1. A train of length 150 meters takes 40.5 seconds to cross a tunnel of length 300 meters. What is the speed of the train in km/hr?
a) 13.33
b) 26.67
c) 40
d) 66.67
Discussion
Explanation:
$$\eqalign{ & {\text{Speed = }}\left( {\frac{{150 + 300}}{{40.5}}} \right)m/\sec \cr & = \left( {\frac{{450}}{{40.5}} \times \frac{{18}}{5}} \right)km/hr \cr & = 40km/hr \cr} $$
2. A 280 meter long train crosses a platform thrice its length in 50 seconds. What is the speed of the train in km/hr?
a) 60.48
b) 64.86
c) 80.64
d) 82.38
Discussion
Explanation:
$$\eqalign{ & {\text{Length of train}} = 280 m \cr & {\text{Length of platform}} \cr & {\text{ = (3}} \times {\text{280) m = 840m}} \cr & {\text{Speed of train}} \cr & {\text{ = }}\left( {\frac{{280 + 840}}{{50}}} \right)m/\sec \cr & = \frac{{1120}}{{50}}m/\sec \cr & = \left( {\frac{{1120}}{{50}} \times \frac{{18}}{5}} \right)km/hr \cr & = 80.64\,km/hr \cr} $$
3. A train 110 meters long is running with a speed of 60 kmph. In what time will it pass a man who is running at 6 kmph in the direction opposite to that in which the train is going?
a) 5 sec
b) 6 sec
c) 7 sec
d) 10 sec
Discussion
Explanation:
$$\eqalign{ & {\text{Speed of train relative to man}} \cr & {\text{ = }}\left( {60 + 6} \right){\text{km/hr}} \cr & = 66\,{\text{km/hr}} \cr & = \left( {66 \times \frac{5}{{18}}} \right)m/\sec \cr & = \left( {\frac{{55}}{3}} \right)m/\sec \cr & {\text{Time taken to pass the man}} = \left( {110 \times \frac{3}{{55}}} \right)\sec \cr & = 6\,\sec \cr} $$
4. Two trains A and B start running together from the same point in the same direction, at the speed of 60 kmph and 72 kmph respectively. If the length of each of the trains is 240 meters, how long will it take for B to cross train A?
a) 1 min 12 sec
b) 1 min 24 sec
c) 2 min 12 sec
d) 2 min 24 sec
Discussion
Explanation:
$$\eqalign{ & {\text{Relative speed}} = (72 - {\text{60) km/hr}} \cr & {\text{ = 12 km/hr}} \cr & = \left( {12 \times \frac{5}{{18}}} \right)m/\sec \cr & = \left( {\frac{{10}}{3}} \right)m/\sec \cr & {\text{Total distance covered}} \cr & {\text{ = Sum of lengths of trains}} \cr & {\text{ = (240 + 240) m}} \cr & {\text{ = 480 m}} \cr & {\text{Time taken}} \cr & {\text{ = }}\left( {480 \times \frac{3}{{10}}} \right)\sec \cr & = 144\sec \cr & = 2\min \,24sec \cr} $$
5. Two trains are moving in opposite directions @60 km/hr and 90 km/hr. Their lengths are 1.10 km and 0.9 km respectively. The time taken by the slower train to cross the faster train in second is?
a) 36
b) 45
c) 48
d) 49
Discussion
Explanation:
$$\eqalign{ & {\text{Relative speed}} \cr & {\text{ = (60 + 90) km/hr}} \cr & {\text{ = }}\left( {150 \times \frac{5}{{18}}} \right){\text{m/sec}} \cr & {\text{ = }}\left( {\frac{{125}}{3}} \right){\text{m/sec}} \cr & {\text{Distance coverd}} \cr & {\text{ = (1}}{\text{.10 + 0}}{\text{.9)km}} \cr & {\text{ = 2 km}} \cr & {\text{ = 2000 m}}{\text{}} \cr & {\text{Required time}} = \left( {2000 \times \frac{3}{{125}}} \right)\sec \cr & = 48{\text{ sec}}\cr} $$
6. A train B speeding with 120 kmph crosses another train C running in the same direction, in 2 minutes. If the lengths of the trains B and C be 100m and 200m respectively, what is the speed (in kmph) of the train C?
a) 111 kmph
b) 123 kmph
c) 127 kmph
d) 129 kmph
Discussion
Explanation:
$$\eqalign{ & {\text{Relative speed of the trains }} \cr & {\text{ = }}\left( {\frac{{100 + 200}}{{2 \times 60}}} \right){\text{m/sec}} \cr & {\text{ = }}\left( {\frac{5}{2}} \right){\text{m/sec}} \cr & {\text{Speed of train B}} \cr & {\text{ = 120 kmph}} \cr & = \left( {120 \times \frac{5}{{18}}} \right){\text{m/sec}} \cr & {\text{ = }}\left( {\frac{{100}}{3}} \right){\text{m/sec}} \cr & {\text{Let the speed of second train be }}x{\text{ m/sec}} \cr & {\text{Then, }} \frac{{100}}{3} - x = \frac{5}{2} \cr & \Rightarrow x = \left( {\frac{{100}}{3} - \frac{5}{2}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{{185}}{6}} \right){\text{m/sec}} \cr & {\text{Speed of second train}} \cr & {\text{ = }}\left( {\frac{{185}}{6} \times \frac{{18}}{5}} \right){\text{ kmph}} \cr & {\text{ = 111 kmph}} \cr} $$
7. What is the speed of a train if it overtakes two persons who are walking in the same direction at the rate of a m/s and (a + 1) m/s and passes them completely in b seconds and (b + 1) seconds respectively?
a) (a + b) m/s
b) (a + b + 1) m/s
c) (2a + 1) m/s
d) $$\frac{{2{\text{a}} + 1}}{2}$$ m/s
Discussion
Explanation:
$$\eqalign{ & {\text{Let the length of the train be }}x{\text{ metres}} \cr & {\text{and its speed be }}y{\text{ m/s}} \cr & {\text{ }}\frac{x}{{y - a}}{\text{ = b}}\,\,{\text{and}}\, \cr & \,\frac{x}{{y - \left( {a + 1} \right)}} = \left( {b + 1} \right) \cr & \Leftrightarrow {\text{ }}x{\text{ = }}b\left( {y - a} \right){\text{ and}} \cr & \,\,\,\,\,\,\,\,\,\,{\text{ }}x = \left( {b + 1} \right)\left( {y - a - 1} \right) \cr & \Leftrightarrow b\left( {y - a} \right) = \left( {b + 1} \right)\left( {y - a - 1} \right) \cr & \Leftrightarrow by - ba = by - ba - b + y - a - 1 \cr & y = \left( {a + b + 1} \right) \cr} $$.
8. A train passes a 50 meter long platform in 14 seconds and a man standing on platform 10 seconds.The speed of the train is?
a) 24 km/hr
b) 36 km/hr
c) 40 km/hr
d) 45 km/hr
Discussion
Explanation:
$$\eqalign{ & {\text{Distance travelled in 14 sec}} \cr & {\text{ = 50 + }}l \cr & {\text{Distance travelled in 10 sec}} = l \cr & {\text{Speed of train}} \cr & {\text{ = }}\frac{{50}}{{14 - 10}}{\text{m/sec}} \cr & {\text{ = }}\frac{{50}}{4} \times \frac{{18}}{5}{\text{km/hr}} \cr & {\text{ = 45 km/hr}} \cr} $$
9. A train is moving at a speed of 132 km/hr. If the length of the train is 110 meters, how long it will take to cross a railway platform 165 meter long?
a) 5 second
b) 7.5 second
c) 10 second
d) 15 second
Discussion
Explanation:
$$\eqalign{ & {\text{Speed = 132 km/hr }} \cr & {\text{ = 132}} \times \frac{5}{{18}}{\text{m/sec}} \cr & {\text{ = }}\frac{{110}}{3}m/\sec \cr & T = \frac{D}{S} \cr & \,\,\,\,\,\, = \frac{{110 + 165}}{{\frac{{100}}{3}}} \cr & \,\,\,\,\,\, = \frac{{3\left( {275} \right)}}{{110}} \cr & \,\,\,\,\,\, = 7.5\sec \cr} $$
10. A train of length 500 feet crosses a platform of length 700 feet in 10 seconds. The speed of the train is?
a) 70 ft/second
b) 85 ft/second
c) 100 ft/second
d) 120 ft/second
Discussion
Explanation:
$$\eqalign{ & {\text{Speed of the train}} \cr & {\text{ = }}\frac{{700 + 500}}{{10}} \cr & {\text{ = 120 ft/second}} \cr} $$
11. A train running at the speed of 60 km/hr crosses a pole in 9 seconds. What is the length of the train?
a) 120 metres
b) 180 metres
c) 324 metres
d) 150 metres
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}} = \left( {60 \times \frac{5}{{18}}} \right){\text{m/sec}} = {\frac{{50}}{3}} {\text{m/sec}} \cr & {\text{Length}}\,{\text{of}}\,{\text{the}}\,{\text{train}} = \left( {{\text{Speed}} \times {\text{Time}}} \right) \cr & {\text{Length}}\,{\text{of}}\,{\text{the}}\,{\text{train}} \cr & = \left( {\frac{{50}}{3} \times 9} \right)m = 150m \cr} $$
12. A train 125 m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is:
a) 45 km/hr
b) 50 km/hr
c) 54 km/hr
d) 55 km/hr
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{relative}}\,{\text{to}}\,{\text{man}} = {\frac{{125}}{{10}}} {\text{ m/sec}} \cr & = {\frac{{25}}{2}} {\text{ m/sec}} \cr & = {\frac{{25}}{2} \times \frac{{18}}{5}} {\text{ km/hr}} \cr & = 45\,{\text{km/hr}} \cr & {\text{Let}}\,{\text{the}}\,{\text{speed}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{be}}\,x\,{\text{km/hr}}. \cr & \text{Then, relative speed} = \left( {x - 5} \right)\,{\text{km/hr}} \cr & x - 5 = 45 \cr & \Rightarrow x = 50\,{\text{km/hr}} \cr} $$
13.The length of the bridge, which a train 130 metres long and travelling at 45 km/hr can cross in 30 seconds, is
a) 200 m
b) 225 m
c) 245 m
d) 250 m
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}} = {45 \times \frac{5}{{18}}} \,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\frac{{25}}{2}} \,{\text{m/sec}} \cr & {\text{Time}} = 30\,{\text{sec}} \cr & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{bridge}}\,{\text{be}}\,x\,{\text{metres}} \cr & {\text{Then}},\,\frac{{130 + x}}{{30}} = \frac{{25}}{2} \cr & 2\left( {130 + x} \right) = 750 \cr & x = 245\,m \cr} $$
14. Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is:
a) 1 : 3
b) 3 : 2
c) 3 : 4
d) None of these
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{speeds}}\,{\text{of}}\,{\text{the}}\,{\text{two}}\,{\text{trains}}\,{\text{be}}\,x\,{\text{m/sec}} \cr & {\text{and}}\,y\,{\text{m/sec}}\,{\text{respectively}}. \cr & {\text{Then,}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{first}}\,{\text{train}} = 27x\,{\text{metres}}, \cr & {\text{and}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{second}}\,{\text{train}} = 17y\,{\text{metres}}. \cr & \frac{{27x + 17y}}{{x + y}} = 23 \cr & 27x + 17y = 23x + 23y \cr & 4x = 6y \cr & \frac{x}{y} = \frac{3}{2} \cr} $$
15. 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, what is the length of the platform?
a) 120 m
b) 240 m
c) 300 m
d) None of these
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}} = {54 \times \frac{5}{{18}}} \,{\text{m/sec}} = 15\,{\text{m/sec}} \cr & {\text{Length}}\,{\text{of}}\,{\text{the}}\,{\text{train}} = \left( {15 \times 20} \right){\text{m}} = 300\,{\text{m}} \cr & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{platform}}\,{\text{be}}\,x\,{\text{metres}} \cr & {\text{Then}},\,\frac{{x + 300}}{{36}} = 15 \cr & x + 300 = 540 \cr & x = 240\,{\text{m}} \cr} $$
16. Two, trains, one from Howrah to Patna and the other from Patna to Howrah, start simultaneously. After they meet, the trains reach their destinations after 9 hours and 16 hours respectively. The ratio of their speeds is
a) 2 : 3
b) 4 : 3
c) 6 : 7
d) 9 : 16
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{us}}\,{\text{name}}\,{\text{the}}\,{\text{trains}}\,{\text{as}}\,{\text{A}}\,{\text{and}}\,{\text{B}}{\text{.}}\,{\text{Then}}, \cr & \left( {{\text{A's}}\,{\text{speed}}} \right):\left( {{\text{B's}}\,{\text{speed}}} \right) \cr & = \sqrt b :\sqrt a \cr & = \sqrt {16} :\sqrt 9 \cr & = 4:3\, \cr} $$
17. A 100 m long train is going at a speed of 60 km/hr. It will cross a 140 m long railway bridge in-
a) 3.6 sec
b) 7.2 sec
c) 14.4 sec
d) 21.6 sec
Discussion
Explanation:
$$\eqalign{ & {\text{Speed }} \cr & {\text{ = }}\left( {60 \times \frac{5}{{18}}} \right){\text{m/sec}} \cr & {\text{ = }}\frac{{50}}{3}{\text{ m/sec}} \cr & {\text{Total distance covered}} \cr & {\text{ = (100 + 140) m = 240 m}} \cr & {\text{Required time}} = \left( {240 \times \frac{3}{{50}}} \right){\text{sec}} \cr & {\text{ = }}\frac{{72}}{5}{\text{sec}} \cr & {\text{ = 14}}{\text{.4 sec}} \cr} $$
18. A train 132 m long passes a telegraph pole in 6 seconds. Find the speed of the train?
a) 70 km/hr
b) 72 km/hr
c) 79.2 km/hr
d) 80 km/hr
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}} = \left( {\frac{{132}}{6}} \right){\text{m/sec}} \cr & {\text{ = }}\left( {22 \times \frac{{18}}{5}} \right){\text{km/sec}} \cr & {\text{ = 79}}{\text{.2 km/hr}} \cr} $$
19. A train running at the speed of 60 kmph crosses a 200 m long platform in 27 seconds. What is the length of the train?
a) 200 meters
b) 240 meters
c) 250 meters
d) 450 meters
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}} = \left( {60 \times \frac{5}{{18}}} \right){\text{m/sec}} \cr & {\text{ = }}\frac{{50}}{3}{\text{m/sec}} \cr & {\text{Time = 27 sec}}{\text{.}} \cr & {\text{Let the length of the train be }}x{\text{ metres}}{\text{.}} \cr & {\text{Then,}}\frac{{x + 200}}{{27}}{\text{ = }}\frac{{50}}{3}{\text{ }} \cr & \Leftrightarrow x + 200 = \left( {\frac{{50}}{3} \times 27} \right) = 450 \cr & \Leftrightarrow x = 450 - 200 = 250{\text{ metres}} \cr} $$
20. A train running at a speed of 90 km/hr crosses a platform double its length in 36 seconds. What is the length of the platform in meters?
a) 200
b) 300
c) 450
d) None of these
Discussion
Explanation:
$$\eqalign{ & {\text{Let the length of the train be x metres}}{\text{.}} \cr & {\text{Then, length of the platform = (2}}x{\text{) metres}}{\text{.}} \cr & {\text{Speed of the train}} \cr & {\text{ = }}\left( {90 \times \frac{5}{{18}}} \right)m/\sec \cr & = 25m/sec \cr & \frac{{x + 2x}}{{25}} = 36 \cr & 3x = 900 \cr & x = 300 \cr & {\text{Hence, length of platform}} \cr & {\text{ = }}2x = \left( {2 \times 300} \right){\text{m}} = 600{\text{m}} \cr} $$
21. A 270 metres long train running at the speed of 120 kmph crosses another train running in opposite direction at the speed of 80 kmph in 9 seconds. What is the length of the other train?
a) 230 m
b) 240 m
c) 260 m
d) 320 m
Discussion
Explanation:
$$\eqalign{ & {\text{Relative}}\,{\text{speed}} = \left( {120 + 80} \right)\,{\text{km/hr}} \cr & = {200 \times \frac{5}{{18}}} \,{\text{m/sec}} \cr & = {\frac{{500}}{9}} \,{\text{m/sec}} \cr & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{other}}\,{\text{train}}\,{\text{be}}\,{\text{x}}\,{\text{metres}}{\text{.}} \cr & {\text{Then,}}\,\frac{{x + 270}}{9} = \frac{{500}}{9} \cr & x + 270 = 500 \cr & x = 230 \cr} $$
22. A goods train runs at the speed of 72 kmph and crosses a 250 m long platform in 26 seconds. What is the length of the goods train?
a) 230 m
b) 240 m
c) 260 m
d) 270 m
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}} = {72 \times \frac{5}{{18}}} \,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 20\,{\text{m/sec}} \cr & {\text{Time}} = 26\,{\text{sec}} \cr & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{be}}\,x\,{\text{metres}}{\text{.}} \cr & {\text{Then}},\,\frac{{x + 250}}{{26}} = 20 \cr & x + 250 = 520 \cr & x = 270 \cr} $$
23. Two trains, each 100 m long, moving in opposite directions, cross each other in 8 seconds. If one is moving twice as fast the other, then the speed of the faster train is:
a) 30 km/hr
b) 45 km/hr
c) 60 km/hr
d) 75 km/hr
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{speed}}\,{\text{of}}\,{\text{the}}\,{\text{slower}}\,{\text{train}}\,{\text{be}}\,x\,{\text{m/sec}} \cr & {\text{Then,}}\,{\text{speed}}\,{\text{of}}\,{\text{the}}\,{\text{faster}}\,{\text{train}} = 2x\,{\text{m/sec}} \cr & {\text{Relative}}\,{\text{speed}} = \,\left( {x + 2x} \right)\,{\text{m/sec}} = 3x\,{\text{m/sec}} \cr & \frac{{ {100 + 100} }}{8} = 3x \cr & 24x = 200 \cr & x = \frac{{25}}{3} \cr & {\text{So,}}\,{\text{speed}}\,{\text{of}}\,{\text{the}}\,{\text{faster}}\,{\text{train}}\, = \frac{{50}}{3}\,{\text{m/sec}} \cr & = {\frac{{50}}{3} \times \frac{{18}}{5}} \,{\text{km/hr}} \cr & = 60\,{\text{km/hr}} \cr} $$
24.Two trains 140 m and 160 m long run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time (in seconds) which they take to cross each other, is:
a) 9
b) 9.6
c) 10
d) 10.8
Discussion
Explanation:
$$\eqalign{ & {\text{Relative}}\,{\text{speed}} = \left( {60 + 40} \right)\,{\text{km/hr}} \cr & = {100 \times \frac{5}{{18}}} \,{\text{m/sec}} \cr & = {\frac{{250}}{9}} \,{\text{m/sec}}. \cr & {\text{Distance}}\,{\text{covered}}\,{\text{in}}\,{\text{crossing}}\,{\text{each}}\,{\text{other}} \cr & = \left( {140 + 160} \right)m = 300\,m \cr & {\text{Required}}\,{\text{time}} = {300 \times \frac{9}{{250}}} \,{\text{sec}} \cr & = \frac{{54}}{5}\,{\text{sec}} \cr & = 10.8\,{\text{sec}} \cr} $$
25. A train 110 metres long is running with a speed of 60 kmph. In what time will it pass a man who is running at 6 kmph in the direction opposite to that in which the train is going?
a) 5 sec
b) 6 sec
c) 7 sec
d) 10 sec
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}}\,{\text{of}}\,{\text{train}}\,{\text{relative}}\,{\text{to}}\,{\text{man}} = \left( {60 + 6} \right)\,{\text{km/hr}} \cr & = 66\,{\text{km/hr}} \cr & = {66 \times \frac{5}{{18}}} \,{\text{m/sec}} \cr & = {\frac{{55}}{3}} \,{\text{m/sec}} \cr & {\text{Time}}\,{\text{taken}}\,{\text{to}}\,{\text{pass}}\,{\text{the}}\,{\text{man}} \cr & = {110 \times \frac{3}{{55}}} {\text{sec}} = 6\,{\text{sec}} \cr} $$.
26.Two trains are running at 40 km/hr and 20 km/hr respectively in the same direction. Fast train completely passes a man sitting in the slower train in 5 seconds. What is the length of the fast train?
a) 23 m
b) $$23\frac{2}{9}$$ m
c) $$27\frac{7}{9}$$ m
d) 29 m
Discussion
Explanation:
$$\eqalign{ & {\text{Relative}}\,{\text{speed}} = \left( {40 - 20} \right)\,{\text{km/hr}} \cr & = \left( {20 \times \frac{5}{{18}}} \right)\,{\text{m/sec}} \cr & = {\frac{{50}}{9}} \,{\text{m/sec}} \cr & {\text{Length}}\,{\text{of}}\,{\text{faster}}\,{\text{train}} = \left( {\frac{{50}}{9} \times 5} \right)\,m \cr & = \frac{{250}}{9}\,m \cr & = 27\frac{7}{9}\,m \cr} $$
27. A train overtakes two persons who are walking in the same direction in which the train is going, at the rate of 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. The length of the train is:
a) 45 m
b) 50 m
c) 54 m
d) 72 m
Discussion
Explanation:
$$\eqalign{ & 2\,kmph = \left( {2 \times \frac{5}{{18}}} \right)\,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{5}{9}\,{\text{m/sec}} \cr & 4\,kmph = \left( {4 \times \frac{5}{{18}}} \right)\,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{10}}{9}\,{\text{m/sec}} \cr & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{be}}\,x\,{\text{metres}}\, \cr & {\text{and}}\,{\text{its}}\,{\text{speed}}\,{\text{be}}\,y\,{\text{m/sec}} \cr & {\text{Then}},\, {\frac{x}{{y - \frac{5}{9}}}} = 9\,{\text{and}}\, {\frac{x}{{y - \frac{{10}}{9}}}} = 10 \cr & 9y - 5 = x\,{\text{and}}\,10\left( {9y - 10} \right) = 9x \cr & \Rightarrow 9y - x = 5\,{\text{and}}\,90y - 9x = 100 \cr & {\text{On}}\,{\text{solving,}}\,{\text{we}}\,{\text{get}}:\,x = 50 \cr & {\text{Length}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{is}}\,50\,m \cr} $$.
28.A train overtakes two persons walking along a railway track. The first one walks at 4.5 km/hr. The other one walks at 5.4 km/hr. The train needs 8.4 and 8.5 seconds respectively to overtake them. What is the speed of the train if both the persons are walking in the same direction as the train?
a) 66 km/hr
b) 72 km/hr
c) 76 km/hr
d) 81 km/hr
Discussion
Explanation:
$$\eqalign{ & 4.5\,{\text{km/hr}} = \left( {4.5 \times \frac{5}{{18}}} \right)\,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{5}{4}\,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1.25\,{\text{m/sec,}}\,{\text{and}} \cr & 5.4\,km/hr = \left( {5.4 \times \frac{5}{{18}}} \right)\,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{3}{2}\,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1.5\,{\text{m/sec}} \cr & {\text{Let}}\,{\text{the}}\,{\text{speed}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{be}}\,x\,{\text{m/sec}} \cr & {\text{Then}},\,\left( {x - 1.25} \right) \times 8.4 = \left( {x - 1.5} \right) \times 8.5 \cr & 8.4x - 10.5 = 8.5x - 12.75 \cr & 0.1x = 2.25 \cr & x = 22.5 \cr & {\text{Speed}}\,{\text{of}}\,{\text{the}}\,{\text{train}} \cr & = \left( {22.5 \times \frac{{18}}{5}} \right)\,{\text{km/hr}} \cr & = 81\,{\text{km/hr}} \cr} $$
29. A train travelling at 48 kmph completely crosses another train having half its length and travelling in opposite direction at 42 kmph, in 12 seconds. It also passes a railway platform in 45 seconds. The length of the platform is
a) 400 m
b) 450 m
c) 560 m
d) 600 m
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{first}}\,{\text{train}}\,{\text{be}}\,x\,{\text{metres}} \cr & {\text{Then,}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{second}}\,{\text{train}}\,{\text{is}}\, {\frac{x}{2}} \,{\text{metres}} \cr & {\text{Relative}}\,{\text{speed}} = \left( {48 + 42} \right)\,{\text{kmph}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {90 \times \frac{5}{{18}}} \right)\,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 25\,{\text{m/sec}} \cr & \frac{{ {x + \left( {x/2} \right)} }}{{25}} = 12 \cr & \frac{{3x}}{2} = 300 \cr & x = 200 \cr & {\text{Length}}\,{\text{of}}\,{\text{first}}\,{\text{train}} = 200\,{\text{m}} \cr & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{platform}}\,{\text{be}}\,y\,{\text{metres}} \cr & {\text{Speed}}\,{\text{of}}\,{\text{the}}\,{\text{first}}\,{\text{train}} \cr & = \left( {48 \times \frac{5}{{18}}} \right)\,{\text{m/sec}} \cr & = \frac{{40}}{3}\,{\text{m/sec}} \cr & \therefore \left( {200 + y} \right) \times \frac{3}{{40}} = 45 \cr & \Rightarrow 600 + 3y = 1800 \cr & \Rightarrow y = 400\,{\text{m}} \cr} $$.
30. Two stations A and B are 110 km apart on a straight line. One train starts from A at 7 a.m. and travels towards B at 20 kmph. Another train starts from B at 8 a.m. and travels towards A at a speed of 25 kmph. At what time will they meet?
a) 9 a.m.
b) 10 a.m.
c) 10.30 a.m.
d) 11 a.m.
Discussion
Explanation:
$$\eqalign{ & {\text{Suppose}}\,{\text{they}}\,{\text{meet}}\,x\,{\text{hours}}\,{\text{after}}\,{\text{7}}\,{\text{a}}{\text{.m}}. \cr & {\text{Distance}}\,{\text{covered}}\,{\text{by}}\,{\text{A}}\, \cr & {\text{in}}\,x\,{\text{hours = 20x}}\,{\text{km}}{\text{.}} \cr & {\text{Distance}}\,{\text{covered}}\,{\text{by}}\,{\text{B}} \cr & \,{\text{in}}\,\left( {x - 1} \right)\,{\text{hours}} = 25\left( {x - 1} \right)\,km \cr & 20x + 25\left( {x - 1} \right) = 110 \cr & 45x = 135 \cr & x = 3 \cr & {\text{So,}}\,{\text{they}}\,{\text{meet}}\,{\text{at}}\,{\text{10}}\,{\text{a}}{\text{.m}}{\text{.}}\, \cr} $$
31. Two trains start at the same time for two station A and B toward B and A respectively. If the distance between A and B is 220 km and their speeds are 50 km/hr and 60 km/hr respectively then after how much time will they meet each other?
a) 2 hr
b) $$2\frac{1}{2}$$ hr
c) 3 hr
d) 1 hr
Discussion
Explanation:
$$\eqalign{ & {\text{Relative speed}} = 60 + 50 \cr & {\text{ = 110 km/h}} \cr & {\text{Time taken}} \cr & {\text{ = }}\frac{{220}}{{110}} \cr & {\text{ = 2 hr}} \cr} $$
32. A train 100 meter long meets a man going in opposite direction at 5 km/h and passes him in 71/5 seconds. What is the speed of the train (in km/hr)?
a) 45 km/h
b) 60 km/h
c) 55 km/hr
d) 50 km/hr
Discussion
Explanation:
$$\eqalign{ & {\text{Relative speed of man & train}} \cr & {\text{ = }}\frac{{100 \times 5}}{{36}} \times \frac{{18}}{5} \cr & {\text{ = 50km/hr}} \cr & {\text{speed of train}} \cr & {\text{ = 50}} - {\text{5}} \cr & {\text{ = 45 km/hr}} \cr} $$
33. A train takes 9 sec to cross a pole. If the speed of the train is 48 kmph, then length of the train is?
a) 150 m
b) 120 m
c) 90 m
d) 80 m
Discussion
Explanation:
$$\eqalign{ & {\text{Time taken by train to cross a pole}} = 9 sec \cr & {\text{Distance covered in crossing a pole}} \cr & {\text{ = length of train}} \cr & {\text{Speed of the train}} = 48 km/h \cr & = \left( {\frac{{48 \times 5}}{{18}}} \right)m/\sec \cr & = \frac{{40}}{3}m/\sec \cr & {\text{Length of the train}} \cr & {\text{ = Speed }} \times {\text{Time}} \cr & {\text{ = }}\frac{{40}}{3} \times 9 \cr & {\text{ = 120 m}} \cr} $$
34. Two trains start at the same time from A and B and proceed toward each other at the sped of 75 km/hr and 50 km/hr respectively. When both meet at a point in between, one train was found to have traveled 175 km more then the other. Find the distance between A and B?
a) 875 km
b) 785 km
c) 758 km
d) 857 km
Discussion
Explanation:
$$\eqalign{ & {\text{Let the trains meet after t hours}} \cr & {\text{Speed of train A}} = 75 km/hr \cr & {\text{Speed of train B}} = 50 km/hr \cr & {\text{Distance covered by train A}} \cr & {\text{ = 75}} \times {\text{t = 75t}} \cr & {\text{Distance covered by train B}} \cr & {\text{ = 50}} \times {\text{t = 50t}} \cr & {\text{Distance}}\,{\text{ = Speed }} \times {\text{Time}} \cr & {\text{According to question}} \cr & 75{\text{t}} - 50{\text{t}} = 175 \cr & \Rightarrow 25{\text{t}} = 175 \cr & \Rightarrow {\text{t}} = \frac{{175}}{{25}} = 7\,{\text{hour}} \cr & {\text{Distance between A and B }} \cr & {\text{ = 75t}} + 50{\text{t}} = 125{\text{t}} \cr & = 125 \times 7 = 875\,{\text{km}} \cr} $$
35. Two trains 180 meters and 120 meters in length are running towards each other on parallel tracks, one at the rate 65 km/hr and another at 55 km/hr. In how many seconds will they be cross each other from the moment they meet?
a) 6 seconds
b) 9 seconds
c) 12 seconds
d) 15 seconds
Discussion
Explanation:
$$\eqalign{ & {\text{Time taken by trains to cross each }} \cr & {\text{other in opposite direction}} \cr & {\text{ = }}\frac{{{l_1} + {l_2}}}{{{\text{relative speed in opposite direction}}}} \cr & {\text{ = }}\frac{{\left( {180 + 120} \right)}}{{\left( {65 + 55} \right)}} \cr & {\text{ = }}\frac{{300}}{{120 \times \frac{5}{{18}}}} \cr & {\text{ = 9 seconds}} \cr} $$
36. A train starts from A at 7 a.m. towards B with speed 50 km/h. Another train starts from B at 8 a.m. with speed of 60 km/h towards A. Both of them meet at 10 a.m. at C. The ratio of the distance AC to BC is?
a) 5 : 6
b) 5 : 4
c) 6 : 5
d) 4 : 5
Discussion
Explanation: The speed of train A is 50km/hr and A starts its journey at 7 AM and reaches C at 10 AM. Total Travel time = 3hr
Distance cover by A in 3hr = 50 × 3 = 150KM
Similarly, the speed of train B is 60km/hr and B starts its journey at 8 AM and reaches C at 10 AM. Total Travel time = 2hr
Distance cover by B in 2hr = 60 × 2 = 120KM
The ratio of the distance between AC : BC
= 150 : 120
= 5 : 4
37. Train A passes a lamp post in 9 seconds and 700 meter long platform in 30 seconds. How much time will the same train take to cross a platform which is 800 meters long? (in seconds)
a) 32 seconds
b) 31 seconds
c) 33 seconds
d) 30 seconds
Discussion
Explanation:
$$\eqalign{ & {\text{Let the length of train be x m}} \cr & {\text{When a train crosses a light }} \cr & {\text{post in 9 second the distance covered}} \cr & {\text{ = length of train }} \cr & \Rightarrow {\text{speed of train = }}\frac{x}{9} \cr & {\text{Distance covered in crossing a}} \cr & {\text{700 meter platfrom in 30 seconds}} \cr & {\text{ = Length of platfrom + length of train}} \cr & {\text{Speed of train = }}\frac{{x + 700}}{30} \cr & \Rightarrow \frac{x}{9} = \frac{{x + 700}}{{30}}\left[ { {\text{Speed = }}\frac{{{\text{Distance}}}}{{{\text{Time}}}}} \right] \cr & \frac{x}{3} = \frac{{x + 700}}{{10}} \cr & 10x = 3x + 2100 \cr & 10x - 3x = 2100 \cr & 7x = 2100 \cr & x = \frac{{2100}}{7} = 300{\text{m}} \cr & {\text{When the length of the platform be 800m,}} \cr & {\text{then time T be taken by train to cross 800m}} \cr & {\text{long platform}} \cr & \frac{x}{9} = \frac{{x + 800}}{T} \cr & Tx = 9x + 7200 \cr & 300T = 2700 + 7200 \cr & 300T = 9900 \cr & T = \frac{{9900}}{{300}} = 33{\text{ seconds}} \cr} $$
38. Train A traveling at 63 kmph can cross a platform 199.5 m long in 21 seconds. How much would train A take to completely cross (from the moment they meet ) train B, 157 m long and traveling at 54 kmph in opposite direction which train A is traveling? (in seconds)
a) 16
b) 18
c) 12
d) 10
Discussion
Explanation:
$$\eqalign{ & {\text{Speed of train A}} \cr & {\text{ = 63 kmph}} \cr & {\text{ = }}\left( {\frac{{63 \times 5}}{{18}}} \right){\text{m/sec}} \cr & {\text{ = 17}}{\text{.5 m/sec}} \cr & {\text{Speed of train B}} \cr & {\text{ = 54 kmph}} \cr & {\text{ = }}\left( {\frac{{54 \times 5}}{{18}}} \right){\text{m/sec = 15 m/sec}} \cr & {\text{If the length of train A be }}x{\text{ metre,}} \cr & {\text{then}} \cr & {\text{Speed of train A}} \cr & {\text{ = }}\frac{{{\text{Length of train + length of platform}}}}{{{\text{Time taken in crossing}}}}{\text{ }} \cr & \Rightarrow 17.5 = \frac{{x + 199.5}}{{21}} \cr & 17.5 \times 21 = x + 199.5 \cr & 367.5 = x + 199.5 \cr & x = 367.5 - 199.5 \cr & 168\,{\text{metres}} \cr & {\text{Relative speed}} \cr & {\text{ = ( Speed train A + Speed train B)}} \cr & {\text{ = (17}}{\text{.5 + 15) m/sec}} \cr & {\text{ = 32}}{\text{.5 m/sec}} \cr & {\text{Required time}} \cr & {\text{ = }}\frac{{{\text{ Length of train A + Length of train B}}}}{{{\text{Relative speed }}}} \cr & = \left( {\frac{{168 + 157}}{{32.5}}} \right){\text{seconds}} \cr & = 10\,{\text{seconds}} \cr} $$
39. A train which is moving at an average speed of 40 km/h reaches its destination on time. When its average speed reduces to 35 km/h, then it reaches its destination 15 minutes late. The distance traveled by the train is?
a) 70 km
b) 80 km
c) 40 km
d) 30 km
Discussion
Explanation:
$$\eqalign{ & {\text{Average speed of train}} = 40 km/hr \cr & {\text{Reach at its destination at on time }} \cr & {\text{New average speed of train}} = 35 km/h \cr & {\text{Time = 15 minutes}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = }}\frac{{15}}{{60}}{\text{hours }} \cr & {\text{Then distance travelled}} \cr & {\text{ = }}\frac{{40 \times 35}}{{40 - 35}}{\text{ }} \times \frac{{15}}{{60}} \cr & {\text{ = }}\frac{{40 \times 35}}{5}{\text{ }} \times \frac{{15}}{{60}} \cr & {\text{ = 70}}\,{\text{km}} \cr} $$
40. A train moves with a speed of 30 kmph for 12 minutes and for next 8 minutes at a speed of 45 kmph. Find the average speed of the train?
a) 37.50 kmph
b) 36 kmph
c) 48 kmph
d) 30 kmph
Discussion
Explanation:
$$\eqalign{ & {\text{Distance = Speed }} \times {\text{Time}} \cr & {\text{Distance covered by train with the}} \cr & {\text{speed of 30 kmph in 12 minutes is }} \cr & {\text{ = 30}} \times \frac{{12}}{{60}} = 6{\text{km}} \cr & {\text{Distance covered by the same train}} \cr & {\text{with the speed of 45 kmph in 8 minutes is }} \cr & {\text{ = 45}} \times \frac{8}{{60}} = 6{\text{km}} \cr & {\text{Average speed}} \cr & {\text{ = }}\frac{{{\text{total distance}}}}{{{\text{total time}}}}. \cr & \Rightarrow \frac{{\left( {6 + 6} \right){\text{km}}}}{{\left( {12 + 8} \right)\min }} = \frac{{12}}{{20}} \times 60 \cr & {\text{ = 36 kmph}} \cr} $$
41. How many seconds will a 500 metre long train take to cross a man walking with a speed of 3 km/hr in the direction of the moving train if the speed of the train is 63 km/hr?
a) 25
b) 30
c) 40
d) 45
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{relative}}\,{\text{to}}\,{\text{man}} \cr & = \left( {63 - 3} \right)\,{\text{km/hr}} \cr & = 60\,{\text{km/hr}} \cr & = \left( {60 \times \frac{5}{{18}}} \right)\,{\text{m/sec}} \cr & = \frac{50}{3}\, \text{m/sec} \cr & {\text{Time}}\,{\text{taken}}\,{\text{to}}\,{\text{pass}}\,{\text{the}}\,{\text{man}} \cr & = \left( {500 \times \frac{3}{{50}}} \right)\,\sec \cr & = 30\,\sec \cr} $$
42. Two goods train each 500 m long, are running in opposite directions on parallel tracks. Their speeds are 45 km/hr and 30 km/hr respectively. Find the time taken by the slower train to pass the driver of the faster one.
a) 12 sec
b) 24 sec
c) 48 sec
d) 60 sec
Discussion
Explanation:
$$\eqalign{ & {\text{Relative}}\,{\text{speed}} \cr & = \left( {45 + 30} \right)\,{\text{km/hr}} \cr & = \left(75 \times \frac{5}{18} \right)\, \text{m/sec} \cr & = {\frac{{125}}{6}} \,{\text{m/sec}} \cr & {\text{We}}\,{\text{have}}\,{\text{to}}\,{\text{find}}\,{\text{the}}\,{\text{time}}\,{\text{taken}}\,{\text{by}}\,{\text{the}} \cr & {\text{slower}}\,{\text{train}}\,{\text{to}}\,{\text{pass}}\,{\text{the}}\,{\text{DRIVER}}\,{\text{of}}\, \cr & {\text{The}}\,{\text{faster}}\,{\text{train}}\,{\text{and}}\,{\text{not}}\,{\text{the}}\,{\text{complete}}\,{\text{train}}{\text{.}} \cr & {\text{So,}}\,{\text{distance}}\,{\text{covered = Length}}\,{\text{of}}\,{\text{the}}\,{\text{slower}}\,{\text{train}}. \cr & {\text{Therefore,}}\,{\text{Distance}}\,{\text{covered = 500}}\,{\text{m}}. \cr & {\text{Required}}\,{\text{time}} = {500 \times \frac{6}{{125}}} \cr & = 24\,\sec \cr} $$
43. Two trains are running in opposite directions with the same speed. If the length of each train is 120 metres and they cross each other in 12 seconds, then the speed of each train (in km/hr) is:
a) 10
b) 18
c) 36
d) 72
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{speed}}\,{\text{of}}\,{\text{each}}\,{\text{train}}\,{\text{be}}\,x\,{\text{m/sec}}. \cr & {\text{Then,}}\,{\text{relative}}\,{\text{speed}}\,{\text{of}}\,{\text{the}}\,{\text{two}}\,{\text{trains}} = 2x\,{\text{m/sec}} \cr & {\text{So}},\,2x = \frac{{ {120 + 120} }}{{12}} \cr & 2x = 20 \cr & x = 10 \cr & {\text{Speed}}\,{\text{of}}\,{\text{each}}\,{\text{train}} = 10\,{\text{m/sec}} \cr & = {10 \times \frac{{18}}{5}} \,{\text{km/hr}} \cr & = 36\,{\text{km/hr}} \cr} $$
44. Two trains of equal lengths take 10 seconds and 15 seconds respectively to cross a telegraph post. If the length of each train be 120 metres, in what time (in seconds) will they cross each other travelling in opposite direction?
a) 10
b) 12
c) 15
d) 20
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}}\,{\text{of}}\,{\text{the}}\,{\text{first}}\,{\text{train}} \cr & = {\frac{{120}}{{10}}} \,{\text{m/sec}} \cr & = 12\,{\text{m/sec}} \cr & {\text{Speed}}\,{\text{of}}\,{\text{the}}\,{\text{second}}\,{\text{train}} \cr & {\frac{{120}}{{15}}} \,{\text{m/sec}} \cr & = 8\,{\text{m/sec}} \cr & {\text{Relative}}\,{\text{speed}} = {12 + 8} = 20\,{\text{m/sec}} \cr & {\text{Required}}\,{\text{time}} \cr & = {\frac{{ {120 + 120} }}{{20}}} \,{\text{ sec}} \cr & = 12\,{\text{sec}} \cr} $$
45. A train 108 m long moving at a speed of 50 km/hr crosses a train 112 m long coming from opposite direction in 6 seconds. The speed of the second train is:
a) 48 km/hr
b) 54 km/hr
c) 66 km/hr
d) 82 km/hr
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{speed}}\,{\text{of}}\,{\text{the}}\,{\text{second}}\,{\text{train}}\,{\text{be}}\,x\,{\text{km/hr}}. \cr & {\text{Relative}}\,{\text{speed}}\, \cr & = \,\left( {x + 50} \right)\,{\text{km/hr}} \cr & = \left[ {\left( {x + 50} \right) \times \frac{5}{{18}}} \right]\,{\text{m/sec}} \cr & = {\frac{{250 + 5x}}{{18}}} \,{\text{m/sec}} \cr & {\text{Distance}}\,{\text{covered}} \cr & = \left( {108 + 112} \right) = 220\,m \cr & \frac{{220}}{{ {\frac{{250 + 5x}}{{18}}} }} = 6 \cr & 250 + 5x = 660 \cr & x = 82\,{\text{km/hr}} \cr} $$.
46. A train 240 m long passes a pole in 24 seconds. How long will it take to pass a platform 650 m long?
a) 65 sec
b) 89 sec
c) 100 sec
d) 150 sec
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}} = {\frac{{240}}{{24}}} \,{\text{m/sec}} = 10\,{\text{m/sec}} \cr & {\text{Required}}\,{\text{time}} \cr & {\text{ = }}\, {\frac{{240 + 650}}{{10}}} \,{\text{sec}}. \cr & = 89\,sec. \cr} $$
47. Two trains of equal length are running on parallel lines in the same direction at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is:
a) 50 m
b) 72 m
c) 80 m
d) 82 m
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{each}}\,{\text{train}}\,{\text{be}}\,x\,{\text{metres}}. \cr & {\text{Then,}}\,{\text{distance}}\,{\text{covered}} = 2x\,{\text{metres}}. \cr & {\text{Relative}}\,{\text{speed}} \cr & = \left( {46 - 36} \right)\,{\text{km/hr}} \cr & = {10 \times \frac{5}{{18}}} \,{\text{m/sec}} \cr & = {\frac{{25}}{9}} \,{\text{m/sec}} \cr & \therefore \frac{{2x}}{{36}} = \frac{{25}}{9} \cr & 2x = 100 \cr & x = 50 \cr} $$
48. A train 360 m long is running at a speed of 45 km/hr. In what time will it pass a bridge 140 m long?
a) 40 sec
b) 42 sec
c) 45 sec
d) 48 sec
Discussion
Explanation:
$$\eqalign{ & {\text{Formula}}\,{\text{for}}\,{\text{converting}}\,{\text{from}}\,{\text{km/hr}}\,{\text{to}}\,{\text{m/s:}} \cr & X\,{\text{km/hr}} = {X \times \frac{5}{{18}}} \,{\text{m/s}} \cr & {\text{Therefore,}}\,{\text{Speed}} \cr & = {45 \times \frac{5}{{18}}} \,{\text{m/sec}} = \frac{{25}}{2}{\text{m/sec}} \cr & {\text{Total}}\,{\text{distance}}\,{\text{to}}\,{\text{be}}\,{\text{covered}} \cr & = \left( {360 + 140} \right)m = 500\,m \cr & {\text{Formula}}\,{\text{for}}\,{\text{finding}}\,{\text{Time}} \cr & = {\frac{{{\text{Distance}}}}{{{\text{Speed}}}}} \cr & {\text{Required}}\,{\text{time}} \cr & = \left( {\frac{{500 \times 2}}{{25}}} \right)\,\sec \cr & = 40\,\sec . \cr} $$
49. Two trains are moving in opposite directions @ 60 km/hr and 90 km/hr. Their lengths are 1.10 km and 0.9 km respectively. The time taken by the slower train to cross the faster train in seconds is:
a) 36
b) 45
c) 48
d) 49
Discussion
Explanation:
$$\eqalign{ & {\text{Relative}}\,{\text{speed}} = \left( {60 + 90} \right)\,{\text{km/hr}} \cr & = {150 \times \frac{5}{{18}}} \,{\text{m/sec}} \cr & = {\frac{{125}}{3}} \,{\text{m/sec}} \cr & {\text{Distance}}\,{\text{covered}} \cr & = \left( {1.10 + 0.9} \right)\,km \cr & = 2\,km \cr & = \,2000\,m \cr & {\text{Required}}\,{\text{time}} = {2000 \times \frac{3}{{125}}} \,{\text{sec}} \cr & = 48\,sec \cr} $$
50. A jogger running at 9 kmph alongside a railway track in 240 metres ahead of the engine of a 120 metres long train running at 45 kmph in the same direction. In how much time will the train pass the jogger?
a) 3.6 sec
b) 18 sec
c) 36 sec
d) 72 sec
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}}\,{\text{of}}\,{\text{train}}\,{\text{relative}}\,{\text{to}}\,{\text{jogger}} \cr & = \left( {45 - 9} \right)\,{\text{km/hr}} \cr & = 36\,{\text{km/hr}} \cr & {36 \times \frac{5}{{18}}} \,{\text{m/sec}} \cr & = 10\,{\text{m/sec}} \cr & {\text{Distance}}\,{\text{to}}\,{\text{be}}\,{\text{covered}} \cr & = \left( {240 + 120} \right)\,m \cr & = 360\,m \cr & {\text{Time}}\,{\text{taken}} = {\frac{{360}}{{10}}} \,{\text{sec}} \cr & = 36\,{\text{sec}} \cr} $$
51. The Ghaziabad - Hapur - Meerut EMU and the Meerut - Hapur - Ghaziabad EMU start at the same time from Ghaziabad and Meerut and proceed towards each other at 16 km/hr and 21 km/hr respectively. When they meet, it is found that one train has traveled 60 km more than the other . The distance between two stations is?
a) 440 km
b) 444 km
c) 445 km
d) 450 km
Discussion
Explanation:
$$\eqalign{ & {\text{At the time of meeting ,}} \cr & {\text{let the distance travelled by the}} \cr & {\text{first train be }}x{\text{ km}}{\text{.}} \cr & {\text{Then distance travelled by the }} \cr & {\text{second train is (}}x{\text{ + 60) km}} \cr & \frac{x}{{16}} = \frac{{x + 60}}{{21}} \cr & \Rightarrow 21x = 16x + 960 \cr & \Rightarrow 5x = 960 \Rightarrow x = 192 \cr & {\text{Distance between two stations}} \cr & {\text{ = (192 + 192 + 60) km}} \cr & {\text{ = 444 km}}{\text{.}} \cr} $$
52. Two trains start simultaneously (with uniform speeds) from two stations 270 km apart, each to the opposite station; they reach their destinations in $$6\frac{1}{4}$$ hours and 4 hours after they meet. The rate at which the slower train travels is :
a) 16 km/hr
b) 24 km/hr
c) 25 km/hr
d) 30 km/hr
Discussion
Explanation:
$$\eqalign{ & {\text{Ratio of speeds}} \cr & {\text{ = }}\sqrt 4 :\sqrt {6\frac{1}{4}} \cr & = \sqrt 4 :\sqrt {\frac{{25}}{4}} \cr & = 2:\frac{5}{2} \cr & = 4:5 \cr }$$
Let the speeds of the two trains be 4x and 5x km/hr respectively
Then time taken by trains to meet each other
$$\eqalign{ & {\text{ = }}\left( {\frac{{270}}{{4x + 5x}}} \right){\text{hr}} \cr & {\text{ = }}\left( {\frac{{270}}{{9x}}} \right){\text{hr = }}\left( {\frac{{30}}{x}} \right){\text{hr}} \cr & {\text{Time taken by slower train to travel}} \cr & {\text{ 270 km = }}\left( {\frac{{270}}{{4x}}} \right){\text{hr}} \cr & \frac{{270}}{{4x}} = \frac{{30}}{x} + 6\frac{1}{4} \cr & \frac{{270}}{{4x}} - \frac{{30}}{x} = \frac{{25}}{4} \cr & \frac{{150}}{{4x}} = \frac{{25}}{4} \cr & 100x = 600 \cr & x = 6 \cr & {\text{Hence speed of slower train}} \cr & {\text{ = 4}}x \cr & = \,24\,{\text{km/hr}} \cr} $$
53. Two trains, A ans B start from stations X and Y towards each other, they take 4 hours 48 minutes and 3 hours 20 minutes to reach Y and X respectively after they meet. If train A is moving at 45 km/hr, then the speed of the train B is?
a) 60 km/hr
b) 64.80 km/hr
c) 54 km/hr
d) 37.5 km/hr
Discussion
Explanation:
$$\eqalign{ & {\text{In these type of questions use the given}} \cr & {\text{below formula to save your valuable time}} \cr & \frac{{{{\text{S}}_1}}}{{{{\text{S}}_2}}}{\text{ = }}\sqrt {\frac{{{{\text{T}}_2}}}{{{{\text{T}}_1}}}} {\text{ }} \cr & {\text{Where }}{{\text{S}}_1}{\text{,}}{{\text{S}}_2}{\text{ and }}{{\text{T}}_1}{\text{, }}{{\text{T}}_2}{\text{ are the respective}} \cr & {\text{speeds and times of the objects}} \cr & \Rightarrow \frac{{45}}{{{{\text{S}}_2}}} = \sqrt {3\frac{1}{3} \div 4\frac{4}{5}} \cr & {\text{ = }}{{\text{S}}_2}{\text{ = 45}} \times \frac{6}{5}{\text{ = 54 km/hr}} \cr & {\text{Required speed = 54 km/hr}} \cr} $$
54. A train passes by a lamp post at platform in 7 sec. and passes by the platform completely in 28 sec. If the length of the platform is 390m, then length of the train (in meters) is?
a) 120 m
b) 130 m
c) 140 m
d) 150 m
Discussion
Explanation: Length of train
$$ = \frac{{{\text{Length}}\,{\text{of}}\,{\text{the}}\,{\text{platform}}}}{{{\text{Difference}}\,{\text{in time}}}}$$ × (Time taken to cross a lamp post)
$$\eqalign{ & = \frac{{390}}{{28 - 7}} \times 7 \cr & = \frac{{390}}{{21}} \times 7 \cr & = \frac{{390}}{3} \cr & = 130\,{\text{m}} \cr} $$
55. A train moving at a rate of 36 km/hr crosses a standing man in 10 seconds. It will cross a platform 55 meters long in?
a) 6 second
b) 7 second
c) $$15\frac{1}{2}$$ second
d) $$5\frac{1}{2}$$ second
Discussion
Explanation:
$$\eqalign{ & {\text{Length of the train}} \cr & {\text{ = Speed }} \times {\text{time}} \cr & {\text{ = 36 km/hr}} \times {\text{10 sec}} \cr & {\text{ = 36}} \times \frac{5}{{18}}{\text{m/s}} \times 10\sec \cr & = 100{\text{ metres}} \cr & {\text{Time taken by train to cross a plateform}} \cr & {\text{ of 55 metre long in time}} \cr & {\text{ = }}\frac{{\left( {100 + 55} \right)}}{{36 \times \frac{5}{{18}}}} \cr & = \frac{{155}}{{10}} \cr & {\text{Time}} = 15\frac{1}{2}\,\sec \cr} $$
56. A train travelling at a speed of 75 mph enters a tunnel 3 1/2 miles long. The train is 1/4 mile long. How long does it take for the train to pass through the tunnel from the moment the front enters to the moment the rear emerges?
a) 2.5 min
b) 3 min
c) 3.5 min
d) 3.5 min
Discussion
Explanation:
$$\eqalign{ & {\text{Total}}\,{\text{distance}}\,{\text{covered}} \cr & = \left( {\frac{7}{2} + \frac{1}{4}} \right)\,{\text{miles}} \cr & = \frac{{15}}{4}\,{\text{miles}} \cr & {\text{Time}}\,{\text{taken}} = \left( {\frac{{15}}{{4 \times 75}}} \right)\,{\text{hrs}} \cr & = \frac{1}{{20}}\,{\text{hrs}} \cr & = \left( {\frac{1}{{20}} \times 60} \right)\,\min \cr & = 3\,\min \cr} $$
57. A train 800 metres long is running at a speed of 78 km/hr. If it crosses a tunnel in 1 minute, then the length of the tunnel (in meters) is:
a) 130
b) 360
c) 500
d) 540
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}} = \left( {78 \times \frac{5}{{18}}} \right)\,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\frac{{65}}{3}} \,{\text{m/sec}} \cr & {\text{Time = }}\,{\text{1}}\,{\text{minute = 60}}\,{\text{second}}. \cr & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{tunnel}}\,{\text{be}}\,x\,{\text{metres}}. \cr & {\text{Then}},\, {\frac{{800 + x}}{{60}}} = \frac{{65}}{3} \cr & 3\left( {800 + x} \right) = 3900 \cr & x = 500 \cr} $$
58. A 300 metre long train crosses a platform in 39 seconds while it crosses a signal pole in 18 seconds. What is the length of the platform?
a) 320 m
b) 350 m
c) 650 m
d) Data inadequate
Discussion
Explanation:
$$\eqalign{ & {\text{Speed}} = {\frac{{300}}{{18}}} \,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{50}}{3}\,{\text{m/sec}} \cr & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{platform}}\,{\text{be}}\,x\,{\text{metres}}{\text{.}} \cr & {\text{Then}}, {\frac{{x + 300}}{{39}}} = \frac{{50}}{3} \cr & 3\left( {x + 300} \right) = 1950 \cr & x = 350\,m. \cr} $$
59. A train speeds past a pole in 15 seconds and a platform 100 m long in 25 seconds. Its length is:
a) 50 m
b) 150 m
c) 200 m
d) Data inadequate
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{be}}\,x\,{\text{metres}} \cr & \,{\text{and}}\,{\text{its}}\,{\text{speed}}\,{\text{by}}\,y\,{\text{m/sec}} \cr & Then,\,\frac{x}{y} = 15\,\,\,\,\,\, \Rightarrow \,\,\,\,\,y = \frac{x}{{15}} \cr & \frac{{x + 100}}{{25}} = \frac{x}{{15}} \cr & 15\left( {x + 100} \right) = 25x \cr & 15x + 1500 = 25x \cr & 1500 = 10x \cr & x = 150m \cr} $$.
60. A train moves past a telegraph post and a bridge 264 m long in 8 seconds and 20 seconds respectively. What is the speed of the train?
a) 69.5 km/hr
b) 70 km/hr
c) 79 km/hr
d) 79.2 km/hr
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{be}}\,x\,{\text{metres}} \cr & \,{\text{and}}\,{\text{its}}\,{\text{speed}}\,{\text{by}}\,y\,{\text{m/sec}} \cr & {\text{Then}},\,\frac{x}{y} = 8\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x = 8y \cr & {\text{Now}},\,\frac{{x + 264}}{{20}} = y \cr & 8y + 264 = 20y \cr & y = 22 \cr & {\text{Speed}} = 22\,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {22 \times \frac{{18}}{5}} \,{\text{km/hr}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 79.2\,{\text{km/hr}} \cr} $$
61. Two trains of lenths 120 m and 90 m are running with speed of 80 km/hr and 55 km/hr respectively towards each other on parallel lines. If they are 90 m apart, after how many seconds they will cross each other?
a) 5.6 sec
b) 7.2 sec
c) 8 sec
d) 9 sec.
Discussion
Explanation:
$$\eqalign{ & {\text{Relative speed}} \cr & {\text{ = (80 + 55)km/hr}} \cr & {\text{ = 135 km/hr}} \cr & {\text{ = }}\left( {135 \times \frac{5}{{18}}} \right)m/\sec \cr & = \left( {\frac{{75}}{2}} \right)m/\sec \cr & {\text{Distance covered}} = (120 + 90 + 90)m \cr & {\text{ = 300m}} \cr & {\text{Required time}} \cr & {\text{ = }}\left( {300 \times \frac{2}{{75}}} \right)\sec \cr & = 8\sec \cr} $$
62. Two trains are coming from opposite directions with speed of 75 km/hr and 100 km/hr on to parallel tracks. At some moment the distance between them is 100km. After T hours, distance between them is again 100 km. T is equal to?
a) 1 hr
b) $$1\frac{1}{7}$$ hr
c) $$1\frac{1}{2}$$ hr
d) 2 hr
Discussion
Explanation:
$$\eqalign{ & {\text{Relative speed}} = (75 + 100)km/hr \cr & {\text{ = 175 km/hr}} \cr & {\text{Time taken to cover 175 km}} \cr & {\text{at relative speed = 1 hr}} \cr & {\text{T = Time taken to cover 200 km}} \cr & {\text{ = }}\left( {\frac{1}{{175}} \times 200} \right)\, \text{hr} \cr & = \frac{8}{7}\, \text{hr} \cr & = 1\frac{1}{7}\, \text{hr} \cr} $$
63. A train, 240 m long, crosses a man walking alone the line in opposite direction at the rate of 3 kmph in 10 seconds. The speed of the train is?
a) 63 kmph
b) 75 kmph
c) 83.4 kmph
d) 86.4 kmph
Discussion
Explanation:
$$\eqalign{ & {\text{Speed of the train relative to man}} \cr & {\text{ = }}\left( {\frac{{240}}{{10}}} \right){\text{m/sec}} \cr & {\text{ = 24 m/sec}} \cr & {\text{ = }}\left( {24 \times \frac{{18}}{5}} \right){\text{ km/sec}} \cr & {\text{ = }}\frac{{432}}{5}{\text{km/hr}} \cr & {\text{Let the speed of the train be x kmph}}{\text{.}} \cr & {\text{Then relative speed = }}\left( {x + 3} \right){\text{kmph}} \cr & x{\text{ + 3 = }}\frac{{432}}{5} \cr & x = \frac{{432}}{5} - 3 \cr & x = \frac{{417}}{5} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 83.4\,{\text{kmph}} \cr} $$
64. Two trains of equal length are running on parallel lines in the same directions at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is?
a) 50 m
b) 72 m
c) 80 m
d) 82 m
Discussion
Explanation:
$$\eqalign{ & {\text{Let the length of each train be }}x{\text{ metres}} \cr & {\text{Then distance covered}} \cr & {\text{ = 2x metres}} \cr & {\text{Relative speed}} \cr & {\text{ = (46}} - {\text{36)km/hr}} \cr & {\text{ = }}\left( {10 \times \frac{5}{{18}}} \right)m/\sec \cr & = \left( {\frac{{25}}{9}} \right)m/\sec \cr & \frac{{2x}}{{36}} = \frac{{25}}{9} \Leftrightarrow 2x = 100 \Leftrightarrow x = 50 \cr} $$
65. Two trains of equal lengths takes 10 seconds and 15 seconds respectively to cross a telegraph post. If the length of each train be 120 miters, in what time ( in seconds) will they cross each other traveling in opposite direction?
a) 10
b) 12
c) 15
d) 20
Discussion
Explanation:
$$\eqalign{ & {\text{Speed of the train}} \cr & {\text{ = }}\left( {\frac{{120}}{{10}}} \right){\text{ m/sec}} \cr & {\text{ = 12 m/sec}} \cr & {\text{Speed of the second train}} \cr & {\text{ = }}\left( {\frac{{120}}{{15}}} \right){\text{ m/sec}} \cr & {\text{ = 8 m/sec}} \cr & {\text{Relative speed}} \cr & {\text{ = (12 + 8)m/sec}} \cr & {\text{ = 20 m/sec}} \cr & {\text{Required time}} \cr & {\text{ = }}\frac{{\left( {120 + 120} \right)}}{{20}}\,\sec \cr & = 12\,\sec \cr} $$
66. A train speeds past a pole in 20 seconds and speeds past a platform 100 meters in length in 30 seconds. What is the length of the train?
a) 100 meters
b) 150 meters
c) 180 meters
d) 200 meters
Discussion
Explanation: Let the length of the train be x meters and its speed be y m/sec.
Then, $$\frac{{\text{x}}}{{\text{y}}}$$ = 20
⇒ y = $$\frac{{\text{x}}}{{20}}$$
$$\frac{{{\text{x}} + 100}}{{30}}$$ = $$\frac{{\text{x}}}{{20}}$$
30x = 20x + 2000
10x = 2000
x = 200 meters
67. The time taken by a train 180 m long, travelling at 42 kmph, in passing a person walking in the same direction at 6 kmph, will be
a) 18 sec
b) 21 sec
c) 24 sec
d) 25 sec
Discussion
Explanation: Speed of train relative to man
= (42 - 6) kmph = 36 kmph
= $$\left( {36 \times \frac{5}{{18}}} \right)$$ m/sec
= 10 m/sec
Time taken to pass the man
= $$\frac{{180}}{{10}}$$ sec
= 18 sec
68. Two trains 200 meters and 150 meters long are running on parallel rails in the same direction at speed of 40 km/hr and 45 km/hr respectively. Time taken by the faster train to cross the slowed train will be:
a) 72 seconds
b) 132 seconds
c) 192 seconds
d) 252 seconds
Discussion
Explanation: Relative speed = (45 - 40) km/hr = 5 km/hr
= $$\left( {5 \times \frac{5}{{18}}} \right)$$ m/sec
= $$\frac{{25}}{{18}}$$ m/sec
Total distance covered = Sum of lengths of trains = (200 + 150) m = 350 m
Time taken
= $$\left( {350 \times \frac{{18}}{{25}}} \right)$$ sec
= 252 seconds
69. A train with 90 km/hr crosses a bridge in 36 seconds. Another train 100 meters shorter crosses the same bridge at 45 km/hr. What is the time taken by the second train to cross the bridge?
a) 61 seconds
b) 62 seconds
c) 63 seconds
d) 64 seconds
Discussion
Explanation: Let the lengths of the train and the bridge be x meters and y meters respectively.
Speed of the first train = 90 km/hr
= $$\left( {90 \times \frac{5}{{18}}} \right)$$ m/sec
= 25 m/sec
Speed of the second train = 45 km/hr
= $$\left( {45 \times \frac{5}{{18}}} \right)$$ m/sec
= $$\frac{{25}}{2}$$ m/sec
Then, $$\frac{{{\text{x}} + {\text{y}}}}{{36}}$$ = 25
⇒ x + y = 900
Required time
$$\eqalign{ & = \left[ {\frac{{\left( {{\text{x}} - 100} \right) + {\text{y}}}}{{\frac{{25}}{2}}}} \right]{\text{sec}} \cr & = \left[ {\frac{{\left( {{\text{x}} + {\text{y}}} \right) - 100}}{{\frac{{25}}{2}}}} \right]{\text{sec}} \cr & = \left( {800 \times \frac{2}{{25}}} \right){\text{sec}} \cr & = 64\,{\text{sec}} \cr} $$.
70. A train 125 m long passes a man, running at 5 kmph in the same direction in which the train is going, in 10 seconds. The speed of the train is:
a) 45 km/hr
b) 50 km/hr
c) 54 km/hr
d) 55 km/hr
Discussion
Explanation: Speed of the train relative to man
$$\eqalign{ & = \frac{{125}}{{10}}{\text{m/sec}} \cr & = \frac{{25}}{2}{\text{m/sec}} \cr & = \left( {\frac{{25}}{2} \times \frac{{18}}{5}} \right){\text{m/sec}} \cr & = 45\,{\text{km/hr}} \cr} $$
Let the speed of the train be x kmph.
Then, relative speed = (x - 5) kmph
x - 5 = 45 or
x = 50 km/hr
71. A man sitting in a train is counting the pillars of electricity. The distance between two pillars is 60 meters, and the speed of the train is 42 km/hr. In 5 hours, how many pillars will he count?
a) 3501
b) 3600
c) 3800
d) None of these
Discussion
Explanation: Distance covered by the train in 5 hours = (42 × 5) km
= 210 km
= 210000 m
Number of pillars counted by the man
= $$\left( {\frac{{210000}}{{60}} + 1} \right)$$
= 3500 + 1
= 3501
72. A 120 meter long train is running at a speed of 90 km/hr. It will cross a railway platform 230 m long in :
a) 4 seconds
b) 7 seconds
c) 12 seconds
d) 14 seconds
Discussion
Explanation: Speed = $$\left( {90 \times \frac{5}{{18}}} \right)$$ m/sec = 25 m/sec
Total distance covered = (120 + 230) m
= 350 m
Required time
= $$\frac{{350}}{{25}}$$ seconds
= 14 seconds
73. A 50 meter long train passes over a bridge at the speed of 30 km per hour. If it takes 36 seconds to cross the bridge, what is the length of the bridge?
a) 200 meters
b) 250 meters
c) 300 meters
d) 350 meters
Discussion
Explanation: Speed = $$\left( {30 \times \frac{5}{{18}}} \right)$$ m/sec = $$\frac{{25}}{3}$$ m/sec
Time = 36 second
Let the length of the bridge be x meters.
Then, $$\frac{{50 + {\text{x}}}}{{36}}$$ = $$\frac{{25}}{3}$$
3(50 + x) = 900
50 + x = 300
x = 250 meters
74. A train takes 5 minutes to cross a telegraphic post. Then the time taken by another train whose length is just double of the first train and moving with same speed to cross a platform of its own length is :
a) 10 minutes
b) 15 minutes
c) 20 minutes
d) Data inadequate
Discussion
Explanation: Let the length of the train be x metres.
Time taken to cover x meters = 5 min
= (5 × 60) sec
= 300 sec
Speed of the train = $$\frac{{\text{x}}}{{300}}$$ m/sec
Length of the second train = 2x meters
Length of the platform = 2x meters
Required time
$$\eqalign{ & = \left[ {\frac{{2{\text{x}} + 2{\text{x}}}}{{\left( {\frac{{\text{x}}}{{300}}} \right)}}} \right]{\text{sec}} \cr & = \left( {\frac{{4{\text{x}} \times 300}}{{\text{x}}}} \right){\text{sec}} \cr & = 1200\,{\text{sec}} \cr & = \frac{{1200}}{{60}}\,{\text{min}} \cr & = 20\,{\text{minutes}} \cr} $$
75. 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, what is the length of the platform?
a) 225 meters
b) 240 meters
c) 230 meters
d) 235 meters
Discussion
Explanation: Speed = $$\left( {54 \times \frac{5}{{18}}} \right)$$ m/sec = 15 m/sec
Length of the train = (15 × 20) m = 300 m
Let the length of the platform be x meters
Then, $$\frac{{{\text{x}} + 300}}{{36}}$$ = 15
x + 300 = 540
x = 240 meters
76. A train 75 m long overtook a person who was walking at the rate of 6 km/hr in the same direction and passed him in $$7\frac{1}{2}$$ seconds. Subsequently, it overtook a second person and passed him in $$6\frac{3}{4}$$ seconds. At what rate was the second person travelling?
a) 1 km/hr
b) 2 km/hr
c) 4 km/hr
d) 5 km/hr
Discussion
Explanation: Speed of the train relative to first man
$$\eqalign{ & = \frac{{75}}{{7.5}}{\text{m/sec}} = 10\,{\text{m/sec}} \cr & = \left( {10 \times \frac{{18}}{5}} \right){\text{km/hr}} = 36\,{\text{km/hr}} \cr} $$
Let the speed of the train be x km/hr.
Then, relative speed = (x - 6) km/hr
x - 6 = 36
x = 42 km/hr
Speed of the train relative to second man
$$\eqalign{ & {\text{ = }}\frac{{75}}{{6\frac{3}{4}}}\,{\text{m/sec}} \cr & = \left( {75 \times \frac{4}{{27}}} \right){\text{m/sec}} \cr & = \frac{{100}}{9}{\text{m/sec}} \cr & = \left( {\frac{{100}}{9} \times \frac{{18}}{5}} \right){\text{km}} \cr & = 40\,{\text{km/hr}} \cr} $$
Let the speed of the second man be y kmph.
Then, relative speed = (42 - y) kmph
42 - y = 40
y = 2 km/hr
77.Two trains are running in opposite directions with the same speed. If the length of each train is 120 meters and they cross each other in 12 seconds, then the speed of each train (in km/hr) is?
a) 10 km/hr
b) 18 km/hrs
c) 72 km/hr
d) 36 km/hr
Discussion
Explanation: Let the speed of each train be x m/sec.
Then, relative speed of the two trains = 2x m/sec
So, 2x = $$\frac{{120 + 120}}{{12}}$$
2x = 20
x = 10
Speed of each train = 10 m/sec
= $$\left( {10 \times \frac{{18}}{5}} \right)$$ km/hr
= 36 km/hr
78. A 150 m long train crosses a milestone in 15 seconds and a train of same length coming from the opposite direction in 12 seconds. The speed of the other train is?
a) 36 kmph
b) 45 kmph
c) 50 kmph
d) 54 kmph
Discussion
Explanation: Speed of first train = $$\frac{{150}}{{15}}$$ m/sec = 10 m/sec
Let the speed of second train be x m/sec
Relative speed = (10 + x) m/sec
$$\frac{{300}}{{10 + {\text{x}}}}$$ = 12
300 = 120 + 12x
12x = 180
x = $$\frac{{180}}{{12}}$$ = 15 m/sec
Hence, speed of other train = $$\left( {15 \times \frac{{18}}{5}} \right)$$ kmph
= 54 kmph
79.A man standing on a platform finds that a train takes 3 seconds to pass him and another train of the same length moving in the opposite direction takes 4 seconds. The time taken by the trains to pass each other will be :
a) $$2\frac{3}{7}$$ seconds
b) $$3\frac{3}{7}$$ seconds
c) $$4\frac{3}{7}$$ seconds
d) $$5\frac{3}{7}$$ seconds
Discussion
Explanation: Let the length of each train be x meters
Then, speed of first train = $$\frac{{\text{x}}}{3}$$ m/sec
Speed of second train = $$\frac{{\text{x}}}{4}$$ m/sec
Required time
$$\eqalign{ & = \left[ {\frac{{{\text{x}} + {\text{x}}}}{{\left( {\frac{{\text{x}}}{3} + \frac{{\text{x}}}{4}} \right)}}} \right]{\text{sec}} \cr & = \left[ {\frac{{2{\text{x}}}}{{\left( {\frac{{7{\text{x}}}}{{12}}} \right)}}} \right]{\text{sec}} \cr & = \left( {2 \times \frac{{12}}{7}} \right){\text{sec}} \cr & = \frac{{24}}{7}{\text{sec}} \cr & = 3\frac{3}{7}{\text{sec}} \cr} $$.
80. Two trains, 130 and 110 meters long, are going in the same direction. The faster train takes one minute to pass the other completely. If they are moving in opposite directions, they pass each other completely in 3 seconds. Find the speed of the faster train.
a) 38 m/sec
b) 42 m/sec
c) 46 m/sec
d) 50 m/sec
Discussion
Explanation: Let the speeds of the faster and slower trains be x m/sec and y m/sec respectively.
Then, $$\frac{{240}}{{{\text{x}} - {\text{y}}}}$$ = 60
x - y = 4 . . . . . . . . (i)
And, $$\frac{{240}}{{{\text{x}} + {\text{y}}}}$$ = 3
x + y = 80 . . . . . . . . (ii)
Adding (i) and (ii)
2x = 84
x = 42
Putting x = 42 in (i), we get: y = 38
Hence, speed of faster train = 42 m/sec