The average speed of a bus is three-fourth the average speed of a train. The train covers 240 km in 12 h . How much distance will the bus cover in 7 h?

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

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 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

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 speed of both trains be V km/h and (V + 6) km/h, respectively
Then, according to the question.
160 = V x 4 + (V + 6) x 4
? 160 = 4V + 4V + 24
? 40 = V + V + 6
? 2V + 6 = 40
? 2V = 34
? V = 17
Hence, speeds of both the trains are 17 km/h and (17 + 6 ) km/h i,e 17 km/h and 23 km/h .

First train leaves at 8 :00 am and second at 9:00 am.
So, first train i,e from P to Q has covered 25 km distance in 1 h.
So, distance left between the station = 145 - 25 = 120 km

Now, trains are travelling in opposite directions,
So, relative speed = 25 + 35 = 60 km/h
Time taken to cover 120 km = 120/60 = 2h
? The time, at which both the trains will meet, is 2 h after second train left i.e., 9 : 00 am + 2h = 11 : 00 am

Let x be the length of the standing train
? Speed of the train = (500 + x)/20
? 100 x (5/18) = (500 + x)/20
? 500 + x = (100 x 5 x 20)/18
? 500 + x = 555.56
? x = 555.56 - 500 = 55.56 m

Relative speed = (32 + 40) = 72 km/h = (72 x 5/18) m/s = 20 m/s
? Length of train = (20 x 6) = 120 m

Total distance to be traveled = 121 + 99 = 220 m
Relative speed = Sum of speeds = 72 km/h
= 72 x (5/18) = 20 m/s
? Required time = 220/20 = 11s

Let length of the train be x m.
Relative length of the train = (x + 600) m
According to the question,
(x + 600)/30 = 30
? x + 600 = 900
? x = 900 - 600 = 300 m