Shortcut 55:
Train crossing a man on another train in same direction
$$T=\frac{L_T}{S_1\sim S_2}$$
Question:
Two trains A and B of length 100m and 200 are travelling at a speed of 72kmph and 54kmph respectively in same directions. How long will train A take to cross a man travelling in train B?
LT = 100, (Length of train crossing the man)
S1 = 72kmph = 72(5/18)mps = 20mps
S2 = speed of the man travelling in train B
S2 = 54kmph = 36(5/18)mps = 10mps
Substitute the values in the above formula, we get
T = (100)/(20-10)
= 10 seconds

Shortcut 56:
Time taken to meet from different stations
$$T=\frac D{S_1+S_2}$$
Question:
Two trains A and B start from stations 1 and 2, 300km apart at 9am and travel towards each other at a speed of 30kmph and 50kmph respectively. At what time they will meet?
D = 300km
S1 = 30kmph; S2 = 50kmph
T = 300/(30+50)
= 3.75 hours
= 3 hours 45 minutes
The time at which the two trains will meet is,
= 9am + 3:45 hours
= 12:45pm

Shortcut 57:
Time taken for trains to meet in same direction
$$T=\frac D{S_1\sim S_2}$$
Train A starts from station at 10am at a speed of 50kmph. Train B starts from the same station at 11am at a speed of 55kmph. How long will train B take to meet train A?
Difference in time of starting between the two trains = 11am – 10am = 1 hour
Distance travelled by train A in 1 hour = 1 x 50
= 50km
D = 50, distance between two trains
S1 = 50; S2 = 55
T = 50/(55 – 50) = 10 hours
The two trains will meet at 11am + 10 hours
= 9pm

Shortcut 58:
Speed ratio
$$S_1:S_2=\sqrt a:\sqrt b$$
a, b – time taken by respective trains travelling in opposite direction, to reach destination after they meet.
Question:
Two trains start from stations A and B. Speed of train from A is 60kmph. The two trains meet each other and take 9 and 4 hours to reach the destined stations. What is the speed of train from B?