Two station P and Q are at a distance of 160 km. Two trains starts moving from P and Q to Q and P respectively and meet each other after 4 h . If speed of the train starting from P is more than that of other train by 6 km/h. then find the speeds of both the trains, respectively.?

If the trains meet after t h.
Relative speed of train = (24 -18) = 6 km/h
? Distance = 27
? t = 27/6 = 9/2 h
? QR distance travel by train which is travelling at a speed of 18 km/h = 18t = 18 x 9/2 = 81 km

Given that, t₁ = 6 s and t₂ = 9 s
Then, time taken by the trains to cross each other = 2t₁ t₂ / (t₂ - t₁)
= (2 x 6 x 9)/(9 - 6) = 36 s

The length of the fast train = Relative speed x Time
= (40 - 20) x (5/18) x 5 = 27⁷/₉ m

Let the speeds of two trains be x and y, respectively .
? Length of 1st train = 54x
Length of the 2nd train = 34y
According to the question.
(54x + 34y) / (x + y) = 46
? 54x + 34y = 46x + 46y
? 27x + 17y = 23x + 23y
? 4x = 6y
? x/y = 3/2
? x : y = 3 : 2

Because of stoppages, train covers 50 km less per hour.
? Time taken taken to cover 50 km = 50/150 h = (50/150) x 60 = 20 min

Let the two trains meet at D km from Mumbai. Then,
3:45 pm + D/50 = 2 : 35 pm + 510 - D/60
? (3:45 pm - 2:35 pm) + (D/50 + D/ 60) = 510/60
? (1¹/₆ h) + D{( 50 + 60 / 50 x 60)} = 17/2
? D(110/3000) = 17/2 - 7/6
? D(110/3000) = (51 - 7)/6 = 44/6
? D = (44/6) x (3000/110) = 200 km

Let the lengths of the trains A and B be a and 2a, respectively.
when a train crosses a pole, it covers the distance equal to its length.
? Required ratio of speeds = a / 25 : 2a /75 = 3 : 2

The distance covered by the train to cross the platform and bridge
= Length of the platform + Length of bridge + Distance between platform and bridge + Length of the train
= 165 + 135 + 30 + 110 = 440 m
Speed of train = 110 /3 m/s
? Required time taken = 440/(110/3) = 12 s

Let time taken to cover the distance = t
? Speed = (2t + 1)
? t(2t + 1) = 1830 ? 2t² + t = 1830
? 2t² + t - 1830 = 0
? t = -1 ± ?(1)² - 4 x 2 x(-1830) / 2 x 2
= -1 ± ?1 + 14640 / 4
= -1 ± 121 / 4
Taking '+' sign t = -1 + 121 / 4 = 120/4 = 30
? Required ratio = (2 x 30 + 1) : 30 = 61 : 30

Average speed of train = 240/12 = 20 km/h
Average speed of bus = (3/4) x 20 = 3 x 5 = 15 km/h
Required distance = 15 x 7 = 15 km/h