1. 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
Explanation:
$$\eqalign{ & {\text{Speed}} = \left( {60 \times \frac{5}{{18}}} \right){\text{m/sec}} = {\frac{{50}}{3}} {\text{m/sec}} \cr & {\text{Length}}\,{\text{of}}\,{\text{the}}\,{\text{train}} = \left( {{\text{Speed}} \times {\text{Time}}} \right) \cr & {\text{Length}}\,{\text{of}}\,{\text{the}}\,{\text{train}} \cr & = \left( {\frac{{50}}{3} \times 9} \right)m = 150m \cr} $$
2. 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
Explanation:
$$\eqalign{ & {\text{Speed}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{relative}}\,{\text{to}}\,{\text{man}} = {\frac{{125}}{{10}}} {\text{ m/sec}} \cr & = {\frac{{25}}{2}} {\text{ m/sec}} \cr & = {\frac{{25}}{2} \times \frac{{18}}{5}} {\text{ km/hr}} \cr & = 45\,{\text{km/hr}} \cr & {\text{Let}}\,{\text{the}}\,{\text{speed}}\,{\text{of}}\,{\text{the}}\,{\text{train}}\,{\text{be}}\,x\,{\text{km/hr}}. \cr & \text{Then, relative speed} = \left( {x - 5} \right)\,{\text{km/hr}} \cr & x - 5 = 45 \cr & \Rightarrow x = 50\,{\text{km/hr}} \cr} $$
3.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
Explanation:
$$\eqalign{ & {\text{Speed}} = {45 \times \frac{5}{{18}}} \,{\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\frac{{25}}{2}} \,{\text{m/sec}} \cr & {\text{Time}} = 30\,{\text{sec}} \cr & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{bridge}}\,{\text{be}}\,x\,{\text{metres}} \cr & {\text{Then}},\,\frac{{130 + x}}{{30}} = \frac{{25}}{2} \cr & 2\left( {130 + x} \right) = 750 \cr & x = 245\,m \cr} $$
4. 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
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{speeds}}\,{\text{of}}\,{\text{the}}\,{\text{two}}\,{\text{trains}}\,{\text{be}}\,x\,{\text{m/sec}} \cr & {\text{and}}\,y\,{\text{m/sec}}\,{\text{respectively}}. \cr & {\text{Then,}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{first}}\,{\text{train}} = 27x\,{\text{metres}}, \cr & {\text{and}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{second}}\,{\text{train}} = 17y\,{\text{metres}}. \cr & \frac{{27x + 17y}}{{x + y}} = 23 \cr & 27x + 17y = 23x + 23y \cr & 4x = 6y \cr & \frac{x}{y} = \frac{3}{2} \cr} $$
5. 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
Explanation:
$$\eqalign{ & {\text{Speed}} = {54 \times \frac{5}{{18}}} \,{\text{m/sec}} = 15\,{\text{m/sec}} \cr & {\text{Length}}\,{\text{of}}\,{\text{the}}\,{\text{train}} = \left( {15 \times 20} \right){\text{m}} = 300\,{\text{m}} \cr & {\text{Let}}\,{\text{the}}\,{\text{length}}\,{\text{of}}\,{\text{the}}\,{\text{platform}}\,{\text{be}}\,x\,{\text{metres}} \cr & {\text{Then}},\,\frac{{x + 300}}{{36}} = 15 \cr & x + 300 = 540 \cr & x = 240\,{\text{m}} \cr} $$
6. 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
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{us}}\,{\text{name}}\,{\text{the}}\,{\text{trains}}\,{\text{as}}\,{\text{A}}\,{\text{and}}\,{\text{B}}{\text{.}}\,{\text{Then}}, \cr & \left( {{\text{A's}}\,{\text{speed}}} \right):\left( {{\text{B's}}\,{\text{speed}}} \right) \cr & = \sqrt b :\sqrt a \cr & = \sqrt {16} :\sqrt 9 \cr & = 4:3\, \cr} $$
7. 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
Explanation:
$$\eqalign{ & {\text{Speed }} \cr & {\text{ = }}\left( {60 \times \frac{5}{{18}}} \right){\text{m/sec}} \cr & {\text{ = }}\frac{{50}}{3}{\text{ m/sec}} \cr & {\text{Total distance covered}} \cr & {\text{ = (100 + 140) m = 240 m}} \cr & {\text{Required time}} = \left( {240 \times \frac{3}{{50}}} \right){\text{sec}} \cr & {\text{ = }}\frac{{72}}{5}{\text{sec}} \cr & {\text{ = 14}}{\text{.4 sec}} \cr} $$
8. 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
Explanation:
$$\eqalign{ & {\text{Speed}} = \left( {\frac{{132}}{6}} \right){\text{m/sec}} \cr & {\text{ = }}\left( {22 \times \frac{{18}}{5}} \right){\text{km/sec}} \cr & {\text{ = 79}}{\text{.2 km/hr}} \cr} $$
9. 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
Explanation:
$$\eqalign{ & {\text{Speed}} = \left( {60 \times \frac{5}{{18}}} \right){\text{m/sec}} \cr & {\text{ = }}\frac{{50}}{3}{\text{m/sec}} \cr & {\text{Time = 27 sec}}{\text{.}} \cr & {\text{Let the length of the train be }}x{\text{ metres}}{\text{.}} \cr & {\text{Then,}}\frac{{x + 200}}{{27}}{\text{ = }}\frac{{50}}{3}{\text{ }} \cr & \Leftrightarrow x + 200 = \left( {\frac{{50}}{3} \times 27} \right) = 450 \cr & \Leftrightarrow x = 450 - 200 = 250{\text{ metres}} \cr} $$
10. 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
Explanation:
$$\eqalign{ & {\text{Let the length of the train be x metres}}{\text{.}} \cr & {\text{Then, length of the platform = (2}}x{\text{) metres}}{\text{.}} \cr & {\text{Speed of the train}} \cr & {\text{ = }}\left( {90 \times \frac{5}{{18}}} \right)m/\sec \cr & = 25m/sec \cr & \frac{{x + 2x}}{{25}} = 36 \cr & 3x = 900 \cr & x = 300 \cr & {\text{Hence, length of platform}} \cr & {\text{ = }}2x = \left( {2 \times 300} \right){\text{m}} = 600{\text{m}} \cr} $$