Chapter 20

Rule 1 Theorem: If the speed of the boat (or the swimmer) isxkm/ hr and if the speed of the stream is y km/hr then, while upstream the effective speed of the boat = (x-y) km/hr.

Illustrative Example Ex.: Speed of a man is 8 km/hr in still water. I f the rate of

current is 3 km/hr, find the effective speed of the man upstream.

Soln: Applying the above theorem, we have effective speed of the man upstream

= (8 -3 = 5) km/hr Note: Normally by speed of the boat or swimmer we mean

the speed of the boat (or swimmer) in still water.

Exercise 1. The speed of a boat in still water is 2 km/hr. I f its speed

upstream be 1 km/hr, then speed of the stream is [Asst. Grade Exam, 1997]

a)2km/hr b)3km/hr c) 1 km/hr d) None of these

2 A boat goes 14 km upstream in 56 minutes. The speed of stream is 2 km/hr. The speed of boat in still water is a)6km/hr b)15km/h> c)14km/hr d)17km/hr

3. Speed of a man is 7 km/hr in still water. I f the rate of current is 2 km/hr, find the effective speed of the man upstream. a)9km/hr b)5 km/hr c) 14 km/hr . d) Data inadequate

1 Speed of a man is 9 km/hr in still water. I f the rate of current is 5 km/hr, find the effective speed of the man upstream.

a)14km/hr b)12km/hr c)4km/hr d)5km/hr

Answers : c ; H i n t : 2 - y = l .-. y = 2 - 1 = 1 km/hr

14x60 - d; Hint: Rate upstream = = 15 km/hr

56 (x-2)=15 .-. x=17km/hr

3 b 4.c

Streams

Rule 2 Theorem: If the speed of the boat (or the swimmer) isx km/ hr and if the speed of the stream is y km/hr then, while downstream the effective speed of the boat = (x+y) km/hr.

Illustrative Example Ex.: Speed of a swimmer is 8 km/hr in still water. I f the rate

of stream is 3 km/hr, find the effective speed of the swimmer downstream.

Soln: Applying the above theorem, we have effective speed of the swimmer downstream

= (8 + 3)= 11 km/hr.

Exercise 1. The speed of a boat in still water is 10 km/hr. I f its speed

downstream be 13 km/hr, then speed of the stream is: a)1.5km/hr b)3km/hr c) 11.5km/hr d)5.75km/hr

2. The rowing speed of man in still water is 20 km/hr. Going downstream, he moves at the rate of 25 km/hr. The rate of stream is a) 45 km/hr b) 2.5 km/hr c) 12.5 km/hr d) 5 km/hr

3. I f a man goes upstream at 6 km/hr and the rate of stream is 2 km/hr, then the man's speed in still water is a)4km/hr b)8km/hr c)2km/hr d)12km/hr

4. A boat goes 12 kms upstream in 48 minutes. The speed of stream is 2 km/hr. The speed of boat in still water is a) 13 km/hr b)2.25km/hr c) 17km/hr d)15km/hr

Answers 1. b;Hint: 10 + y = 13 .-. y = 13-10 = 3km/hr 2. d 3 . b

12x60 4.c;Hint: =x-2 .-. x=15 + 2 = 17km/hr

4o

Rule 3 Theorem: Ifx km per hour be the man's rate in still water,

1 then x = (man's rate with current + his rate against current) ie "A man's rate in still water is half the sum of his rates with and against the current."

4 8 8 P R A C T I C E B O O K ON Q U I C K E R MATHS

Illustrative Example Ex.: A man can row upstream at 10 km/hr and downstream

at 16 km/hr. Find the man's rate in still water. Soln: Detail Method: Let the speed of the man in still water

be x km/hr and speed of the stream be y km/r. According to the question, x+y= 16.... ( i ) andx-y = 10 (ii) Adding eqn (i) with eqn (ii), we have 2x = 26 :.x= 13 km/hr .-. Speed of the man in still water = 13 km/hr. Quicker Method: Applying the above theorem, we have

Rate of man in still water = 1 (l 6 +10) = 13 km/hr. 2

Exercise 1. A man can row downstream at the rate of 14 km/hr and

upstream at 5 km/hr. Find man's rate in still water. a)9.5km/hr b)8km/hr c)8.5km/hr d)9km/hr

2. A man can row downstream at the rate of 16 km/hr and upstream at 11 km/hr. Find man's rate in still water. a) 14 km/hr b) 13.5 km/hr c) 14.5 km/hr d) 15.5 km/hr

3. The speed of a boat in still water is 12 km per hour. Going downstream it moves at the rate of 19 km per hour. The speed of the boat against the stream is km/hr. a) 5 km/hr b) 3 km/hr c) 8 km/hr d) Data inadequate

4. A man can row 15 km downstream in 3 hours and 5 km

upstream in 2 hours. His speed in still water is

km/hr. a)4km/hr b)4.5 km/hr c)3.5 km/hr d) Data inadequate

5. A man can row with the stream at 10 km/hr and against the stream at 5 km/hr. Man's rate in still water is a) 5 km/hr b) 2.5 km/hr c) 7.5 km/hr d) 15 km/hr

6. A boat goes 40 km upstream in 8 hours and a distance of 36 km downstream in 6 hours. The speed of the boat in standing water is a) 6.5 km/hr b) 6 km/hr c) 5.5 km/hr d) 5 km/hr

7. I f a man rows at the rate of 5 km/hr in still water and his rate against the current is 3.5 km/hr, then the man's rate along the current is a) 8.5 km/hr b) 6.5 km/hr c) 6 km/hr d) 4.25 km/hr

8. A man can row 44 km downstream in 4 hours. I f the man's rowing rate in still water is 8 km/hr, then find in what time will he cover 25 km upstream? a) 5 hours b) 6 hours c) 4.5 hours d) 4 hours

9. A man can row his boat with the stream at 6 km/hr and against the stream at 4 km/hr. The man's rate is a) 1 km/hr b)5km/hr c)8km/hr d)6km/hr

10. A man rows 40 km upstream in 8 hours and a distance of

36 km downstream in 6 hours, then the speed of man in still water is a) 0.5 km/hr b) 5.5 km/hr c) 6 km/hr d) 5 km/hr

11. I f a man's downstream rate is 10 km/hr, and the rate of stream is 1.5 km/hr, then the man's upstream rate is a) 13 km/hr b) 10 km/hr c) 3 km/hr d) 7 km/hr

12. I f a man rows at 8 km/hr in still water and his upstream rate is 5 km/hr, then the man's rate along the current (downstream) is a)21km/hr b)8km/hr c)16km/hr d) l lkm/hr

Answers L a 2.b

3. a; Hint: (x + 1 9 ) - =12 2

\ x = 24--19 = 5 km/hr

4. c 5.c 6.c

7. b; Hint: (x +3 .5 )1 =5 . x = 1 0 --3.5 = 6.5 km/hr

8. a; Hint: Man's rate in still water = [man's rate with

current + his rate against current]

1[44 25" o r ' 8 = 2 | T + T .-. t = 5 hours.

9. b 10. b

11. d; Hint: Rate of stream = (downstream rate - up-

stream rate)

or, 1.5= l ( l 0 - x ) .-. x = 7km/hr

12. d; Hint: Man's rate in still water = (downstream rate

+ upstream rate)

or,8= -(S+x)*U km/hr

Rule 4 Theorem: Ifx km per hour be the rate ofthe current, theny

1 = (man's rate with current -his rate against current) ie

"The rate of the current is half the difference between the rate of the man with and against the current"

Illustrative Example Ex.: A man can row upstream at 10 km/hr and downstream

at 16 km/hr. Find the rate of the current.

Streams 489

Soln: Detail Method: Let the speed of the man in still water be x km/hr and the rate of the current be y km/hr According to the question, Effective speed of man downstream

= x + y = 16 km/hr.... (i) Effective speed of man upstream

= xy 10 km/hr.... (ii) Subtracting eqn (ii) from eqn (i), we have 2y = 6 km/hr or, y = 3 km/hr .-. Speed of the current = 3 km/hr Quicker Method: Applying the above theorem,

Rate of current = l ( l 6 -10) = 3 km/hr

Exercise 1. A man rows upstream 16 km and downstream 27 km tak-

ing 5 hours each time. What is the velocity of current? a)2km/hr b)2.1 km/hr c) 1.1 km/hr d) None of these

1 2. A boat moves downstream at the rate of one km in 7

2 minutes and upstream at the rate of 5 km an hour. What is the velocity of current? a) 1.3km/hr b) 1.2km/hr c)1.6km/hr d) 1.5km/hr

3. A person rows a kilometre down the stream in 10 min-utes and upstream in 30 minutes. Find the velocity of the stream. a) 1 km/hr b)2km/hr c)3km/hr d)4km/hr

4. A man can row three quarters of a km against the stream in 11 minutes 15 seconds and return in 7 minutes 30 seconds. Find the speed of the man in still water and also the speed of the stream a) 5 km/hr, 2 km/hr b) 5 km/hr, 1 km/hr c) 6 km/hr, 2 km/hr d) 4 km/hr, 1 km/hr

5. A boat's man goes 48 km downstream in 8 hours and returns back in 12 hours. Find the speed of the boat in still water and the rate of the stream. a) 5 km/hr, 1 km/hr b) 10 km/hr, 2 km/hr c) 6 km/hr, 1.5 km/hr d) None of these

6. A boat moves with a speed of 11 km per hour and along the stream and 7 km per hour against the stream. The rate of the stream is km/hr. a) 1 km/hr b)1.5km/hr c)2km/hr d)2.5km/hr

7. A man rows upstream 11 km and downstream 26 km tak-ing 5 hours each time. The velocity of the current is

km/hr. a) 1 km/hr b) 1.3km/hr c)1.5km/hr d)2.5km/hr

8. A boat moves downstream at the rate of 1 km in 6 min-utes and upstream at the rate of 1 km in 10 minutes. The speed of the current is

a)2km/hr b) 1 km/hr c)1.5km/hr d)2.5km/hr 9. The speed of a boat downstream is 15 km/hr and the

speed of the stream is 1.5 km/hr. The speed of the boat upstream is a) 13.5 km/hr b) 16.5 km/hr c) 12 km/hr d) 8.25 km/hr

10. I f a man's rate with the current is 12 km/hr and the rate of current is 1.5 km/hr, then the man's rate against the cur-rent is a) 9 km/hr b) 6.75 km/hr c) 5.25 km/hr d) 7.5 km/hr

11. A man can swim downstream at 8 km/hr and upstream at 2 km/hr. Find man's rate in still water and the speed of current. a) 5 km, 2 km/hr b) 5 km, 1 km/hr c) 6 km, 3 km/hr d) 5 km, 3 km/hr

12. A man rows upstream 20 km and downstream 30 km tak-ing 5 hours each. Find the speed of current. a)2km/hr b) 1 km/hr c) 1.5 km/hr d) None of these

13. A boat man can row 2 km against the stream in 20 min-utes and return in 15 minutes. Find the rate of rowing in still water and the speed of current. a) 7 km/hr, 2 km/hr b) 6 km/hr, 2 km/hr c) 7 km/hr, 1 km/hr d) 7.5 km/hr, 1.5 km/hr

14. A boat moves downstream at the rate of 12 km/hr and upstream at 4 km/hr. Find the speed of the boat in still water and also the speed of current. a) 8 km/hr, 4 km/hr b) 4 km/hr, 2 km/hr c) 6 km/hr, 3 km/hr d)3km/hr, 1.5km/hr

. 7 15. A man can row downstream at the rate of 2 metres per

second and upstream at the rate of 5 km/hr. Find the man's rowing rate in still water and speed of current, a) 8 km/hr, 3 km/hr b) 7.5 km/hr, 3 km/hr c) 7.5 km/hr, 2.5 km/hr d) None of these

16. A boatman can row 1 km against the stream in 22

minutes and return in 15 minutes. Find the rate of cur-rent. a)lkm/hr b)2km/hr c)1.5km/hr d) 1.3km/hr

17. A man can row 30 km downstream in 2 hours and 15 km upstream in 5 hours. Find the man's rowing rate in still water and speed of current. a) 9 km/hr, 6 km/hr b) 8 km/hr, 5 km/hr c) 9 km/hr, 5 km/hr d) Data inadequate

18. A man can row 60 km downstream in 6 hours. I f the speed of the current is 3 km/hr, then find in what time will he be able to cover 16 km upstream? a) 4.5 hours b) 4 hours c) 5 hours d) 5.5 hours

19. A person rows 2 km downstream in 20 minutes and up-stream in one hour. Find the velocity of the stream. a)2.1 km/hr b)3.1 km/hr c)2km/hr d) 1.5km/hr

20. A boatman rows 64 km downstream in 8 hours and re-turns back in 16 hours. Find the speed of the boat in still

4 9 0 P R A C T I C E B O O K ON Q U I C K E R MATHS

water and the rate of the stream. a) 6 km/hr, 2 km/hr b)4km/hr, 1 km/hr c) 5 km/hr, 1.5 km/hr d) Data inadequate

21. A boat goes 100 km downstream in 10 hours, and 75 km upstream in 15 hours. The speed of the stream is

a)7km/hr b)5km/hr c)3km/hr d) 2 - km/hr

22. A man rows 40 km upstream in 8 hours and a distance of 36 km downstream in 6 hours, then speed of the stream is a)0.5km/hr b)5.5km/hr c)6km/hr d)5km/hr

23. A man can row three quarters of a km against the stream

1 , 1 in 11 minutes and return in 7 minutes. Find the

4 2 speed of the man in still water. What is the speed of the stream? a) 5 km/hr, 1 km/hr b) 6 km/hr, 2 km/hr c) 4 km/hr, 1 km/hr d) None of these

24. A man rows upstream 13 km and downstream 28 km tak-ing 5 hours each time. What is the velocity of the currrent? a)1.5km/hr b)3 km/hr c) 2.5 km/hr d) Data inadequate

25. A boat is rowed down a river at 10 km/hr and up the river

Velocity of the current = ( 8 - 5) km/hr =1.5 km/hr

3. b; Hint: Rate downstream = * 60 6 km/hr

Rate upstream = * 60 =2 km/hr

Velocity of the stream = (6 - 2) = 2 km/hr

3 4 r\ 4. b; Hint: Rate upstream = 4 x ^ x 6 0 = 4 km/hr

3 2 Rate downstream = x x 6 0 = 6 km/hr

4 15

Speed of the man in still water (6 + 4)i = 5 km/hr

(See Rule 3)

Speed of the stream = (6 - 4)~ = 1 km/hr

48 5. a; Hint: Rate downstream = = 6 km/hr

o at 4 km/hr. Find the velocity of the river.

a) 2 km/hr

c) 2 km/hr

b) 2 - km/hr O

d) 2 - km/hr o

26. In 3 hours a boat can be rowed 9 km upstream or 18 km downstream. Find the speed of the boat in still water and the rate at which the stream is running, a) 4.5 km/hr, 1.5 km/hr b) 5 km/hr, 3 km/hr c) 6 km/hr, 4 km/hr d) Data inadequate

Answers

16 1. c; Hint: Man's rate upstream = km/hr

27 Man's rate downstream = km/hr

Velocity of the current -J / 2 7 _ 1 6 2\ 5

2. d; Hint: Rate downstream =

Rate upstream = 5 km/hr

r_2 15

x60

km/hr= 1.1 km/hr

km/hr = 8 km/hr

48 Rate upstream = = 4 km/hr

6.c 7.c 8. a

9. c;Hint: ( l 5 - y ) l = 1.5 ... y = 15-3 = 12km/hr

10. a 11.d 12.b 13.c 14.a 15.c 16.a 17. a

18. b; Hint: 3 = 1 60 _ 16

6 t t = 4 hours

19. c 20. a 21. d 22. a

3 60 23. a; Hint: The boat travels with stream at x = 6 km/hr

4 7 I 2

The boat travels against the stream at 3 60 x r = 4 km/hr 4 , , 1 l l x

4

.-. speed of man in still water = ^-(6 + 4) = 5 km/hr and

speed of stream = ^ ( 6 - 4 ) = 1 km/hr

24. a 25. d 26. a

Streams 491

an/hr

rand

Rule 5 Theorem: Ifx km be the rate of stream and a man takes n times as long to row up as to row down the river, then the

(n+\)

rate of the man in still water is given by x I _ j J km/hr.

Illustrative Example Ex.: A man takes twice as long to row up as to row down

the river. I f the rate of river is 4 km/hr, find the rate of the man in still water. Detail Method: Let rate of man in still water be x km/hr Then,

x + 4 = 2(x-4) or,x= 12 km/hr Quicker Method: Applying the above theorem, we have

Soln:

the speed of the man in still water = 4 2 + 1

. 2 - 1 ,

= 4 x 3 = 12km/hr.

Exercise 1. A man can row 4.5 km/hr in still water and he finds that it

takes him twice as long to row up as to row down the river. Find the rate of stream. a)2km/hr b)1.5km/hr c)2.5km/hr d) 1.75km/hr

2. A man can row 6 km/hr in still water. It takes him twice as long to row up as to row down the river. Find the rate of the stream. a)2km/hr b)3km/hr c)1.5km/hr d) 1 km/hr

3. A man can row at the rate of 3.5 km/hr in still water. I f the

time taken to row a certain distance upstream is 2 1

times as much as to row the same distance downstream, find the speed of the current. a)2.5km/hr b)1.5km/hr c)3km/hr d)1.25km/hr A man can row 4 km/hr in still water and he finds that it takes him twice as long to row up as to row down the river. Find the rate of stream. a) 1.3km/hr b)2km/hr c) 1 km/hr d) 1.5 km/hr A man can row at the rate of 4 km/hr in still water. I f the time taken to row a certain distance upstream is 3 times as much as to row the same distance downstream, find the speed of the current. a)3km/hr b) 1.5km/hr c) 1 km/hr d)2km/hr

A person can row 7 km an hour in still water and he

finds that it takes him twre as long to row up as to row down the river. Find the rate of the stream. a)2.5km/hr b)2km/hr c)3km/hr d) 1.5km/hr

7. A man swimming in a stream which flows 1.5 km/hr finds that in a given time he can swim twice as far with the stream as he can against it, at what rate does he swim? a)4km/hr b)4.5km/hr c)5km/hr d)3.5km/hr

Answers

L b ; Hint: 2 + 1 2 - 1

4.5

4.5 . '* = = L5 km/hr

2.a 3.b 4. a 5.d 6. a 7.b

Rule 6 Theorem: If the speed of the boat in still water is x km/hr and the rate of current is y km/hr, then the distance trav-elled downstream in 'T' hours is (x +y)Tkm Le. Distance travelled downstream=Downstream Rate X Time. And the distance travelled upstream in 'T' hours is (x-y)T km ie Distance travelled upstream = Upstream Rate x Time.

Illustrative Example Ex.: The speed of a boat in still water is 8 km/hr and the

rate of current is 4 km/hr. Find the distance travelled downstream and upstream in 5 minutes.

Soln: Applying the above theorem,

5 Distance travelled downstream = (8 + 4):

60 = l k m .

Distance travelled upstream = (8-4)x = km. v ; 60 3

Exercise 1. The speed of a boat in still water is 15 km/hr and the rate

of current is 3 km/hr. The distance travelled downstream in 12 minutes is a)3.6km b)2.4km c) 1.2km _ d) 1.8km

2. Speed of a boat in standing water is 7 km/hr and the speed of the stream is 1.5 km/hr. A distance of 7.7 km, going upstream is covered in a) 1 hr 15minb) 1 hr 12 mine) 1 hr24mind)2hr6min

3. The speed of a boat in still water is 15 km/hr and the rate of current is 13 km/hr. Find the distance travelled down-stream in 15 minutes. a)7km b)8km c)7.5km d)7.6km

4. A man can row upstream 32 km in 4 hours. I f the speed of current is 2 km/hr, find how much he can go downstream in 6 hours. a) 70 km b)72km c)64km d)81km

5. A man can row upstream 36 km in 6 hours. I f the speed of a man in still water is 8 km/hr, find how much he can go downstream b 10 hours. a) 150km b)80km c)90km d) 100km

4 9 2 P R A C T I C E B O O K ON Q U I C K E R MATH

The speed of a boat in still water is 5 km/hr and the rate of current is 1 km/hr. Find the distance travelled in 20 minutes in (i) downstream, (ii) upstream. a)2km, 1.33km b)3km, 1.33 km c) 1.5 km, 1 km d) Data inadequate The speed of a boat in still water is 4 km/hr and the speed of current is 2 km/hr. I f the time taken to reach a certain distance upstream is 9 hours, find the time it wil l take to go to same distance downstream. a)2hrs b)2.5hrs c)3.5hrs d)3hrs A person can swim in still water at 4 km/hr. I f the speed of water is 2 km/hr, how many hours will the man take to swim back against the current for 6 km.

IUTI Exam, 1990]

a) 3 b)4 1

0 4 - d) Data inadequate

9. A man can row in still water at 7 km/hr and the rate of stream is 3.5 km/hr. A distance of 10.5 km in going up-stream is covered in a)1.5hrs b) 1 hr c)3hrs d)15hrs

10. The speed of a boat in still water is 15 km/hr and the rate of stream is 5 km/hr the distance travelled downstream in 24 minutes is

a)4km b)8km c)6km d)16km

Answers

1. a;

2. c;

3. a

4. b;

5-d;

6. a 7. d;

12 18 Hint: Required distance = (l 5 + 3 ) = = 3.6 km

V ; 6 0 5

Hint: (7-1.5)T = 7.7 :.T = = 1 hr 24 minutes v ' 5.5

32 Hint: Downstream rate = = 8 km/hr

4 Speed of man in still water = 8 + 2=10 km/hr Now, the required distance = (10 + 2)6 = 72 km

36 Hint: Speed of current = - r = 2 km/hr

6 .-. required distance = (8 + 2) 10 = 100 km

Hint: Distance = (4 - 2)9 = 18 km

.-. Required time = 18

4 + 2 = 3hrs

8. a 9.c 10. b

Rule 7 Theorem: A man can row x km/hr in still waters. If in a stream which isflowing aty km/hr, it takes him z hrs to row to a place and back, the distance between the two places is

A 2x

Distance' Total Time :\speed

OR

in still water)' - (Speed of current)']

2 x Speed in stilt water

Note: The speed of a boat in still water is x km/hr and f.-. speed of the stream is y km/hr. A man rows to a place at i distance ofDkm and comes back to the starting point thi

the total time taken by him is 2Dx

hours OR Ton

time 2 x Distance x Speed in still water

(Speed in still water)2 - (Speed in current)2

Illustrative Examples Ex. 1: The speed of a boat in still water is 6 km/hr and

speed of the stream is 1.5 km/hr. A man rows to ; place at a distance of 22.5 km and comes back to th= starting point. Find the total time taken by him.

Soln: Detail Method: Boat's upstream speed = 6 - 1.5 = 4.5 km/hr Boat's downstream speed = 6+1.5 = 7.5 km/hr

22.5 22.5 Total time = + - 5 + 3 = 8 hrs.

4.5 7.5 Quicker Method: Applying the above theorem, w : have

2x22.5x6 the total time = ,2 6 2 - ( l . 5 ) 2

: 8 hours.

Ex. 2: A man can row 6 km/hr in still water. When the river is running at 1.2 km/hr, it takes him 1 hour to row to a place and back. How far isthe place? Detail Method: Man's rate downstream = (6 + 1.2) k, hr = 7.2 km/hr Man's rate upstream = (6 - 1.2) km/hr = 4.8 km/hr Let the required distance be x km. Then

Soln:

X + X - 1 12 4^8" or, 4.8.x + 7.2* = 7.2x4.8

7.2x4.8 = 2.88 km.

12 Quicker Method: Applying the above theorem, we have

the required distance = l x 6 -Q.2)21

2x6 36-1.44

12 = 3-0.12 = 2.88 km

Exercise 1. A man rows 8 km/hr in still water. I f the river is running at

2 km/hr, it takes 32 minutes to row to a place and back.

m A

MATHS S t r e a m s 493

How far is the place? a)1.5km b)2.5km c)2km d)3km A man can row 5 km/hr in still water. I f the river is run-ning at 1.5 km/hr, it takes him 1 hour to row to a place and back. How far is the place? a)2km b)2.5km c)2.275km d)2.175km A man can row 8 km per hour in still water. I f the river is running at 2 km an hour, it takes him 48 minutes to row to a place and back, how far is the place? a)5km b)4km c)2km d)3km A man can row 5 km per hour in still water. I f the river is running at 1 km an hour, it takes him 75 minutes to row to a place and back. The place is at a distance of km from the starting point. a)3km b)4km c)5km d)2km A man can row 5 km/hr in still water. I f the river is run-ning at 1 km/hr, it takes him 1 hour to row to a place and back. How far js the place? a)2.5km b)2.4km c)3km d)3.6km A boat travels upstream from B to A and downstream from A to B in 3 hours. I f the speed of the boat in still water is 9 km/hr and the speed of the current is 3 km/hr, the distance between A and B (in km) is

[BSRB Bank PO Exam, 1990] a) 4 b)6 c)8 d) 12 A man can row 6 km/hr in the still water. I f the river is running at 2 km/hr, it takes him 3 hours to row to a place and back. How far is the place? a)8km b)12km c)9km d)6km A man can row 4 km/hr in still water i f the river is running at 2 km/hr, it takes 6 hours to row to a place and back. How far is the place? a)6km b)8km c)9km d)9.5km A boat's crew rowed down a stream from A to B and up

1 again in 7 hours. I f the stream flows at 3 km/hr and

the crew can row in still water at 5 km/hr, find the dis-tance from A to B. a) 16km b)8km c)14km d)12km A boat's crew rowed down a stream from A to B and up

again in 9 hours. I f the stream flows at 2 ^ km/hr and

the crew can row in still water at 4 km/hr, find the

distance from A to B.

a) 14km b)13km c)12km d)16km

_ 1 A man can row 7 km/hr in still water. I f in a river

2

running at 1 km/hr, it takes him 50 minutes to row to a

place and back, how far off is the place? a)2km b)3km c)4km d)6km

Answers

2.ZX8 l.c;Hint: 2 2

2.c 9.d

8^ 3.d 10. a

32 2 ~60 4. a 11.b

.-. D = 2km

5.b 6.d 7. a 8.c

Rule 8 Theorem: If in a stream running atx km/hr, a motorboat goes D km upstream and back again to the starting point in 'T' hours, then the speed of the motorboat in still water is

D + ^D2+(Tx)2 km/hr.

Illustrative Example Ex.: In a stream running at 2 km/hr, a motorboat goes 10

km upstream and back again to the starting point in 55 minutes. Find the speed of the motorboat in still water.

Soln: Detail Method: Let the speed of the motorboat in still water be x km/hr.

10 x + 2

10 55 60

or, 240*=11*' - 44

or, l l x 2 - 240* -44 = 0

.-. (x -22) ( lbc + 2 ) = 0

So, x = 22 km/hr (neglect the -ve value) .-. speed of the motorboat in still water = 22 km/hr. Quicker Method: Applying the above theorem, we have the speed of the motorboat in still water

10 + (,0)2+||lx2

n 12

60 + 61 6 11 12

121 12 = x = 22 km/hr.

11 6

Exercise 1. The current of a stream runs at the rate of 4 km an hour.

A boat goes 6 km and back to its starting point in 2 hours. Find the speed of the boat in still water. a)8km/hr b)9km/hr c)6km/hr , d)4km/hr A motor boat can travel at 10 km/hr in still water. It trav-elled 91 km downstream in a river and then returned,

2.

4 9 4 P R A C T I C E B O O K ON Q U I C K E R MA

taking altogether 20 hours. Find the rate of flow of river. a)6km/hr b)2km/hr c)3km/hr d)4km/hr

3. The rate of flow of river water is 4 km/hr. A boat goes 6 km and back to the starting point in 2 hours. Find the speed of the boat in still water. a)6km/hr b)8km/hr c)9km/hr d)10km/hr

4. In a stream running at 2 km/hr, a motorboat goes 12 km

upstream and back again to the starting point in 2

hours. Find the speed of the motorboat in still water. a) 15 km/hr b)12km/hr c) 10 km/hr d) None of these

Answers l .a

2. c; Hint: 91 + v / 91 2 +(20x) 2

20 = 10

or, 4Q0xz =11881-8281 = 3600

or, xl = 9 x = 3 km/hr 3 . b x 4.c

Rule 9 Theorem: A man can row x, km upstream and y] km down-

stream in T} hours. Also, he can row x2 km upstream and

y2 km downstream in T2 hours. Then, the rate of the cur-rent and speed of the man in still water is calculated by the use of multiple cross-multiplication method as given be-low.

Step I: Arrange the given figures in the following form Upstream Downstream Time

yi

Upstream speed of man = km/hr

Downstream speed of man = xly2-x2yl V *1*2 ~x2^\ )

km/hr

Step II : Now to calculate the speed of man and current, use the following formula,

Speed of man = [upstream speed of man + down-

| Remember Rule 3]

[Downstream speed ofman

stream speed of man]

and speed of stream =

Upstream speed of man] [Remember Rule 4]

Soln:

Note: 1. How do the denominators of the above two for-differ? For upstream speed we use the figures of: stream speed and time and for downstream spec use the figures of upstream speed and time. 2. Numerators remain the same in both formula:

Illustrative Example Ex.: A man can row 30 km upstream and 44 km

stream in 10 hrs. Also, he can row 40 km upstrea-55 km downstream in 13 hrs. Find the rate of the rent and the speed of the man in still water. Detail Method: Let, upstream rate = x km/hr and downstream

= y km/hr

30 44 , n 40 55 , T h e n >T + 7 = 1 a n d T + 7 = 1 3 or,30u + 44v=10

40u + 55v=13

1 1 Where u = and v =

x y

l l Solving, we get u = and v =

.-. x = 5 and y = 11

5 + 11 u rate in still water = - = 8 km/hr.

Rate of current = 11-5

= 3 km/hr.

Quicker Method: (By use of multiple cross-multiplicar. Step I: Arrange the given figures in the following forr

Upstrearn Downstream 30 44 40 , 55

Upstream speed of man

30x55-40x44 -110 55x10-44x13 -22

Downstream speed of man

30x55-40x44 _ -110 ~ 30x13-40x10 ~ -10

= 5

= 11

StepD:

5 + 11 Speed of man = - = 8 km/hr.

and speed of stream = - - = 3 km/hr.

Exercise 1. A man can row 15 km upstream and 22 km down

5 hrs. Also, he can row 22 km upstream and 2"

downstream in 6 hrs. Find the rate of the current and 2

the speed of the man in still water, a) 11 km/hr, 5 km/hr b) 8 km/hr, 3 km/hr c) 5 km/hr, 2 km/hr d) None of these

1 A man can row 45 km upstream and 66 km downstream in 15 hrs. Also, he can row 66 km upstream and 82.5 km

downstream in 19 hrs. Find the rate of the current and 2

the speed of the man in still water. a) 8 km/hr, 3 km/hr b) 11 km/hr, 3 km/hr c) 11 km/hr, 8 km/hr d) Data inadequate A man can row 60 km upstream and 88 km downstream in 20 hrs. Also, he can row 80 km upstream and 110 km downstream in 26 hrs. Find the rate of the current and the speed of the man in still water. a) 12 km/hr, 4 km/hr b) 16 km/hr, 6 km/hr c) 8 km/hr, 3 km/hr d) None of these

\nswers : b 2. a 3.c

Rule 10 Theorem: A man rows a certain distance downstream in x hours and returns the same distance iny hrs. If the stream flows at the rate of z km/ltr then the speed of the man in still

mater is given by _ x km/hr. Or, Speed in still water =

Rate of stream (Sum of upstream and downstream time) Difference of upstream and downstream time

unlir.

Illustrative Example L\.: Ramesh can row a certain distance downstream in 6

hours and return the same distance in 9 hours. I f the stream flows at the rate of 3 km per hour find the speed of Ramesh in still water.

vjln: Detail Method: Let the speed of Ramesh in still water

be x km/hr. Then his upstream speed = (x-3) km/hr

and downstream speed = (x + 3) km/hr. Now, we are given that up and down journey are equal, therefore,

(x + 3)6 = (x-3)9

or, 6x + 18 = 9 x - 2 7

or, 3x = 45

.'. x = 15 km/hr Quicker Method: By the above theorem, we have

3(9 + 6) Ramesh's speed in still water = T T

9 6

Exercise 1. Ramesh can row a certain distance downstream in 6 hours

and return the same distance in 9 hours. I f the speed of Ramesh in still water is 12 km/hr,find the speed of the stream. a) 2.4 km/hr b) 2 km/hr c) 3 km/hr d) Data inadequate

2. A can row a certain distance down a stream in 6 hours and return the same distance in 9 hours. I f the stream

flows at the rate of 2 km/hr, find how far he can row in 4

an hour in still water?

a) 11 km/hr

c) 1 1 - km/hr 4

b) 12 km/hr

d) Data inadequate

3. Ajay can row a certain distance downstream in 5 hours and return the same distance in 7 hours. I f the stream flows at the rate of 2 km per hour find the speed of Ajay in still water. a) 12 km/hr b) 10 km/hr c) 18 km/hr d) 16 km/hr

4. Rohit can row a certain distance downstream in 8 hours and return the same distance in 12 hours. I f the stream flows at th&rate of 5 km per hour find the speed of Rohit in still water.

a) 20 km/hr b) 30 km/hr c) 15 km/hr d) 25 km/hr

Answers

La;Hint: zj

2.c

9 + 6 9 - 6 ,

3. a 4.d

12 z = 2.4 km/hr

Rule 11 Theorem: If a man can row at a speed of x km/hr in still water to a certain upstream point and back to the starting point in a river which flows at y km/hr, then the averge speed for total journey (up + down) is given by

(x + y^x-y) km/hr.

OR

Average speed for the total journey

Upstream rate x Downstream rate

15 km/hr.

Speed in still water

Illustrative Example Ex.: A man can row at a speed of 5 km/hr in still water to a

4 9 6 P R A C T I C E B O O K ON Q U I C K E R MATH

certain upstream point and back to the starting point in a river which flows at 2 km/hr. Find his average speed for total journey.

Soln: Detail Method: Letthe distance bexkm.

Illustrative Example

Average speed = Total Distance

Total Time

2x 2x 2xx2\

(5 + 2 H 5 - 2 ) x x + -7 3

lOx

= = 4 - km/hr.

Quicker Method: Applying the above theorem, we have

(5 + 2X5-2) 21 . 1 average speed = 5~ 5 ' a T 1 ^ i r -

Exercise 1. A man can row at a speed of 4.5 km/hr in still water to a

certain upstream point and back to the starting point in a river which flows at 1.5 km/hr. Find his average speed for total journey. a) 4 km/hr b) 6 km/hr c) 4.5 km/hr d) 5 km/hr

2. A man row at a speed of 8 km/hr in still water to a certain distance upstream and back to the starting point in a river which flows at 4 km/hr. Find his average speed for total journey. a) 8 km/hr b) 6 km/hr c) 4 km/hr d) 10 km/hr

3. A man can row at a speed of 10 km/hr in still water to a certain upstream point and back to the starting point in a river which flows at 4 km/hr. Find his average speed for total journey. a) 2 km/hr b) 3 km/hr c) 1.5 km/hr d) Data inadequate

4. A man can row at a speed of 15 km/hr in still water to a certain upstream point and back to the starting point in a river which flows at 3 km/hr. Find his average speed for total journey.

b) 6 km/hr d) 2 km/hr

a) 9 km/hr c) 3 km/hr

Answers l .a 2.b j . a 4.d

Rule 12 Theorem: A man can row x km/hr in still water. If the river is running aty km/hr, it takes T hours more in upstream than to go downstream for the same distance, then the dis-

V -v 2 V 2v

tance is given by km.

Ex.:

Soln:

A man can row 6 km/hr in still water. I f the rive-running at 2 km/hr, it takes 3 hours more in upstrei-than to go downstream for the same distance. H far is the place? Detail Method: Let the distance of the place be x k -According to the question,

= 3 6 - 2 6 + 2

o r , - - 8 = 3

.-. x = 8*3 = 24km Quicker Method: Applying the above theorem. have the required distance

_ (6 2 - 2 2 )3 32x3 2x2 ~ 4

: 8x3 = 24 km.

Exercise 1. A man can row

ning at 1 km/hr, go downstream place? a) 24 km b)

2. A man can row ning at 3 km/hr, go downstream place? a) 48 km b)

3. A man can row ning at 4 km/hr, go downstream place? a) 16 km b)

4. A man can row ning at 3 km/hr, go downstream place? a) 30 km b)

Answers l .a 2.d 3.

5 km/hr in still water. I f the river is rur-it takes 2 hours more in upstream than:: for the same distance. How far is tht

20km c)18km d)16km 7 km/hr in still water. I f the river is rur-it takes 6 hours more in upstream than: for the same distance. How far is tht

36 km c)42km d)40km 8 km/hr in still water. I f the river is run-it takes 1 hour more in upstream than t: for the same distance. How far is thi

12 km c)8km d)6km 9 km/hr in still water. I f the river is run-it takes 3 hours more in upstream than tc for the same distance. How far is the

36 km d)24km d) None of these

d 4.b

Miscellaneous A boat takes 3 hours to travel from place M to N down-stream and back from N to M upstream. I f the speed 0: he boat in still water is 4 km, what is the distance be-tween the two places? [ BSRB Delh i PO, 20001 a)8km b)12km c) 6 km d) Data inadequate A man rows to a place 48 km distant and back in 14 hours. He finds that he can row 4 km with the stream in

ATHS S t r e a m s 497

the same time as 3 km against the stream. Find the rate of the stream? a) 1 km/hr b)2km/hr c)1.5km/hr d)2.5km/hr

I P, Q and R are the three towns on a river which flows uniformly. Q is equidistant from P and R. I row from P to Q and back in 10 hours and I can row from P to R in 4 hours. Compare the speed of my boat in still water with that of the river.

a)5:3 b)4:3 c)6:5 d)7:3

Answers '.. d; Let the distance between M and N and the speed of

current in still water be d km and x km/hr respectively. d d ,

According to the question, ^ + ^ + - - 3

In the above equation we have only one equation but two variables. Hence can't be determined,

1 a; Suppose that the man takes x hours to cover 4 km downstream and x hours to cover 3 km upstream.

48* 48* , 1 Then, ~A~ + ~y = U o r X = 2

.-. Rate upstream = 6 km/hr and rate downstream;

km/hr

.-. Rate of the stream = (8 - 6)-^- = 1 km/hr 2

3. a;

[See Rule 4]

_2 R I can row from P to R in 4 hours .-. I can row from P to Q in 2 hours But 1 can row from P to Q and back in 10 hours. .-. I can row from Q to P in (10 - 2 =) 8 hours Hence in rowing with the current I take 2 hours and in rowing against the current I take 8 hours, the dis-tance being same in both the cases. Now, distance being the same the 'down rate' and the 'up rate' are inversely proportional to the times. .-. down rate : up rate = 8:2 = 4 : 1 .-. speed of boat in still water : speed of river

= ( 4 + l ) : ( 4 - l ) = 5:3.