19. A train travelling at 36 kmph completely crosses another train having half its length and travelling in the opposite direction at 54 kmph, in 12 seconds. If it also passes a railway platform in ${1\displaystyle\frac{1}{2}}$ minutes, the length of the platform is :
a. 560 metres
b. 620 metres
c. 700 meres
d. 750 metres

Answer: C

Explanation:
Let the length of slower train be $x$ metres and the length of faster train be $\left( {\displaystyle\frac{x}{2}} \right)$ meters.
Relative speed = (36 + 54)km/hr =$\left({90 \times \displaystyle\frac{5}{{18}}} \right)$ m/sec = 25 m/sec
$\displaystyle\frac{{3x}}{{2 \times 25}} = 12 \Rightarrow 3x = 600 \Rightarrow x = 200$ m
Length of slower train = 200 m
Let the length of platform be $y$ metres
Then, $\displaystyle\frac{{200 + y}}{{36 + \displaystyle\frac{5}{{18}}}} = 90 \Rightarrow 200 \times y = 900$ or
y = 700 m
Length of platform = 700 m

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20. A train 100 metres in length passes a milestone in 10 seconds and another train of the same length travelling in opposite direction in 8 seconds. The speed of the second train is :
a. 36 kmph
b. 48 kmph
c. 54 kmph
d. 60 kmph

Answer: C

Explanation:
Speed of first train = $\left( {\displaystyle\frac{{100}}{{10}}} \right)$ m/sec = 10 m/sec
Let the speed of 2nd rain be x m/sec
Relative speed = (10 + x) m/sec
${\displaystyle\frac{{200}}{{10 + x}} = 8 \Rightarrow 200 = 80 + 8x \Rightarrow x = 15}$
Speed of 2nd train = 15 m/sec = $\left( {15 \times \displaystyle\frac{{18}}{5}} \right)$ km/hr = 54 km/hr.

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21. Two trains are running in opposite directions with speed of 62 kmph and 40 kmph respectively. If the length of one train is 250 metres and they cross each other in 18 seconds, the length of the other train is :
a. 145 metres
b. 230 metres
c. 260 metres
d. Cannot be determined

Answer: C

Explanation:
Let the length of the another train = x meres
Their relative speed = (62 + 40) km/hr = $\left( {102 \times \displaystyle\frac{5}{{18}}}\right)$m/sec = $\left( {\displaystyle\frac{{85}}{3}} \right)$
$\displaystyle\frac{{250 + x}}{{\displaystyle\frac{{85}}{3}}} = 18 \Rightarrow \displaystyle\frac{{3(250 + x)}}{{85}} = 18$
$ \Rightarrow 250 + x = 510 \Rightarrow x = 260$
Length of another train = 260 m

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22. A train 100 metres long moving at a speed of 50 kmph crosses a train 120 metres long coming from opposite direction in 6 seconds. The speed of second train is :
a. 132 kmph
b. 82 kmph
c. 60 kmph
d. 50 kmph

Answer: B

Explanation:
Let the speed of the second train be x km/hr.
Relative speed = (50 + x) km/hr = $\left[ {(50 + x) \times \displaystyle\frac{5}{{18}}} \right]$ m/sec =$\left( {\displaystyle\frac{{250 + 5x}}{{18}}} \right)$ m/sec.
$ \Rightarrow \displaystyle\frac{{100 + 120}}{{\displaystyle\frac{{250 + 5x}}{{18}}}} = 6$ or
$220 \times 18 = 6(250 + 5x)$ or 30x = 3960-1500 or x = $\displaystyle\frac{{2460}}{{30}} = 82$
Speed of the second train = 82 m/s.

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23. Two stations A and B are 110 kms.apart on a straight line. One train starts from A at 7 a.m and travels towards B at 20 km per hour speed. Another train starts from B at 8.a.m and travels towards A at a speed of 25 km per hour. At what time will they meet?
a. 9 a.m
b. 10 a.m
c. 11 a.m
d. None of these.

Answer: B

Explanation:
Suppose they meet x hrs after 7 a.m.
Distance covered by A in x hrs = $(20 \times x)$ km
Distance covered by B in (x - 1) hrs = 25(x - 1) km
20x + 25(x - 1)=110 or
45x = 135 or x = 3
So, they meet at 10 a.m

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24. 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. 72 metres
b. 54 metres
c. 50 metres
d. 45 metres

Answer: C

Explanation:
Let the length of the train be $x$ km and its speed $y$ km/hr. Then, speed relative to first man = $(y-2)$ km/hr.
Speed relative to second man = $(y-4)$ km/hr.
$\displaystyle\frac{x}{{y-2}} = \displaystyle\frac{9}{{60 \times 60}}$ and $\displaystyle\frac{x}{{y - 4}} = \displaystyle\frac{{10}}{{60 \times 60}}$
$9y - 18 = 3600x$ or $10y - 40 = 3600x$
So, $9y -18 = 10y-40$ or $y =22$
$\displaystyle\frac{x}{{22 - 2}} = \displaystyle\frac{9}{{3600}}$ or x = $\displaystyle\frac{{20 \times 9}}{{3600}} = \displaystyle\frac{1}{{20}}$
= $\left( {\displaystyle\frac{1}{{20}} \times 1000} \right)$m = 50 m

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25. A man standing on a railway platform notices that a train going in one direction takes 10 seconds to pass him and other train of the same length takes 15 seconds to pass him. Find the time taken by the two trains to cross each other when they are running in the opposite directions.
a. 12
b. 14
c. 18
d. 25

Answer: A

Explanation:
Use the following formula:
Time taken for the trains to cross each other = $\displaystyle\frac{{{\rm{2ab}}}}{{{\rm{a + b}}}}{\rm{ = }}\displaystyle\frac{{{\rm{2 \times 10 \times 15}}}}{{{\rm{25}}}}$ = 12 seconds

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26. A man standing on a railway platform notices that a train going in one direction takes 9 seconds to pass him and other train of the same length takes 6 seconds to pass him. Find the time taken by the two trains to cross each other when they are running in the same direction. a. 15 b. 30 c. 32 d. 36

Answer: D

Explanation:
Use the following formula:
Time taken for the trains to cross each other = $\displaystyle\frac{{{\rm{2ab}}}}{{{\rm{a - b}}}}{\rm{ = }}\displaystyle\frac{{{\rm{2 \times 9 \times 6}}}}{{\rm{3}}}$ = 36 seconds