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21. The distance between two stations, Delhi and Agra is 300 km. A train starts at 3:10 pm from Delhi and moves towards Agra at an average speed of 10 km/hr. Another train starts from Agra at 3:50 pm and moves towards Delhi at an average speed of 30 km/hr. How far from Delhi will the two trains meet and at what times.
  A.  From Delhi trains will meet after: 80 km; Trains meet at: 11:10 pm
  B.  From Delhi trains will meet after: 90 km; Trains meet at: 11:30 pm
  C.  From Delhi trains will meet after: 70 km; Trains meet at: 10:20 pm
  D.  From Delhi trains will meet after: 75 km; Trains meet at: 11:40 pm
     
   
View Answer

Shortcut:
The distance between two stations, X and Y is D km. A train starts from X and moves towards Y at an average speed of a km/hr. If an another train starts from Y, t hours earlier than the train at X, and moves towards X at an average speed of b km/hr, then the distance from X where the two trains will meet, is (D + tb) x
a / a + b
km.
Here, D = 300, a = 10, b = 30, t = 3.50 pm − 3.10 pm = 40 minutes or
40 / 60
or
2 / 3
hours
Using these values in the shortcut, we get:
Required distance = (300 +
2 / 3
x 30) x
10 / 10 + 30

= (300 + 2 x 10) x
10 / 40
= (300 − 20) x
1 / 4

= 320 x
1 / 4
= 80
Hence, the required distance is 80 km.
Time taken by X to cover 80 km =
80 / 10
= 8 hours
Hence, train will meet after 80 km from Delhi at 11:10 pm

22. A train which travels at the uniform rate of 5 m/sec leaves Gurgaon for Ghaziabad at 6 am. At what distance from Gurgaon will it meet a train which leaves Ghaziabad for Gurgaon at 6:20 am and traves one-third faster than it does, the distance from Gurgaon to Ghaziabad is 41 km?
  A.  18 km
  B.  21 km
  C.  24 km
  D.  15 km
     
   
View Answer

Shortcut:
The distance between two stations, X and Y is D km. A train starts from X and moves towards Y at an average speed of a km/hr. If an another train starts from Y, t hours earlier than the train at X, and moves towards X at an average speed of b km/hr, then the distance from X where the two trains will meet, is (D + tb) x
a / a + b
km.
a = 5 m/sec or 5 x
18 / 5
= 18 km/hr
b = 18 +
1 / 3
x 18 = 18 + 6 = 24 km/hr
Here, D = 41, a = 18, b = 24, t = 6.20 am − 6.00 am = 20 minutes or
20 / 60
or
1 / 3
hours
Using these values in the shortcut, we get:
Required distance = (41 +
1 / 3
x 24) x
18 / 18 + 24

= (41 + 8) x
18 / 42
= 49 x
3 / 7
= 7 x 3 = 21
Hence, the required distance is 21 km.

23. A train going 40 km /hr leaves Delhi for Ajmer 500 km far at 9 pm. Another train going 60 km an hour leaves Ajmer for Delhi at the same time, when and where will they pass each other?
  A.  230 km from Delhi, 1:15 am
  B.  200 km from Delhi, 2:00 am
  C.  190 km from Delhi, 1:00 am
  D.  210 km from Delhi, 1:30 am
     
   
View Answer

Shortcut:
The distance between two stations, X and Y is D km. A train starts from X and moves towards Y at an average speed of a km/hr. If an another train starts from Y, t hours earlier than the train at X, and moves towards X at an average speed of b km/hr, then the distance from X where the two trains will meet, is (D + tb) x
a / a + b
km.
Here, D = 500, a = 40, b = 60, t = 0 hours
Using these values in the shortcut, we get:
Required distance = (500 + 0 x 30) x
40 / 40 + 60

= 500 x
40 / 100
= 5 x 40 = 200
Hence, the required distance is 200 km.
Time taken by X to cover 200 km =
200 / 40
= 5 hours
Hence, train will meet after 200 km from Delhi at 2:00 am

24. A starts from Agra to Delhi and walks at the rate of 24 km/hr. B starts from Delhi 4 hours later and walks towards Agra at the rate of 16 km/hr, if they meet in 18 hours after B started, find the distance from Agra to Delhi.
  A.  816 km
  B.  800 km
  C.  850 km
  D.  844 km
     
   
View Answer

Shortcut:
The distance between two stations, X and Y is D km. A train starts from X and moves towards Y at an average speed of a km/hr. If an another train starts from Y, t hours earlier than the train at X, and moves towards X at an average speed of b km/hr, then the distance from X where the two trains will meet, is (D + tb) x
a / a + b
km.
The distance from B where the two meet = 16 x 18 = 288 km
∴ the distance from A where they meet = D − 288 km
Here, D = ?, a = 24, b = 16, t = 4
Using these values in the shortcut, we get:
Required distance = (D + 4 x 16) x
24 / 24 + 16

D − 288 = D + 64 x
24 / 40

40 x (D − 288) = (D + 64) x 24
5 x (D − 288) = (D + 64) x 3
5D − 1440 = 3D + 192
5D − 3D = 1440 + 192
2D = 1632
D =
1632 / 2
= 816
Hence, the required distance is 816 km.

25. Walking
2 / 3
of his usual speed, a person gets 10 minutes late to his office. Find his usual time to cover the distance.
  A.  30 minutes
  B.  20 minutes
  C.  15 minutes
  D.  45 minutes
     
   
View Answer

Shortcut:
If the new speed of a person is
c / d
of the usual speed, then the change in time taken to cover the same distance is
(
d / c
− 1
)
x usual time or, usual time is
Change in time / d/c − 1
hours.
Note: A person improves his timing or becomes late depends on the −ve or +ve sign respectively of the above expression.
Here,
c / d
=
2 / 3
, change in time = 10
Using these values in the shortcut, we get:
Usual time =
10 / d/c − 1

=
10 / 3/2 − 1

=
10 / (3 − 2)/2

=
10 x 2 / 1
= 10 x 2 = 20
Hence, the usual time is 20 minutes.

26. Running
3 / 2
of his usual speed, a person improves his timing by 10 minutes. Find his usual time to cover the distance.
  A.  30 minutes
  B.  25 minutes
  C.  35 minutes
  D.  28 minutes
     
   
View Answer

Shortcut:
If the new speed of a person is
c / d
of the usual speed, then the change in time taken to cover the same distance is
(
d / c
− 1
)
x usual time or, usual time is
Change in time / d/c − 1
hours.
Note: A person improves his timing or becomes late depends on the -ve or +ve sign respectively of the above expression.
Here,
c / d
=
3 / 2
, change in time = 10
Using these values in the shortcut, we get:
Usual time =
10 / 2/3 − 1

=
10 / (2 − 3)/3

=
10 x 3 / −1
= −10 x 3 = −30
Here −ve sign shows that the person improves his timing. Hence in the question improved time has been given.
Hence, the usual time is 30 minutes.

27. A train travelling 15 km/hr leaves Delhi at 10 am and another train travelling 25 km/hr starts at 2 pm in the same direction. How many km from Delhi will they be together?
  A.  130 km
  B.  110 km
  C.  125 km
  D.  150 km
     
   
View Answer

Shortcut:
If a train travelling a km/hr leaves a place and t hours later another train travelling b km/hr, where b > a, in the same direction, then they will be together after travelling
a x b x t / b − a
km from the starting place.
Here, a = 15, b = 25, t = 2 pm − 10 am = 4 hours
Using these values in the shortcut, we get:
Required distance =
15 x 25 x 4 / 25 − 15

=
15 x 25 x 4 / 10
= 3 x 25 x 2 = 150
Hence, the trains will be together after 150 km from Delhi.

28. A train leaves Calcutta at 8:30 am and travels 20 km an hour, another train leaves Calcutta at noon and travels 27 km/hr, when and where will the second train overtake the first?
  A.  210 km; 10:00 am
  B.  190 km; 9:30 pm
  C.  270 km; 1:30 am
  D.  160 km; 8:00 pm
     
   
View Answer

Shortcut:
If a train travelling a km/hr leaves a place and t hours later another train travelling b km/hr, where b > a, in the same direction, then they will be together after travelling
a x b x t / b − a
km from the starting place.
Here, a = 20, b = 27, t = 12 pm − 8:30 am = 3 hours 30 minutes or
7 / 2
hours
Using these values in the shortcut, we get:
Required distance =
20 x 27 x 7/2 / 27 − 20

=
10 x 27 x 7 / 7
= 10 x 27 = 270
Hence, the second train will overtake the first train after 270 km from Calcutta.
Required time =
270 / 20
=
27 / 2
= 13.5 hours
∴ Required answer = 12 noon + 13.5 hours = 1:30 am

29. A man starts at 6 am and travels at the rate of 2
1 / 4
km/hr. At 7:15 am a coach starts from the same place and follows the man travelling at the rate of 3
1 / 2
km/hr, at what o'clock will the coach overtake the man?
  A.  9:00 am
  B.  9:30 am
  C.  8:30 am
  D.  8:45 am
     
   
View Answer

Shortcut:
If a train travelling a km/hr leaves a place and t hours later another train travelling b km/hr, where b > a, in the same direction, then they will be together after travelling
a x b x t / b − a
km from the starting place.
Here, a = 2
1 / 4
or
9 / 4
, b = 3
1 / 2
or
7 / 2
, t = 7:15 am − 6:00 am = 1 hour 15 minutes or 5/4 hours
Using these values in the shortcut, we get:
Required distance =
9/4 x 7/2 x 5/4 / 7/2 − 9/4

=
9/4 x 7/2 x 5/4 / 5/4
=
9 x 7 / 4 x 2
=
63 / 8

Hence, the coach train will overtake the man after
63 / 8
km.
Required time =
63/8 / 9/4
=
63 x 4 / 8 x 9
=
7 / 2
= 3.5 hours or 3 hours 30 minutes
∴ Required answer = 6 am + 3.30 hours = 9:30 am

30. A starts from Delhi to Noida (12 km) at 6 am walking 2 km/hr. B starts from Delhi an hour later and reaches Noida one hour before A, where did they meet?
  A.  6 km from Delhi
  B.  3 km from Delhi
  C.  8 km from Delhi
  D.  5 km from Delhi
     
   
View Answer

Shortcut:
If a train travelling a km/hr leaves a place and t hours later another train travelling b km/hr, where b > a, in the same direction, then they will be together after travelling
a x b x t / b − a
km from the starting place.
Time taken by A to cover the distance of 12 km =
12 / 2
= 6 hours
Time taken by B to cover the same distance = 6 − 2 = 4 hours
∴ b =
12 / 4
= 3 km/hr
Here, a = 2, b = 3, t = 1 hour
Using these values in the shortcut, we get:
Required distance =
2 x 3 x 1 / 3 − 2

=
2 x 3 x 1 / 1
= 2 x 3 = 6
Hence, they will meet 6 km from Delhi.

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