Boats and Streams

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Since you all know that Mathematics is one of the most important part of competitive exams, here we shall discuss Some Quick Study Notes on  "Boats and Streams"We hope that they will help you all in the upcoming competitive exams like  ,SSC CGL  . Hope you all like the post.

QUICK STUDY NOTES ON BOATS AND STREAMS :

1. Still Water:  If the water is not moving then it is called still water.

2. Stream: Moving water of the river is called stream.

3. Downstream: The direction along the stream is called downstream.

4. Upstream: The direction opposite the stream is called upstream.

5. If the speed of a boat in still water is 'u' km/hr and the speed of the stream is 'v' km/hr, then

Speed of boat downstream  = (u + v) km/hr 

Speed of boat upstream  = (u - v) km/hr 

6.If the speed downstream is 'a' km/hr and the speed upstream is 'b' km/hr, then

Speed of boat in still water  =  1/2(a + b) km/hr

Speed of stream  = 1/2(a - b) km/hr 

Shortcut Formula :

Rule - 1: A man can row certain distance downstream in t1 hours and returns the same distance upstream in t2 hours. If the speed of stream is y km/h, then the speed of man in still water is given by

= y{(t2 + t1)/(t2 - t1)} km/hr

Rule - 2: A man can row in still water at x km/h. In a stream flowing at y km/h, if it takes him t hours to row to a place and come back, then the distance between two places is given by

= t(x^2 - y^2)/2x km/hr

Rule- 3: A man can row in still water at x km/h. In a stream flowing at y km/h, if it takes t hours more in upstream than to go downstream for the same distance, then the distance is given by

=  t(x^2 - y^2)/2y km/hr

Rule- 4: A man can row in still water at x km/h. In a stream flowing at y km/h, if he rows the same distance up and down the stream, then his average speed is given by

=  (UpstreamSpeed x DownstreamSpeed)/(Man's speed in still water) km/hr

={(x - y) * (x + y)}/{x} km/hr

= (x^2 - y^2)/x km/hr

1. A motorboat, whose speed in 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is:

A. 4

B. 5

C. 6

D. 10

2. Speed of a boat in standing water is 14 kmph and the speed of the stream is 1.2 kmph. A man rows to a place at a distance of 4864 km and comes back to the starting point. The total time taken by him is:

A. 700 hours

B. 350 hours

C. 1400 hours

D. 1010 hours

3. A boat running upstream takes 8 hours 48 minutes to cover a certain distance, while it takes 4 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current respectively?

A. 2 : 1

B. 3 : 2

C. 8 : 3

D. Cannot be determined

4. An individual can row a ship d km upstream and the identical distance downstream in 5 hours quarter-hour. Additionally, he can row the boat 2d km upstream in 7 hours. How lengthy will it take to row the identical distance 2d km downstream?

A. 3/2 hours 

B. 7 hours

C. 7 (¼)

 4. 7/2 hours

5.  A man takes twice as long to row a distance against the stream as to row the same distance in favour of the stream. The ratio of the speed of the boat (in still water) and the stream is:

A. 3 : 1

B. 1 : 3

C. 1 : 2

D. 2 : 1

6. A man can row at 5 kmph in still water. If the velocity of current is 1 kmph and it takes him 1 hour to row to a place and come back, how far is the place?

A. 2.4 km

B. 2.5 km

C. 3 km

D. 3.6 km

7. A man can row three-quarters of a kilometre against the stream in 11 1/4 minutes and down the stream in 7 1/2minutes. The speed (in km/hr) of the man in still water is:

A. 4 kmph

B. 5 kmph

C. 6 kmph

D. 8 kmph

8. A boatman goes 2 km against the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 5 km in stationary water?

A. 40 minutes

B. 1 hour

C. 1 hr 15 min

D. 1 hr 30 min

9. Speed of a boat in standing water is 9 kmph and the speed of the stream is 1.5 kmph. A man rows to a place at a distance of 105 km and comes back to the starting point. The total time taken by him is:

A. 16 hours

B. 18 hours

C. 20 hours

D. 24 hours

10. A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 mph, the speed of the stream is:

A. 4 mph

B. 2.5 mph

C. 3 mph

D. 2 mph

ANSWERS AND SOLUTION :
1(B)Explanation:
Let the speed of the stream be x km/hr. Then,
Speed downstream = (15 + x) km/hr,
Speed upstream = (15 - x) km/hr.
So 30/(15 + x) + 30/(15 - x) = 4 1/2
=> 900/(225 - x^2) =  9/2
=> 9x^2 = 225
=>x^2 = 25
=> x = 5 km/hr

2(A)Explanation :
Speed downstream = (14 + 1.2) = 15.2 kmph
Speed upstream = (14 - 1.2) = 12.8 kmph
Total time taken = 4864/15.2 + 4864/12.8 = 320 + 380 = 700 hours

3(C)Explanation:
Let the man's rate upstream be x kmph and that downstream be y kmph.
Then, distance covered upstream in 8 hrs 48 min = Distance covered downstream in 4 hrs.
= ( x x 8 4/5) = ( y x 4)
= 44/5 x = 4y
=> y = 11/5 x
So,  Required ratio = (y + x)/2 : (y - x)/2
= (16x/5 x 1/2) : (6x/5 x 1/2)
= 8/5 : 3/5
= 8 : 3

4(D)Explanation :
Let the speeds of boat and stream be x and y km/hr respectively.
Then, Fee downstream = (x + y) km/hr and Fee upstream = (x – y) km/hr
Given d / (x+y) + d / ( x-y ) =5 hrs quarter-hour = 21/4 hours
and 2d / (x+y) = 7 => d / (x+y)=7/4 => 2d / (x+y)=7/2
Therefore, he takes 7/2 hours to row 2d km distance downstream.

5(A)Explanation :
Let speed upstream = x
Then, speed downstream = 2x
Speed in still water = (2x+x)/2 = 3x/2
Speed of the stream = (2x-x)/2 = x/2
Speed in still water : Speed of the stream = 3x/2 : x/2 = 3 : 1

6(A)Explanation:
Speed downstream = (5 + 1) kmph = 6 kmph.
Speed upstream = (5 - 1) kmph = 4 kmph.
Let the required distance be x km.
Then, x/6 + x/4 = 1
=>  2x + 3x = 12
=>  5x = 12
=> x = 2.4 km.

7(B)Explanation :
Distance = 3/4 km
 Time taken to travel upstream = 11 1/4 minutes = 45/4 minutes
 = 45/(4×60)hours = 3/16 hours
Speed upstream = Distance/Time=(3/4)/(3/16) = 4 km/hr
Time taken to travel downstream = 7 1/2 minutes = 15/2 minutes
= 15/(2×60) hours = 1/8 hours
Speed downstream = Distance/Time=(3/4)/(1/8)= 6 km/hr
Rate in still water = (6+4)/2=10/2=5 kmph

8(C)Explanation:
Rate downstream = (1/10 x 60 )km/hr = 6 km/hr.
Rate upstream = 2 km/hr.
Speed in still water = 1/2(6 + 2) km/hr = 4 km/hr.
So  Required time =  (5/4) hrs = 1 1/4hrs = 1 hr 15 min.

9(D)Explanation:
Speed upstream = 7.5 kmph.
Speed downstream = 10.5 kmph.
Total time taken = 105/7.5 + 105/10.5) hours = 24 hours.

10(D)Explanation :
Speed of the boat in still water = 10 mph
Let speed of the stream be x mph
Then, speed downstream = (10+x) mph
speed upstream = (10-x) mph
Time taken to travel 36 miles upstream - Time taken to travel 36 miles downstream = 90/60 hours
=>36/(10-x) - 36/(10+x) = 3/2
=>12/(10-x) - 12(10+x) = 1/2
=>24(10+x)-24(10-x) = (10+x)(10-x)
=>240 + 24x - 240 + 24x=(100-x^2)
=>48x = 100 - x^2
=>x^2 + 48x -100 = 0
=>(x+50)(x-2)=0
=>x =  - 50 or 2
Since x can not be negative, x = 2 mph

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