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41. Two places X and Y are 162 km apart. A train leaves X for Y and at the same time another train leaves Q for P. Both the trains meet 6 hrs after they start moving. If the train trvelling from X to Y travels 8 km/hr faster than the other train, find the speed of the two trains.
  A.  Faster's speed 27.5 km/hr; Slower's speed 10.5 km/hr
  B.  Faster's speed 20.5 km/hr; Slower's speed 11.5 km/hr
  C.  Faster's speed 15 km/hr; Slower's speed 8.5 km/hr
  D.  Faster's speed 17.5 km/hr; Slower's speed 9.5 km/hr
     
   
View Answer

Shortcut:
Two places X and Y are Z km apart. A train leaves X for Y and at the same time another train leaves Y for X. Both the trains meet 't' hours after they start moving. If the train travelling from X to Y travels 'a' km/hr faster or slower than the other train, then the speed of the faster train is
Z + ta / 2t
km/hr and the slower train is
Z − ta / 2t
km/hr.
Here, Z = 162, t = 6, a = 8
Using these values in the shortcut, we get:
Speed of faster train =
162 + (6 x 8) / 2 x 6

=
162 + 48 / 2 x 6
=
210 / 2 x 6
= 17.5
Hence, the speed of faster train is 17.5 km/hr.
Speed of slower train =
162 − (6 x 8) / 2 x 6

=
162 − 48 / 12
=
114 / 12
= 9.5
Hence, the speed of the slower train is 9.5 km/hr.

42. Two trains P and Q start from Mumbai and Goa to Goa and Mumbai respectively. After passing each other they take 4 hours 48 minutes and 3 hours and 20 minutes to reach Goa and Mumbai respectively. If the train from Mumbai is moving at 45 km/hr then find the speed of the other train.
  A.  54 km/hr
  B.  55 km/hr
  C.  50 km/hr
  D.  60 km/hr
     
   
View Answer

Shortcut:
Two trains X and Y start from M and N towards N and M respectively. After passing each other they take t1 hours and t2 hours to reach M and N respectively. If the train from N is moving a km/hr, then the speed of the other train is a
√(
t1 / t2
)
km/hr
Here, a = 45, t1 = 4
4 / 5
or
24 / 5
, t2 = 3
1 / 3
or
10 / 3

Using these values in the shortcut, we get:
Speed of other train = 45
√(
24/5 / 10/3
)

= 45
√(
24 x 3 / 10 x 5
)
= 45
√(
3 x 4 x 3 / 5 x 5
)
= 45 x
3 x 2 / 5
= 9 x 6 = 54
Hence, the speed of other train is 54 km/hr.

43. The speeds of two trains are in the ratio 3:4. They are going in opposite directions along parallel tracks. If each takes 3 seconds to cross a telegraph post, find the time taken by the trains to cross each other completely?
  A.  5 seconds
  B.  6 seconds
  C.  3 seconds
  D.  8 seconds
     
   
View Answer

Shortcut:
The speed of two trains are in the raio a : b. They are moving in the opposite directions on parallel tracks. The first train crosses a telegraph pole in 'm' seconds whereas the second train crosses a telegraph pole in 'n' seconds. Time taken by the trains to cross each other completely is in
ma + nb / a + b
seconds.
Here, a = 3, b = 4, m = n = 3
Using these values in the shortcut, we get:
Required time =
3 x 3 + 3 x 4 / 3 + 4

=
9 + 12 / 7
=
21 / 7
= 3
Hence, the trains will cross each other in 3 seconds.

44. The speeds of two trains are in the ratio of 7:9. They are moving on the opposite directions. The first train crosses a telegraph pole in 4 seconds whereas the second train crosses the pole in 6 seconds. Find the time taken by the trains to cross each other completely.
  A.  44/7 seconds
  B.  42/9 seconds
  C.  45/8 seconds
  D.  41/8 seconds
     
   
View Answer

Shortcut:
The speed of two trains are in the raio a : b. They are moving in the opposite directions on parallel tracks. The first train crosses a telegraph pole in 'm' seconds whereas the second train crosses a telegraph pole in 'n' seconds. Time taken by the trains to cross each other completely is in
ma + nb / a + b
seconds.
Here, a = 7, b = 9, m = 4, n = 6
Using these values in the shortcut, we get:
Required time =
4 x 7 + 6 x 9 / 7 + 9

=
28 + 54 / 16
=
82 / 16
=
41 / 8

Hence, the trains will cross each other in
41 / 8
seconds.

45. A train with speed 90 km/hr crosses a bridge in 36 seconds. Another train 100 metres shorter crosses the same bridge at 45 km/hr. Find the time taken by the second train to cross the bridge.
  A.  74 seconds
  B.  64 seconds
  C.  54 seconds
  D.  65 seconds
     
   
View Answer

Shortcut:
A train with 'a' km/hr crosses a bridge in 'M' seconds. Another train 'N' metres shorter crosses the same bridge at 'b' km/hr. Time taken by the second train to cross the bridge is M
(
a / b
)
−
(
N / b x 5/18
)
seconds.
Here, a = 90, M = 36, N = 100, b = 45
Using these values in the shortcut, we get:
Required time = 36
(
90 / 45
)
−
(
100 / 45 x 5/18
)

= 36 x 2 −
100 x 18 / 45 x 5
= 72 − 4 x 2 = 72 − 8 = 64
Hence, the required time to cross the bridge is 64 seconds.

46. Train X leaves Delhi for Mumbai at 11 am, running at the speed of 60 km/hr. Train Y leaves Delhi for Mumbai by the same route at 2 pm on the same day, having speed 72 km/hr. At what time the two trains meet each other?
  A.  5 am on next day
  B.  10 am on next day
  C.  8 am on next day
  D.  3 am on next day
     
   
View Answer

Distance covered by train A before the train B leaves Mumbai = 60 x 3 = 180 km
∴ Time taken to cross each other =
180 / 12
= 15 hours
∴ Required time = 2 pm + 15 hours = 5 am next day

47. A train leaves Goa at 4 pm and travels at speed of 30 km/hr. The goods train leaves Goa at 9 pm and travels on the parallel line with speed of 45 km/hr, When will the second train overtake the first?
  A.  455 km from Goa
  B.  400 km from Goa
  C.  450 km from Goa
  D.  500 km from Goa
     
   
View Answer

The first train has started 5 hours before the second.
Therefore, (30 x 5 = 150) km away when the second train starts.
Therefore the second train has to gain 150 km on the first, at the rate of 15 (45 − 30) km an hour.
Second train gains 15 km in 1 hour on the first.
∴ Second train gains 150 km in 10 hours on the first.
∴ the time required is 10 hours after the second train starts.
Hence, the second overtakes the first at a distance of 45 x 10 = 450 km from Goa.

48. A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, find the length of the platform.
  A.  280 metres
  B.  220 metres
  C.  200 metres
  D.  240 metres
     
   
View Answer

Shortcut:
A train passes by a stationary man standing on the platform or a pole in t1 seconds and passes by the platform completely in t2 seconds. If the length of the platform is L metres, then the length of the train is
t1 x L / t2 − t1
metres and the speed of the train is
L / t2 − t1
m/sec.
Speed of the train = 54 km/hr = 54 x
5 / 18
= 15 m/sec
Here, t1 = 20, t2 = 36, L = ?
Using these values in the shortcut, we get:
15 =
L / 36 − 20

15 =
L / 16

L = 15 x 16 = 240
Hence, the length of the train is 240 metres.

49. A train 75 metres long overtook a man who was walking at speed of 6 km/hr and passed him in 18 seconds. Again, the train overtook a second person in 15 seconds. What was the speed of second person?
  A.  9 km/hr
  B.  3 km/hr
  C.  5 km/hr
  D.  13 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 = ?, s2 = 6, t1 = 15, t2 = 18
Using these values in the shortcut, we get:
Length of the train =
(6 − s1) x 15 x 18 / 18 − 15
x
5 / 18

75 =
(6 − s1) x 15 x 18 / 3
x
5 / 18

75 = (6 − s1) x 5 x 5
75 = 150 − 25s1
25 s1 = 150 − 75
s1 =
75 / 25
= 3
Hence, the speed of second person is 3 km/hr.

50. Two trains of length 110 metres and 90 metres are running on parallel lines in same direction with speed of 35 km/hr and 40 km/hr respectively. When will they pass each other?
  A.  144 seconds
  B.  154 seconds
  C.  134 seconds
  D.  140 seconds
     
   
View Answer

Relative speed = (40 − 35) = 5 km/hr
Converting km/hr into m/sec
5 x
5 / 15
=
25 / 18
m/sec
Time taken by the trains in passing each other = Time taken to cover ( 110 +90) m at
25 / 18
m/sec =
200 x 18 / 25
= 144 sec

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