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21. 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. Find the total time taken by him.
  A.  8 hours
  B.  5 hours
  C.  9 hours
  D.  6 hours
     
   
View Answer

Shortcut:
The speed of boat in still water is 'a' km/hr and the speed of stream is 'b' km/hr. A man rows to a place at a distance of D km and comes back to the starting point then the total time taken by him
2 x D x a / a2 − b2
hours
Here, D = 22.5, a = 6, b = 1.5
Required time =
2 x 22.5 x 6 / (6)2 − (1.5)2

=
2 x 22.5 x 6 / 36 − 2.25

=
2 x 22.5 x 6 / 33.75
= 8
Hence, total time taken by him is 8 hours.

22. 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 is the place?
  A.  4 km
  B.  2.88 km
  C.  5.2 km
  D.  3.44 km
     
   
View Answer

Shortcut:
A man can row a km/hr in still water. If in a stream which is flowing at b km/hr, it takes him D hours to row to a place and back, the distance between the two places is
D(a2 − b2) / 2a
km
Here, D = 1, a = 6, b = 1.2
Required distance =
1 x [(6)2 − (1.2)2] / 2 x 6

=
36 − 1.44 / 12
= (3 − 0.12) = 2.88
Hence, total distance is 2.88 km.

23. A man rows 8 km/hr in still water. If the river is running at 2 km/hr, it takes 32 minutes to row to a place and back. How far is the place?
  A.  7 km
  B.  2 km
  C.  5 km
  D.  4 km
     
   
View Answer

Shortcut:
A man can row a km/hr in still water. If in a stream which is flowing at b km/hr, it takes him D hours to row to a place and back, the distance between the two places is
D(a2 − b2) / 2a
km
Here, D = 32, a = 8, b = 2
Required distance =
32/60 x [(8)2 − (2)2] / 2 x 8

=
8/15 x (64 − 4) / 16
=
8/15 x 60 / 16
=
8 x 4 / 16
= 2
Hence, total distance is 2 km.

24. A boat travels upstream from B to A and downstream from A to B in 3 hours. If 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 is:
  A.  12 km
  B.  10 km
  C.  15 km
  D.  14 km
     
   
View Answer

Shortcut:
A man can row a km/hr in still water. If in a stream which is flowing at b km/hr, it takes him D hours to row to a place and back, the distance between the two places is
D(a2 − b2) / 2a
km
Here, D = 3, a = 9, b = 3
Required distance =
3 x [(9)2 − (3)2] / 2 x 9

=
3 x [81 − 9] / 18
=
3 x 72 / 18
= 3 x 4 = 12
Hence, total distance is 12 km.

25. 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.
  A.  20 km/hr
  B.  22 km/hr
  C.  25 km/hr
  D.  18 km/hr
     
   
View Answer

Shortcut:
If a a stream running at a km/hr, a motorboat goes Z km upstream and back again to the starting point in 't' hours, then the speed of the motorboat is still water is
Z + √[(Z)2 + (ta)2] / t
km/hr
Here, Z = 10, a = 2, t =
55 / 60
or
11 / 12

Required speed =
10 + √[(10)2 + (11/12 x 2)2] / 11/12

=
10 + √[(100 + (11/6)2] / 11/12

=
10 + √[(100 + 121/36] / 11/12

=
10 + √[(3600 + 121)/36] / 11/12

=
10 + √(3721/36) / 11/12

=
10 + 61/6 / 11/12

=
(60 + 61)/6 / 11/12

=
121/6 / 11/12

=
121 x 12 / 6 x 11
= 11 x 2 = 22
Hence, the required speed is 22 km/hr.

26. A motor boat can travel at 10 km/hr in still water. It travelled 91 km downstream in a river and then returned,taking altogether 20 hours. Find the rate of flow of river.
  A.  3 km/hr
  B.  6 km/hr
  C.  5 km/hr
  D.  2 km/hr
     
   
View Answer

Shortcut:
If a stream running at a km/hr, a motorboat goes Z km upstream and back again to the starting point in 't' hours, then the speed of the motorboat in still water is
Z + √[(Z)2 + (ta)2] / t
km/hr
Here, Z = 91, a = ?, t = 20, Speed in still water = 10
Speed of motorboat in still water =
91 + √[(91)2 + (20 x a)2] / 20

10 =
[91 + √(8281 + 400a)] / 20

10 x 20 = 91 + √(8281 + 400a2)
200 − 91 = √(8281 + 400a2)
109 = √(8281 + 400a2)
Squaring both sides, we get
11881 = 8281 + 400a2
11881 − 8281 = 400a2
3600 = 400a2
a2 =
3600 / 400
= (3)2 = 3
Hence, the rate of current is 3 km/hr.

27. A man can row 30 km upstream and 44 km downstream in 10 hours. Also, he can row 40 km upstream and 55 km downstream in 13 hours. Find the rate of the current and the speed of the man in still water.
  A.  Rate in still water: 9km/hr; Rate of current: 3 km/hr
  B.  Rate in still water: 7km/hr; Rate of current: 4 km/hr
  C.  Rate in still water: 5km/hr; Rate of current: 1 km/hr
  D.  Rate in still water: 8km/hr; Rate of current: 3 km/hr
     
   
View Answer

Shortcut:
A man can row a1 km upstream and b1 km downstream in t1 hours. Also, he can row a2 km upstream and b2 km downstream in t2 hours. Then, the rate of the currrent and speed of the man in still water is calculated by the use of multiple cross-multiplication method as in 2 steps:

Step 1 :Arrange the given figures in the following form

Upstream Downstream Time
a1 b1 t1
a2 b2 t2
Upstream speed of man =
a1b2 − a2b1 / b2t1 − b1t2
km/hr
Downstream speed of man =
a1b2 − a2b1 / a1t2 − a2t1
km/hr
Step 2: To calculate the speed of man and current, use the following formula.

Speed of man =
1 / 2
(upstream speed of man + downstream speed of man)
Speed of stream =
1 / 2
(downstream speed of man − upstream speed of man)

Here, a1 = 30, a2 = 40, b1 = 44, b2 = 55, t1 = 10, t2 = 13

Upstream Downstream Time
30 44 10
40 55 13

Upstream speed of man =
30 x 55 − 40 x 44 / 55 x 10 − 44 x 13

=
1650 − 1760 / 550 − 572
=
− 110 / − 22
= 5
Hence, the upstream speed of man is 5 km/hr.
Downstream speed of man =
30 x 55 − 40 x 44 / 30 x 13 − 40 x 10

=
1650 − 1760 / 390 − 400
=
− 110 / − 10
= 11
Hence, the downstream speed of man is 11 km/hr.
Speed of man =
1 / 2
(5 + 11) =
16 / 2
= 8
Hence, the speed of man is 8 km/hr.
Speed of stream =
1 / 2
(11 − 5) =
6 / 2
= 3
Hence, the speed of stream is 3 km/hr.

28. A man can row 15 km upstream and 22 km downstream in 5 hours. Also, he can row 20 km upstream and 27.5 km downstream in 6
1 / 2
hours. Find the rate of the current and the speed of the man in still water.
  A.  6 km/hr, 3 km/hr
  B.  10 km/hr, 2 km/hr
  C.  8 km/hr, 3 km/hr
  D.  9 km/hr, 2 km/hr
     
   
View Answer

Shortcut:
A man can row a1 km upstream and b1 km downstream in t1 hours. Also, he can row a2 km upstream and b2 km downstream in t2 hours. Then, the rate of the currrent and speed of the man in still water is calculated by the use of multiple cross-multiplication method as in 2 steps:

Step 1 :Arrange the given figures in the following form

Upstream Downstream Time
a1 b1 t1
a2 b2 t2
Upstream speed of man =
a1b2 − a2b1 / b2t1 − b1t2
km/hr
Downstream speed of man =
a1b2 − a2b1 / a1t2 − a2t1
km/hr
Step 2: To calculate the speed of man and current, use the following formula.

Speed of man =
1 / 2
(upstream speed of man + downstream speed of man)
Speed of stream =
1 / 2
(downstream speed of man − upstream speed of man)

Here, a1 = 15, a2 = 20, b1 = 22, b2 = 27.5, t1 = 5, t2 = 6
1 / 2
or
13 / 2


Upstream Downstream Time
15 22 5
20 27.5 13/2

Upstream speed of man =
15 x 27.5 − 20 x 22 / 27.5 x 5 − 22 x 13/2

=
412.5 − 440 / 137.5 − 143
=
− 27.5 / − 5.5
= 5
Hence, the upstream speed of man is 5 km/hr.
Downstream speed of man =
15 x 27.5 − 20 x 22 / 15 x 13/2 − 20 x 5

=
412.5 − 440 / 97.5 − 100
=
− 27.5 / − 2.5
= 11
Hence, the downstream speed of man is 11 km/hr.
Speed of man =
1 / 2
(13 + 5.72) =
18.72 / 2
= 8
Hence, the speed of man is 8 km/hr.
Speed of stream =
1 / 2
(11 − 5) =
6 / 2
= 3
Hence, the speed of stream is 3 km/hr.

29. A man can row 60 km upstream and 88 km downstream in 20 hours. Also, he can row 80 km upstream and 110 km downstram in 26 hours. Find the rate of the current and the speed of the man in still water.
  A.  3 km/hr, 2 km/hr
  B.  8 km/hr, 3 km/hr
  C.  9 km/hr, 4 km/hr
  D.  6 km/hr, 3 km/hr
     
   
View Answer

Shortcut:
A man can row a1 km upstream and b1 km downstream in t1 hours. Also, he can row a2 km upstream and b2 km downstream in t2 hours. Then, the rate of the currrent and speed of the man in still water is calculated by the use of multiple cross-multiplication method as in 2 steps:

Step 1 :Arrange the given figures in the following form

Upstream Downstream Time
a1 b1 t1
a2 b2 t2
Upstream speed of man =
a1b2 − a2b1 / b2t1 − b1t2
km/hr
Downstream speed of man =
a1b2 − a2b1 / a1t2 − a2t1
km/hr
Step 2: To calculate the speed of man and current, use the following formula.

Speed of man =
1 / 2
(upstream speed of man + downstream speed of man)
Speed of stream =
1 / 2
(downstream speed of man − upstream speed of man)

Here, a1 = 60, a2 = 80, b1 = 88, b2 = 110, t1 = 20, t2 = 26

Upstream Downstream Time
60 88 20
80 110 26

Upstream speed of man =
60 x 110 − 80 x 88 / 110 x 20 − 88 x 26

=
6600 − 7040 / 2200 − 2288
=
− 440 / − 88
= 5
Hence, the upstream speed of man is 5 km/hr.
Downstream speed of man =
60 x 110 − 80 x 88 / 60 x 26 − 80 x 20

=
6600 − 7040 / 1560 − 1600
=
− 440 / − 40
= 11
Hence, the downstream speed of man is 11 km/hr.
Speed of man =
1 / 2
(5 + 11) =
16 / 2
= 8
Hence, the speed of man is 8 km/hr.
Speed of stream =
1 / 2
(11 − 5) =
6 / 2
= 3
Hence, the speed of stream is 3 km/hr.

30. A man can row 7
1 / 2
km/hr in still water. If in a river running at 1
1 / 2
km/hr, it takes him 50 minutes to row to a place and back, how far off is the place?
  A.  3 km
  B.  1 km
  C.  5 km
  D.  2 km
     
   
View Answer

Shortcut:
A man can row a km/hr in still water. If in a stream which is flowing at b km/hr, it takes him D hours to row to a place and back, the distance between the two places is
D(a2 − b2) / 2b
km
Here, D =
50 / 60
or
5 / 6
, a =
15 / 2
, b =
3 / 2

Required distance =
5/6 x [(7.5)2 − (1.5)2] / 2 x 15/2

=
5/6 x (56.25 − 2.25) / 15
=
5/6 x 54 / 15
=
5 x 9 / 15
= 3
Hence, total distance is 3 km.

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