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11. A boy is walking at a speed of 15 km per hour. After every km he takes rest for 15 minutes. How much time will he take to cover a distance of 45 km.
  A.  9 hours
  B.  11 hours
  C.  14 hours
  D.  16 hours
     
   
View Answer

Shortcut:
A person is walking at a speed of a km/hr. After every kilometre, if he takes rest for t hours, then the time he will take to cover a distance of z km is
z / a
+ (z − 1)t hr.
Here, a = 15, z = 45, t =
15 / 60
or
1 / 4

Using these values in the shortcut, we get:
Required time =
45 / 15
+ (45 − 1) x
1 / 4

= 3 + 44 x
1 / 4
= 3 + 11 = 14
Hence, the required time is 14 hr.

12. A boy rides at the rate of 10 km per hour, but stops 15 minutes to change hours at the end of every 6th kilometre. How long will he take to go a distance of 50 kilometres?
  A.  4 hours
  B.  8 hours
  C.  5 hours
  D.  7 hours
     
   
View Answer

Shortcut:
A person is walking at a speed of a km/hr. After every kilometre, if he takes rest for t hours, then the time he will take to cover a distance of z km is
z / a
+ (z − 1)t hr.
Here instead of every kilometre, every 6th kilometre is given. Hence, we should first calcualte number of rest i.e
50 / 6
or 8
Here, a = 10, z = 50, (z − 1) = 8, t =
15 / 60
or
1 / 4

Using these values in the shortcut, we get:
Required time =
50 / 10
+ 8 x
1 / 4
= 5 + 2 = 7
Hence, the required time is 7 hr.

13. A tourist rides his bike at the rate of 15 km per hour. But stops 10 minutes to change bike at the end of every 30th kilometre. How long will he take to go a distance of 210 km?
  A.  15 hours
  B.  12 hours
  C.  18 hours
  D.  10 hours
     
   
View Answer

Shortcut:
A person is walking at a speed of a km/hr. After every kilometre, if he takes rest for t hours, then the time he will take to cover a distance of z km is
z / a
+ (z − 1)t hr.
Here instead of every kilometre, every 30th kilometre is given. Hence, we should first calculate number of rest i.e
210 / 30
or 7 − 1 = 6
Here, a = 15, z = 210, (z − 1) = 6, t =
10 / 60
or
1 / 6

Using these values in the shortcut, we get:
Required time =
210 / 15
+ 6 x
1 / 6
= 14 + 1 = 15
Hence, the required time is 15 hr.

14. A boy covers a certain distance between his hourse and office on scooter. Having an average speed of 15 km/hr, he is late by 10 minutes. However, with a speed of 20 km/hr, he reaches his office 5 minutes earlier. Find the distance between his hourse and office.
  A.  13 km
  B.  15 km
  C.  16 km
  D.  11 km
     
   
View Answer

Shortcut:
A person covers a certain distance between two points. Having an average speed of a km/hr, he is late by a1 hours. However, with a speed of b km/hr he reaches his destination b1 hours earlier. The distance between the two points is
ab(a1 + b1) / b − a
km.
or
Distance =
Products of two speeds / Difference of two speeds
x Difference between arrival time
Here, a = 15, a1 =
10 / 60
or
1 / 6
, b = 20, b1 =
5 / 60
or
1 / 12

Using these values in the shortcut, we get:
Required distance =
15 x 20(1/6 + 1/12) / b − a

=
15 x 20(2 + 1)/12 / 5

=
15 x 20 x 3/12 / 5

=
15 x 20 x 1/4 / 5
= 3 x 5 = 15
Hence, the required distance is 15 km.

15. Raju travels at the rate of 3km/hr and he reaches 15 minutes late. If he travels at the rate of 4 km/hr, he reaches 15 minutes earlier. Find the distance he has to travel.
  A.  5 km
  B.  8 km
  C.  3 km
  D.  6 km
     
   
View Answer

Shortcut:
A person covers a certain distance between two points. Having an average speed of a km/hr, he is late by a1 hours. However, with a speed of b km/hr he reaches his destination b1 hours earlier. The distance between the two points is
ab(a1 + b1) / b − a
km.
or
Distance =
Products of two speeds / Difference of two speeds
x Difference between arrival time
Here, a = 3, a1 =
15 / 60
or
1 / 4
, b = 4, b1 =
15 / 60
or
1 / 4

Using these values in the shortcut, we get:
Required distance =
3 x 4(1/4 + 1/4) / 4 − 3

=
3 x 4(1 + 1)/4 / 1

= 3 x 4 x
2 / 4

= 3 x 4 x
1 / 2
= 3 x 2 = 6
Hence, the required distance is 6 km.

16. A man walking at a speed of 12 km/hr reaches his office 20 minutes late. Next time he increases his speed by 3 km/hr, but still he is late by 10 minutes. Find the distance between his office and house.
  A.  10 km
  B.  13 km
  C.  9 km
  D.  15 km
     
   
View Answer

Shortcut:
A person walking at a speed of a km/hr reaches his destination a1 hours late. Next time he increases his speed by b km/hr, but still he is late by b1 hours. The distance of his destination from his house is (a + b)(a1 − b1)
a / b
km.
or
Distance =
Products of two speeds / Addition of two speeds
x Total time taken
Here, a = 12, a1 =
20 / 60
or
1 / 3
, b = 3, b1 =
10 / 60
or
1 / 6

Using these values in the shortcut, we get:
Required distance = (12 + 3) x
(
1 / 3
−
1 / 6
)
x
12 / 3

= 15 x
(2 − 1) / 6
x 4
= 15 x
1 / 6
x 4 = 5 x 2 = 10
Hence, the required distance is 10 km.

17. A man goes to his office at a speed of 2 km/hr and reaturn to the home at a speed of 3 km/hr. If he takes 10 hours in all, what is the distance between the home and office?
  A.  13 km
  B.  15 km
  C.  12 km
  D.  10 km
     
   
View Answer

Shortcut:
A person goes to a destination at a speed of a km/hr and returns to his place at a speed of b km/hr. If he takes t hours in all, the distance between his place and destination is
ab / a + b
x t km.
or
Distance =
Products of two speeds / Addition of two speeds
x Total time taken
Here, a = 2, b = 3, t = 10
Using these values in the shortcut, we get:
Required distance =
2 x 3 / 2 + 3
x 10
=
6 / 5
x 10 = 6 x 2 = 12
Hence, the required distance is 12 km.

18. A bike completes a certain journey in 5 hours. It covers half the distance at 20 km/hr and the rest at 30km/hr. The total journey is of:
  A.  108 km
  B.  120 km
  C.  125 km
  D.  115 km
     
   
View Answer

Shortcut:
If a man covers a journey in t hours and the first half of a1 km/hr and second half at a2 km/hr, then the distance is
2 x t x a1 x a2 / a1 + a2
km.
Here, a1 = 20, a2 = 30, t = 5
Using these values in the shortcut, we get:
Required distance =
2 x 5 x 20 x 30 / 20 + 30

=
2 x 5 x 20 x 30 / 50
= 2 x 2 x 30 = 120
Hence, the required distance is 120 km.

19. A bike does a journey in 2 hours, the first half at 5 km/hr and the second half at 15 km/hr. Find the distance.
  A.  20 km
  B.  18 km
  C.  12 km
  D.  15 km
     
   
View Answer

Shortcut:
If a man covers a journey in t hours and the first half of a1 km/hr and second half at a2 km/hr, then the distance is
2 x t x a1 x a2 / a1 + a2
km.
Here, a1 = 5, a2 = 15, t = 2
Using these values in the shortcut, we get:
Required distance =
2 x 2 x 5 x 15 / 5 + 15

=
2 x 2 x 5 x 15 / 20
= 15
Hence, the required distance is 15 km.

20. The distance between two stations, Delhi and Agra is 300 km. A train starts at 4 pm from Delhi and moves towards Agra at an average speed of 80 km/hr. Another train starts from Agra at 3:20 pm and moves towards Delhi at an average speed of 120 km/hr. How far from Delhi will the two trains meet and at what time?
  A.  Time taken: 1 hour 12 minutes; Trains Trains meets at 5:24 pm
  B.  Time taken: 2 hour 6 minutes; Trains Trains meets at 5:16 pm
  C.  Time taken: 1 hour 6 minutes; Trains Trains meets at 5:06 pm
  D.  Time taken: 2 hour 6 minutes; Trains Trains meets at 5:26 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 = 80, b = 120, t = 4.00 pm − 3.20 pm = 40 minutes or
40 / 60
or
2 / 3
hours
Using these values in the shortcut, we get:
Required distance = (300 −
2 / 3
x 120) x
80 / 80 + 120

= (300 − 2 x 40) x
80 / 200

= (300 − 80) x
2 / 5

= 220 x
2 / 5
= 44 x 2 = 88
Hence, the required distance is 88 km.
Time taken by X to cover 88 km =
88 / 80
= 1.1 hour or 1 hour 6 minutes
Hence, time taken will be 1hour and 6 minutes and the train meets at 5:06 pm

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