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21. Two trains of the length 100 metres and 125 meters respectively with different speeds pass a static pole in 4 seconds and 7 seconds respectively. In what time will cross each other when they are moving in same direction?
  A.  30 sec
  B.  31.5 sec
  C.  35 sec
  D.  38 sec
     
   
View Answer

Shortcut:
Two trains of the length L1 and L2 m respectively with different speeds pass a static pole in t1 seconds and t2 seconds respectively. When they are moving in the same direction, they will cross each other in
(L1 + L2)t1t2 / t2L1 − t1L2
seconds.
Here, L1 = 100, L2 = 125, t1 = 4, t2 = 7
Using these values in the shortcut, we get:
=
(100 + 125) x 4 x 7 / 100 x 7 − 4 x 125

=
225 x 4 x 7 / 700 − 500

=
225 x 4 x 7 / 200
=
45 x 7 / 10
= 31.5
Hence, the trains will cross each other in 31.5 seconds.

22. Two trains of the length 100 metres and 125 meters respectively with different speeds pass a static pole in 4 seconds and 7 seconds respectively. In what time will cross each other when they are moving in opposite direction?
  A.  5 sec
  B.  7 sec
  C.  5.25 sec
  D.  3 sec
     
   
View Answer

Shortcut:
Two trains of the length L1 and L2 m respectively with different speeds pass a static pole in t1 seconds and t2 seconds respectively. When they are moving in the opposite direction, they will cross each other in
(L1 + L2)t1t2 / t2L1 + t1L2
seconds.
Here, L1 = 100, L2 = 125, t1 = 4, t2 = 7
Using these values in the shortcut, we get:
=
(100 + 125) x 4 x 7 / 100 x 7 + 4 x 125

=
225 x 4 x 7 / 700 + 500

=
225 x 4 x 7 / 1200
=
45 x 7 / 60
= 5.25
Hence, the trains will cross each other in 5.25 seconds.

23. Two trains of length 100 metres and 80 metres respetively run on parallel lines of rails. When running in the same direction the faster train passes the slower one in 18 seconds, but when the are running in opposite directions with the same speeds as earlier, they pass each other in 9 seconds. Find the speed of each train.
  A.  8 m/s
  B.  10 m/s
  C.  6 m/s
  D.  5 m/s
     
   
View Answer

Shortcut:
Two trains of the length L1 and L2 m respectively run on parallel lines. When running in the same direction the faster train passes the slower one in t1 seconds, but when they are running in opposite directions with the same speeds as earlier, they pass each other in t2 seconds. Then the speed of faster train is
(L1 + L2) / 2
x
(t1 + t2) / t1t2
m/sec and the speed of slower train is
(L1 + L2) / 2
x
(t1 − t2) / t1t2
m/sec
Here, L1 = 100, L2 = 80, t1 = 18, t2 = 9
Using these values in the shortcut, we get:
Speed of faster train =
(100 + 80) / 2
x
18 + 9 / 18 x 9

=
180 / 2
x
27 / 162
= 90 x
9 / 54
= 90 x
1 / 6
= 15 m/sec
Hence, the speed of faster train is 15 m/sec.
100 + 80 / 2
x
18 − 9 / 18 x 9
=
180 / 2
x
9 / 18 x 9
= 90 x
1 / 18
= 5
Hence, the speed of the slower train is 5 m/sec.

24. Two trains can run on parallel lines of rails at the speed of 54 km/hr and 36 km/hr respectively. When running in the opposite direction they pass eachother in 10 seconds, but when the are running in same directions, a person sitting in the faster train observes that he passes the other train in 30 seconds. Find the length trains.
  A.  Slower train, 150 m; Faster train, 100 m
  B.  Slower train, 120 m; Faster train, 90 m
  C.  Slower train, 160 m; Faster train, 110 m
  D.  Slower train, 140 m; Faster train, 120 m
     
   
View Answer

Shortcut:
Two trains can run at the speed of s1 km/hr and s2 km/hr respectively on parallel tracks. Whey they are running in opposite directions they pass each other in t1 seconds. When they are running in the same direction, a person sitting in the faster train observes that he passes the other train in t2 seconds. The length of the faster train is
5 / 18
x [t1(s1 + s2) − t2(s1 − s2)] metres and the length of the slower train is
5 / 18
x t2(s1 − s2) metres.
Here, s1 = 54, s2 = 36, t1 = 10, t2 = 30
Using these values in the shortcut, we get:
Length of faster train =
5 / 18
x [10(54 + 36) − 30(54 − 36)]
=
5 / 18
x [10 x 90 − 30 x 18]
=
5 / 18
x [900 − 540]
=
5 / 18
x 360 = 5 x 20 = 100
Hence, the length of the faster train is 100 m.
Length of slower train =
5 / 18
x 30(54 − 36) =
5 / 18
x 30 x 18 = 5 x 30 = 150
Hence, the length of the slower train is 150 m.

25. A train overtakes two persons who are walking in the same direction of the train, at the rate of 2 km/hr and 4 km/hr and passes them completely in 9 and 10 seconds respectively. Find the speed and the length of the train.
  A.  Speed 25 km/hr; Length 52 m
  B.  Speed 20 km/hr; Length 52 m
  C.  Speed 22 km/hr; Length 50 m
  D.  Speed 24 km/hr; Length 60 m
     
   
View Answer

Shortcut:
A train overtakes two persons who are walking in the same direction as the train is moving, at the rate of s1 km/hr and s2 km/hr and passes them completely in t1 seconds and t2 seconds respectively. Speed of the train is
s2t2 − s1t1 / t2 − t1
km/hr and the length of the train is
(s2 − s1)t1t2 / t2 − t1
x
5 / 18
metres.
Here, s1 = 2, s2 = 4, t1 = 9, t2 = 10
Using these values in the shortcut, we get:
Speed of the train =
4 x 10 − 2 x 9 / 10 − 9

=
40 − 18 / 1
= 22
Hence, the speed of the train is 22 km/hr.
Length of train =
(4 − 2) x 9 x 10 / 10 − 9
x
5 / 18

=
2 x 9 x 10 / 1
x
5 / 18
= 10 x 5 = 50
Hence, the length of the train is 50 m.

26. A train 75 metres long over took a person who was walking at the rate of 6 km an hour, and passes him in 15/2 seconds. Subsequently it overtook a second person and passed him in 27/4 seconds. At what rate was the second person travelling?
  A.  3 km/hr
  B.  2 km/hr
  C.  5 km/hr
  D.  10 km/hr
     
   
View Answer

Shortcut:
A train overtakes two persons who are walking in the same direction as the train is moving, at the rate of s1 km/hr and s2 km/hr and passes them completely in t1 seconds and t2 seconds respectively. Speed of the train is
s2t2 − s1t1 / t2 − t1
km/hr and the length of the train is
(s2 − s1)t1t2 / t2 − t1
x
5 / 18
metres.
Speed of the second person will be less than the first. Because train takes more time to overtake first person than the second.
Here, s1 = ?, s2 = 6, t1 = 27/4, t2 = 15/2
Using these values in the shortcut, we get:
75 =
(6 − s1)27/4 x 15/2 / 15/2 − 27/4
x
5 / 18

75 =
(6 − s1)27/4 x 15/2 / 7.5 − 6.75
x
5 / 18

75 =
(6 − s1)27 x 15 x 100 / 4 x 2 x 75
x
5 / 18

75 =
(6 − s1)9 x 15 / 2
x
5 / 18

75 =
(6 − s1)9 x 15 x 5 / 18 x 2

75 x 18 x 2 = (6 − s1)9 x 15 x 5
75 x 18 x 2 = (6 − s1)675
75 x 18 x 2 = 4050 − 675 s1
675 s1 = 4050 − 2700
s1 =
1350 / 675
= 2
Hence, the speed of second person is 2 km/hr.

27. A train passes two persons who are walking in the opposite direction of train at the speed of 5 m/s and 10 m/s in 6 seconds and 5 seconds respectively. Find the length of the train and speed of the train.
  A.  Speed 18 m/sec; Length 155 m
  B.  Speed 22 m/sec; Length 160 m
  C.  Speed 20 m/sec; Length 150 m
  D.  Speed 25 m/sec; Length 140 m
     
   
View Answer

Shortcut:
A train overtakes two persons who are walking in the opposite direction to the train, at the rate of s1 m/sec and s2 m/sec in t1 seconds and t2 seconds respectively. Speed of the train is
s2t2 − s1t1 / t1 − t2
m/sec and the length of the train is
(s2 − s1)t1t2 / t1 − t2
metres.
Here, s1 = 5, s2 = 10, t1 = 6, t2 = 5
Using these values in the shortcut, we get:
Speed of the train =
10 x 5 − 5 x 6 / 6 − 5
= 50 − 30 = 20
Hence, the speed of the train is 20 m/sec.
Length of the train =
(10 − 5)6 x 5 / 6 − 5
= 5 x 6 x 5 = 150
Hence, the length of the train is 150 metres.

28. If a train overtakes two persons what are walking in the same direction of train at the rate of 2 km/hr and 4 km/hr and passes them completely in 9 and 10 seconds. Find the length of train and its speed in km per hour.
  A.  60 m, 32 km/hr
  B.  45 m, 25 km/hr
  C.  55 m, 20 km/hr
  D.  50 m, 22 km/hr
     
   
View Answer

Shortcut:
A train overtakes two persons who are walking in the same direction as the train is moving, at the rate of s1 km/hr and s2 km/hr and passes them completely in t1 seconds and t2 seconds respectively. Speed of the train is
s2t2 − s1t1 / t2 − t1
km/hr and the length of the train is
(s2 − s1)t1t2 / t2 − t1
x
5 / 18
metres.
Here, s1 = 2, s2 = 4, t1 = 9, t2 = 10
Using these values in the shortcut, we get:
Speed of the train =
4 x 10 − 2 x 9 / 10 − 9
= 40 − 18 = 22
Hence, the speed of the train is 22 km/hr.
Length of train =
(4 − 2) x 9 x 10 / 10 − 9
x
5 / 18

=
2 x 9 x 10 / 1
x
5 / 18
= 10 x 5 = 50
Hence, the length of the train is 50 m.

29. A train 100 metres in length passes a pole in 10 seconds and another train of the same length travelling in opposite direction in 8 seconds. Find the speed of the second train.
  A.  50 km/hr
  B.  54 km/hr
  C.  52 km/hr
  D.  56 km/hr
     
   
View Answer

Shortcut:
A train l1 metres in length passes a pole in t1 seconds and another train of the length l2 metres travelling in opposite direction in t2 seconds. Then the speed of the second train is
[
l1
t1 − t2 / t1 x t2
+
l2 / t2
]
m/sec
Here, l1 = 100, l2 = 100, t1 = 10, t2 = 8
Using these values in the shortcut, we get:
Speed of the second train =
[
100 x
10 − 8 / 10 x 8
+
100 / 8
]

=
[
100 x
2 / 10 x 8
+
25 / 2
]

=
100 / 10 x 4
+ 12.5
=
5 / 2
+ 12.5
= 2.5 + 12.5 = 15 m/sec
Converting into m/sec into km/hr:
15 x
18 / 5
= 3 x 18 = 54
Hence, the speed of the train is 54 km/hr.

30. A train crosses 210 metres and 122 metres long bridge in 25 seconds and 17 seconds respectively. Find the length and speed of the train.
  A.  Length 45 metres; Speed 12 m/sec
  B.  Length 55 metres; Speed 11 m/sec
  C.  Length 60 metres; Speed 21 m/sec
  D.  Length 65 metres; Speed 11 m/sec
     
   
View Answer

Shortcut:
If a train crosses l1 metre and l2 metre long bridge or platform or tunnel in t1 sec and t2 seconds respectively, then the length of the train is
l1t2 − l2t1 / t1 − t2
metres and the speed of the train is
l1 − l2 / t1 − t2
m/sec
Here, l1 = 210, l2 = 122, t1 = 25, t2 = 17
Using these values in the shortcut, we get:
Length of the train =
210 x 17 − 122 x 25 / 25 − 17

=
3570 − 3050 / 8
=
520 / 8
= 65
Hence, the length of the train is 65 m.
Speed of the train =
210 − 122 / 25 − 17
=
88 / 8
= 11
Hence, the speed of the train is 11 m/sec.

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