction of the stream, them it is called downstream.

#### Basic Rule of Boats and Streams

Rule 1: If the speed of a boat or swimmer in still water is x km/h and speed of the stream is y km/h, then

 Speed of boat or swimmer in downstream = ( x + y ) km/h

Example: A man can row with a speed of 8 km/h in still water. If the speed of stream is 3 km/h, what will be his speed with the stream.
Solution: Given, speed of man in still water = 8 km/h
speed of stream = 3 km/h
∴ downstream speed = ( x + y )
= 8 + 3
= 11 km/h

Rule 2: If the speed of a boat or swimmer in still water is x km/h and speed of the stream is y km/h, then

 Speed of boat or swimmer in upstream = ( x - y ) km/h

Example: If the speed of a boat in still water is 5 km/h and the rate of stream is 2 km/h, then find upstream speed of the boat.
Solution: Given, speed of boat = 5 km/h
speed of stream = 2 km/h
∴ upstream speed = x - y
= 5 - 2
= 3 km/h

Rule 3: If the speed of a boat in downstream is x km/h and speed of the boat in upstream is y km/h, then

 Speed of boat in still water = 1 (speed in downstream + speed in upstream) 2

Example: A man can row a boat in downstream at 14 km/h and in upstream at 10 km/h. Find the speed of the boat that the man can row in still water.
Solution: Given, speed in upstream = 10 km/h
speed in downstream = 14 km/h

 speed of boat in still water = 1 (speed in downstream + speed in upstream) 2
 speed of boat in still water = 1 (14 + 10) 2
 speed of boat in still water = 1 (24) 2

= 12 km/h

Rule 4: If the speed of a boat in downstream is x km/h and speed of the boat in upstream is y km/h, then

 Speed of stream = 1 (speed in downstream - speed in upstream) 2

Example: Ram can row upstream at 11 km/h and downstream at 17 km/h. Find the rate of the current.
Solution:- Given, speed in upstream = 11 km/h
Speed in downstream = 17 km/h

 Speed of stream = 1 (speed in downstream - speed in upstream) 2
 Speed of stream = 1 (17 - 11) 2
 Speed of stream = 1 (6) 2

= 3 km/h

Example: A boat covers 56 km in 8 hours in downstream motion. And return to the same place in 14 hours. Find the speed of boat and Stream.
Solution:

 speed = distance time

 speed in downstream = 56 = 7 km/h 8
 speed in upstream = 56 = 4 km/h 14

 speed of boat = 1 (speed in downstream + speed in upstream) 2

 = 1 (7 + 4) 2
 = 1 × 11 2

= 5.5 km/h
 Speed of stream = 1 (speed in downstream - speed in upstream) 2
 Speed of stream = 1 (7 - 4) 2
 Speed of stream = 1 × 3 2

= 1.5 km/h

Rule 5: If speed of stream is x and a boat takes t times longer to row up as compared to row down the river, then

 Speed of boat in still water = x (t + 1) (t - 1)

Example: A can row 10 km/h in still water. It takes his thrice as long as to row up as to row down the river. Find the speed of stream.
Solution: Let the rate of current be y km/h.
Speed of A in still water = 10 km/h
Speed in downstream = ( 10 + y ) km/h
Speed in upstream = ( 10 - y ) km/h

 Time = Distance Speed
 t = d …..(1) 10 + y
 3t = d …..(2) 10 - y

Putting the value of t from Eq.(1) in Eq.(2), we get
 Speed of boat in still water = x (t + 1) (t - 1)
 3 × d + y = d - y 10 10

or, 3 ( 10 - y ) = 10 + y
or, 30 - 3y = 10 + y
or, 30 - 10 = 3y + y
or, 20 = 4y
∴ y = 20/4 = 5
∴ Speed of stream = 5 km/h
By Rule ,
Here, Speed of A = 10 km/h and t = 3
 Speed of boat in still water = x (t + 1) (t - 1)
 or, 10 = x (3 + 1) (3 - 1)
 or, 10 = x × 4 2

or, 10 = x × 2
∴ x = 10/2 = 5 km/h
= speed of stream

Rule 6: A man rows a certain distance downstream in t1 hour and returns the same distance in t2 hour. When the stream flows at the rate of y km/h, then

 Speed of the man in still water = y (t1 + t2) (t2 - t1)

Example: Rohan can row a certain distance downstream in 6 h and can return the same distance in 8 h. If the stream flows at the rate of 4 km/h, then find the speed of Rohan in still water.
Solution: Let the speed of Rohan in still water be x km/h.
Speed of stream = 4 km/h
Speed in downstream = ( x + 4 ) km/h
Speed in upstream = ( x - 4 ) km/h
According to the question,
Distance traveled in upstream = Distance traveled in upstream
Distance = Speed × Time ( x - 4 ) × 8 = ( x + 4 ) × 6
or, 8x - 32 = 6x + 24
or, 8x - 6x = 32 + 24
or, 2x = 8 ∴ x = 56/2 = 28
∴ speed of Rohan in still water = 28 km/h
By Rule ,

 Speed of the man in still water = y (t1 + t2) (t2 - t1)
 = 4 (6 + 8) (8 - 6)
 = 4 × (14) (2)

= 4 × 7
= 28 km/h

Rule 7: When boat or swimmer's speed in still water is x km/h and river is flowing with a speed of y km/h and time taken to cover a certain distance upstream is t more than the time taken to cover the same distance downstream, then

 Distance = t (x² - y²) 2y

Example: A boat speed in still water is 8 km/h, while river is flowing with a speed of 4 km/h and time taken to cover a certain distance upstream is 5 hours more than time taken to cover the same distance downstream. Find the distance.

Solution: Let the distance be D km
given, speed of boat in still water = 8 km/h
Speed of water = 4 km/h
Speed in downstream = 8 + 4 = 12 km/h
Speed in upstream = 8 - 4 = 4 km/h
According to the question,
D/4 - D/12 = 5
or, 3D - D /12 = 5
or, 2D/12 = 5
or, 2D = 12 × 5
or, 2D = 60
∴ D = 60/2 = 30 km
By Rule,
Here, x = 8 km/h, y = 4 km/h and t = 5 hours

 Distance = t (x² - y²) (speed in downstream + speed in upstream) 2y

 = 5 (8² - 4²) 2 × 4
 = 5 (64 - 16) 8
 = 5 × 48 8

= 5 × 6
= 30 km

Rule 8: If boat or swimmer's speed in still water is x km/h and river is flowing with a speed of y km/h, then average speed in going to a certain place and coming back to

 starting point is given by ( x + y ) ( x - y ) km/h x

Example: A rows in still water with a speed of 7 km/h to go to a certain place and to come back. Find his average speed for the whole journey, if the river is flowing with a speed of 3 km/h.

Solution: Given, speed of A in still water = 7 km/h
Speed of stream = 3 km/h
Speed in upstream = 7 - 3 = 4 km/h
Speed in downstream = 7 + 3 = 10 km/h
Let the distance in one direction be D km.

 Time taken in upstream = D 4
 Time taken in downstream = D 11

 Average speed = Total distance Total time taken in travel

 = D + D D + D 10 10
 = 2D 10D + 4D 40
 = 2D × 40 14D

= 40/7
= 5.7 km/h
By Rule,
Here, x = 7 km/h and y = 3 km/h
 Average speed = (x + y) (x - y) km/h x
 = (7 + 3)(7 - 3) km/h 7

= 10 × 4 /7
= 40/7
= 5.7 km/h

Rule 9: If a boat covers D km distance in t1 hour along the direction of river and it covers same distance in t2 hour against the direction of river, then

 Speed of boat in still water = D t2 + t1 2 t1.t2

 Speed of the stream = D t2 - t1 2 t1.t2

Example: A boat covers 24 km in 2 hour with downstream and covers the same distance in 4 hour with upstream. Then, find the speed of boat in still water and speed of stream.

Solution: Given, Distance = 24 km

 Speed = distance/time

 Speed in downstream = 24 2

= 12 km/h
Speed in upstream = 24/4 = 6 km/h

 ∴ Speed of boat in still water = 1 (speed in downstream + speed in upstream) 2

 = 1 (12 + 6) 2
 = 1 × 18 2

= 9 km/h

 ∴ Speed of stream = 1 (speed in downstream - speed in upstream) 2

 = 1 (12 - 6) 2
 = 1 × 6 2

= 3 km/h
By Rule,
Here, D = 24 km,
t1 = 2 hour and
t2 = 4 hour
 ∴ Speed of boat in still water = D t2 + t1 2 t1.t2
 = 24 4 + 2 2 2 × 4
 = 12 × 6 8
 = 12 × 3 4

= 9 km/h
 Speed of the stream = D t2 - t1 2 t1.t2
 = 24 4 - 2 2 2 × 4
 = 12 × 2 8

= 3 km/h