Test Level 1: Speed, Time and Distance - 1 - CAT MCQ

From question,
Length of the train (L) will be S × T.
L = (36 km/h) × (8 sec) = 36 × 1000/3600 × 8 = 80 m

Let length of slower train-1 be L₁ and length of faster train-2 be L₂.
Relative speed when trains run in opposite directions = (54 + 36) × 5/18 = 25 m/s
So, 10 = (L₁ + L₂)/25
L₁ + L₂ = 250 ........(1)
Relative speed when trains run in the same direction = (54 - 36) × 5/18 = 5 m/s
Now,
20 = L₁/5
L₁ = 100 m ..........(2)
Put (2) in (1),
L₂ = 150 m

Given, both of them travel the same distance.
∴ Speed will be inversely proportional to time.
So, A and B will take 3x minutes and 2x minutes, respectively, to cover the same distance.
Given, 3x – 2x = 10
x = 10
So, A will take 30 minutes to cover the distance.
If A doubles his speed, then the time taken will be halved i.e. 15 minutes.

Let the distance be the LCM of 40 and 30, i.e. 120 km.
So, without stoppages, time taken by train = 120/40 = 3 hr
With stoppages, time taken by train = 120/30 = 4 hr
Thus, in a total time of 4 hours, duration for which the train stops = 4 - 3 = 1 hr
Thus, in a total time of 1 hour, it would stop for 1/4 hour, i.e. 15 minutes.

When two trains cross each other while moving in the opposite direction or when a train overtakes another train, the distance covered by the express train is equal to the sum of the lengths of both the trains.
Case 1: When the trains travel in opposite directions:
Distance travelled = Sum of lengths of the two trains = Relative speeds of the trains × Time taken
In this case, relative speeds = 72 + 45 = 117 km/hr (Relative speed when two objects travel in opposite directions = Sum of the individual speeds)
Time taken = Half a minute = 30 seconds 
As the speed is in km/hr and time is in seconds, let us convert the speed into m/sec to ensure congruency in the units. 

117 km/hr =  =  m/sec
Therefore, distance covered = Sum of the lengths of the two trains =  = 975 metres
Case 2: When the trains are travelling in the same direction:
Distance travelled = Sum of the lengths of the two trains = Relative speed × Time taken
Relative speed = (72 - 45) = 27 km/hr (When two objects travel in the same direction, their relative speed is equal to the difference between their individual speeds.)
As the lengths of the two trains will remain the same, irrespective of their direction of travel, the distance travelled will be the same in both the cases = 975 m 
Converting 27 km/hr into m/sec, we get  =  m/sec 
975 m =  × Time taken (in seconds) 
Time taken =  = 130 seconds

The ratio of time for the travel is 4:3 (Sinhagad to Deccan Queen). Hence, the ratio of speeds would be 3:4.
Since, the sum of their average speeds is 70 kmph, their respective speeds would be 30 and 40 kmph respectively.
Use alligation to get the answer as 34.28 kmph.

Since, the ratio of speeds is 3:5, the ratio of times would be 5:3.
The difference in the times would be 2 (if looked at in the 5:3 ratio context.)
Further, since Ram takes 30 minutes longer, 2 corresponds to 30.
Hence, using unitary method, 5 will correspond to 75 and 3 will correspond to 45 minutes.
Hence at 10 kmph, Bharat would travel 7.5 km.

If you assume that the initial stretch of track is covered by the two trains in time t each, the following figure will give you a clearer picture.

From the above figure, we can deduce that, t/3 = 12/t.
Hence, t² = 36, gives us t = 6.
Hence, the distance between Kolkata to the starting point is covered by the Calcutta Mail in 6 hours, while the same distance is covered by the Bombay Mail in 3 hours.
Hence, the ratio of their speeds would be 1:2. Hence, the Bombay Mail would travel at 96 kmph.

When the train from Khandala starts off, the train from Lonavala will already have covered 50 kms.
Hence, 550 km at a relative speed of 60 kmph will take 550/60 hrs.
From this, you can get the answer as:
50 + (550/60) * 25 = 279.166 km.

Since he gains 2 hours by driving both ways (instead of walking one way) the time taken for driving would be 2 hours less than the time taken for walking.
Hence, he stands to lose another two hours by walking both ways.
Hence his total time should be 8 hrs 45 minutes.