## Problems on Trains Questions and Answers Part-6

1. 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

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}

2. 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

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}

3. 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

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}

4. 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

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}

5. 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

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}

6. 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

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}

7. 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

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}

8. 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

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}

9. 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

\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}.
\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}