Boats and Streams concept of upstream and downstream with solved examples

 Boat and Stream:

Stream means Current and  The speed of the water. 

When we say the speed of a boat it means the speed of the boat in still water.

Upstream: when a boat moves against the direction of the stream it is called upstream.

Downstream: When a boat moves in the direction of the stream it is called downstream.

Important points:

i. The speed of the boat is always greater than steam.

ii. If the speed of the boat is x and the speed of the steam is y. then

Speed of boat in upstream = x-y km/hr

Speed of boat in downstream = x+y km/hr

iii. Speed of boat = Speed of a boat in upstream + speed of a boat in downstream)/2

     Speed of boat = Speed of a boat in downstream - the speed of the boat in upstream )/2

Problems Based on Boat and Stream:

Ex.1 A boatman rows 1 km in 5 minutes, along the stream, and 6 km in 1 hour against the stream. The speed of the stream is?

Sol. Speed of current  = (downstream - upstream)/2

(12-6)/2 = 3 km/hr

Speed of boat along the stream

The boat travels 1 km in 5 minutes therefore it travels 12 km in 60 minutes.

Ex.2 A boat covers 12 km upstream and 18 km downstream in 3 hours while it covers 36 km upstream and 24 km downstream in 6.5 hours. what is the speed of the current?

Sol.Speed of the boat in still water = x km/hr

Speed of the current = y km/hr

Time taken by the boat in upstream = distance/speed = 12/(x-y)

Time taken by the boat downstream = 18/(x+y)

3 =  12/(x-y)  + 18/(x+y) ..... i

Similarily,

13/2 =  36/(x-y) + 24/(x+y) ..... ii

Multiply equation I by 3 and subtract by ii

54/(x+y) - 24/(x-y) = 9- 13/2

30/(x+y)  = 5/2

x+y = 12 = Speed of boat downstream

Put x+y = 12 in equation i

12/(x-y) + 18/12  = 3

x-y = 8 km/hr =  speed of boat in upstream

speed of current = (12-8)/2 = 2 km/hr

Ex.3 A swimmer swims from a point A against a current for 5 minutes and then swims backward in favor of the current for the next 5 minutes and comes to point B. if AB is 100 meters, the speed of the current is km/hr.

Sol. Let distance covered in upstream = d

Total distance from A to C to B  = 100+d

Speed of swimmer in upstream = x-y m/min

Speed of swimmer in downstream = x+y m/min

Time is taken by the swimmer in upstream = 5 = d/(x-y)

d = 5(x-y) ....... i 

Again, Downstream = (100+d)/(x+y) = 5 ......... ii

Put the value of d in equation ii.

[100+5(x-y)]/(x+y) = 5

100 + 5x-5y = 5x+5y

10y = 100

y = 10 m/min =  6 km/hr

Ex.4 A boat 12 km downstream and come back to the starting point in 3 hours. If the speed of the current is 3 km/hr.  Then the speed in km/hr of the boat in still water is?

Sol. Let the speed of the boat in still water x km/hr

Total time

12/(x+3) + 12/(x-3) = 3

12[x-3+x+3/(x+3)(x-3)]

4*2x = x2 -9

x2- 8x- 9 = 0

x(x-9) + 1(x-9) = 0

x = 9  x =3 

Ex.5 A man swims downstream a distance of 15 km/hr. If the speed of the current is 5 km/hr. The time taken by the man to swim the same distance upstream is?

Sol. Let the speed of the boat in still water = x

Speed of current = y

Speed of Boat upstream = x-y

Speed of Boat downstream = x+y

Distance = Speed * Time

(x-y) * 2t = (x+ y) * t 

2x - 2y = x + y

x/y = 3/1 

3:1 Ans.

Ex.6 In a fixed time, a boy swims double the distance along with the current. If the speed of the current is 3 km/hr. The speed of the boy in still water?

Sol. Let the rate of swimming in still water = x km/hr.

Rate of swims in downstream = x+3 km/hr

Rate of swims in upstream = x-3 km/hr

According to the questions

D = speed * time

(x+3)t = (x-3)2t

x= 9 km/hr

Ex.7 The Speed of a boat in still water is 6 km/hr and the Speed of the stream is 1.5 km/hr. A man rows to a place at a distance of 22.5 km and comes back to the starting point. The total time is taken by himself.

Sol. Rate downstream = 6+1.5 = 7.5 km/hr

Rate upstream = 6-1.5 = 4.5 km/hr

Time = distance/ speed

Required time = 22.5/7.5  + 22.5/4.5 = 3+5 = 8 hours

Ex.8 A motorboat covers a certain distance downstream in a river in 3 hours. It covers the same distance upstream in 3 hours and a half. If the speed of water is 1.5 km/hr. The speed of a boat in still water is?

Sol. Let the speed of the boat in still water = x

distance = d

Rate Upstream = x - 1.5 km/hr

Rate Downstream = x+1.5 km/hr

d/(x+1.5) = 3

d(x-1.5) = 7/2

 On diving equations we get.

(x-1.5)/(x+1.5) = 6x+9

7x - 10.5 = 6x + 9

x = 19.5 km/hr

Ex.9 Two boats A and B towards each other from two places, 108 km apart speed of the boat A and B in still water are 12 km/hr and 15 km/hr respectively. If A proceeds down and B up the stream, they will meet them after?

Sol. Let the speed of the stream is x km/hr 

and boats meet after t hours.

According to the questions.

(12+x)t + (15-x)t = 108

27t = 108

t = 108/27 = 4 hours.

Ex.10 Speed of motorboat in still water is 45 km/hr. If the motorboat travels 80 km along the stream in 1 hour 20 minutes, then the time taken to cover the same distance against the stream will be.

Sol Let the Speed of current = x km/hr

Speed of boat downstream = x+45 km/hr

According to the questions

8/(x+45) = 1 hours 20 minutes = 4/3 hours

4x+180 = 240

x = 15 km/hr

Rate Upstream = 45-15 = 30 km/hr

Required time = 80/30 = 8/3 = 2 hour 40 minutes

Ex.11The speed of a boat along the stream is 12 km/hr and against the stream is 8 km/hr. The time taken by the boat to sail 24 km in still water is.

Sol. Speed of boat in still water is x km/hr

Speed of current = y km/hr

x+y=12

x-y=8

2x=20

x=10

Required time = 24/10 = 2.4 hours