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A train travels at a certain speed for 30 minutes. After that due to a malfunction, it travels at twothirds of the original speed and reaches its destination 1 hr 30 mins late. Had the malfunction happened after travelling another 60 km, it would have been only 1 hour late. Find the initial speed of the train.
The difference between times in two cases = 30 minutes
This difference is only due to the reason that train travelled 60 km distance at 2/3 of its original speed.
Let train initially took t minutes to cover 60 km at its original speed.
According to given condition,
(3t/2)  t = 30
t = 60 minutes
i.e the train initially covered 60 kms in 60 minutes
i.e the initial speed of train is 60 kmph.
A car travels along a straight line and covers onethird of the total distance at a speed of 4 m/s. Out of the total time taken to cover the remaining distance, it travels at a speed of 2m/s for half the time and at a speed of 6 m/s for the remaining half of time. What is the average speed of the car?
Let the total distance be 12d m.
So, the car covers a distance of 4d m at a speed of 4 m/s.
Time taken to cover this distance = (4d/4) s = d s
Let the car take a time of 2t s to cover the remaining 8d m.
So, it travels at a speed of 2 m/s for t seconds and at a speed of 6 m/s for the other t seconds.
Thus, we have 2t + 6t = 8d or t = d.
Now, average speed = total distance/total time = 12d/(d + 2t) = 12d/(d + 2d) = 12d/3d = 4 m/s
Hence, answer option 1 is correct.
Singapore is 3 hours ahead of Chennai and New Jersey is 11 hours behind India. A flight from Chennai took off at 6 a.m. and reached Singapore at 12 noon and a flight with the same speed takes off from Chennai at 9 a.m. If the distance between Chennai and New Jersey is exactly 4 times the distance between Chennai and Singapore, at what time does the flight, which takes off from Chennai at 9 a.m, reach New Jersey at its local time? (Consider that all the given timings are local timings.)
The time taken by the flight to fly from Chennai to Singapore is 3 hours.
(Actually, it is 12  6 = 6 hours according to the local time, but Singapore is 3 hours ahead of Chennai.)
So, to fly from Chennai to New Jersey, it takes 3 ^{} 4 = 12 hours (because it is 4 times more distant).
So, the flight which takes off at 9 a.m. in Chennai will reach New Jersey at 9 p.m. according to Chennai time. But then, the local time will be 10 a.m. because it is 11 hours behind.
Saurav and Sachin are running on a 500 m circular track. Their speeds are 8 km/hr and 5 km/hr, respectively. After what time will they meet for the first time at the starting point if they start simultaneously in the same direction?
Let us first calculate the time Saurav and Sachin take to cover one full circle.
Time taken by Saurav = _{} = ^{} = 225 seconds
Time taken by Sachin = _{} = 360 seconds
Hence, after every 225 seconds, Saurav would be at the starting point and after every 360 seconds, Sachin would be at the starting point. The time when they will be together again at the starting point simultaneously for the first time would be the smallest multiple of both 225 and 360, which is the LCM of 225 and 360.
Hence, they would both be together at the starting point for the first time after (LCM of 225, 360) = 1800 seconds = 0.5 hour.
Thus, every half an hour, they would meet at the starting point.
Two men are walking towards each other alongside a railway line. They both start at opposite ends of a part of the railway track. One man is walking faster than the other. They pass each other 720 yards from the nearer of the two ends. When each man reaches the other end of the track, they stop for a cigarette. Both men take exactly the same amount of time to smoke a cigarette. Then, they turn around and head back to the original starting position. This time they meet 400 yards from the other end of the track. What is the length of the track?
When the men first meet, they are 720 yards from one end of the track. The combined distance they have travelled must equal the length of the track.
When each man reaches the other end, the combined distance must be twice the length of the track.
On meeting on the return, they must have travelled a total of three times the length of the track.
Since one man has gone 720 yards on the first meeting and when they meet again, he has gone three times as far, he must have travelled 2160 yards.
Since he is 400 yards from where he started, the length of the track must be 2160  400 yards = 1760 yards.
SlowRun Express runs between Bangalore and Mumbai. For the up as well as the down journey, the train leaves the starting station at 10:00 p.m. everyday and reaches the destination at 11:30 p.m. after three days.
If Mr. Hanni travelled by SlowRun Express, how many SlowRun Express trains did Mr. Hanni cross during his journey?
Let us assume that Mr. Hanni sits in the train on Monday. So, he is scheduled to reach on Thursday at 11:30 p.m.
When the train starts from Mumbai, the first SlowRun Express which he crosses is the one which is scheduled to reach Mumbai at 11:30 p.m, that is the same day (Monday). This train started 3 days ago, i.e. on Friday.
Now, on his way, he will cross the trains which started on Saturday, Sunday and Monday, and the trains which started from Bangalore when Mr. Hanni was himself in the train, i.e. the trains started from Bangalore on Tuesday, Wednesday and Thursday.
Thus, Mr. Hanni must have crossed 7 SlowRun Express trains during his journey.
Train A and Train B are running on parallel tracks with the speeds of 72 kmph and 54 kmph, respectively. If they move in the same direction, Mr. Ravi, who sits in the slower train, observes that the faster train passes him in 18 seconds. If they move in opposite directions, both the trains pass each other in 24 seconds. Find the length of the slower train.
Time is given in seconds, so speeds should be metre per second.
Speed of the faster train = 72 × 5/18^{ }= 20 m/s
Speed of the slower train = 54 × 5/18^{ }= 15 m/s
Let the length of the trains be x and y m, respectively.
Total distance to be covered = Sum of the lengths of the two trains = x + y
When they move in opposite directions:
Relative speed = Sum of their speeds = 20 + 15 = 35 m/s
Time taken = 24 seconds
Distance = x + y
Distance = Relative Speed × Time
⇒ x + y = 35 × 24 = 840 m
When they move in the same direction:
Time taken = 18 seconds
Here, the faster train passes Mr. Ravi only, so distance is equal to the length of the faster train.
Distance = Length of the faster train
Distance = Relative Speed × Time
⇒ x = 5 × 18 = 90 m
We have, x + y = 840 and x = 90 m
⇒ y = 750 m
So, the length of the slower train is 750 m.
Ram and Shyam run around a circular track of length 600 metres. They start simultaneously from the same point and in the same direction. Ram, who runs faster, crosses Shyam in the middle of the 5^{th} round. If Ram and Shyam were to run a 3 km race, how much start in terms of distance should Ram give Shyam so that they finish the race in the same time?
Ram and Shyam run around a circular track.
Ram crosses Shyam in the middle of the 5^{th} lap, i.e. when Ram has run four and a half laps, he has covered a distance which is 1 lap greater than that covered by Shyam.
Therefore, when Ram runs 9/2 laps, Shyam runs 7/2 laps.
This is the same as saying when Ram runs 9 laps, Shyam runs 7 laps, i.e. in a race that is 9 laps long, Ram can give Shyam a start of 2 laps.
So, if the race is of 3000 metres long track, then Ram can give Shyam a start of (2/9) × 3000 = 666.67 metres.
In an event, A rows his boat upstream in a river flowing with a steady speed of 6 km/hr and then returns to the same point without rowing (i.e. only with the speed of river). If the total distance covered by A is 42 km and the time taken in rowing the boat upstream is 4 hours more than his return journey, find the speed of A in still water.
Given:
Total distance covered = 42 km
River speed = 6 km/hr
Distance covered in both directions = 21 km
According to the question,
Suppose the speed of boat in still water be x km/hr.
Then, Time taken in rowing upstream = Time taken in returning to the same point + 4 hr
Therefore, the speed of the boat in still water is 8.8 km/hr.
Hence, this is the required solution.
A boat takes 6 hours to travel from A to B (upstream). The river flows at the rate of 2 kmph. How long will the boat take to travel from B to A (downstream), if the distance from A to B is 36 km?
Upstream:
Let t be the time taken by the boat to travel from A to B = 6 hours
Let V_{1} be the speed of the boat travelling upstream = x
Let V_{2} be the speed of the stream = 2 kmph
Distance = 36 km
V_{1 } V_{2 }= 36/6^{ }= 6 kmph
V_{1} = 8 kmph
Downstream:
V_{1} = 8 kmph
V_{2 }= 2 kmph
Net speed = V_{1 }+ V_{2 }= 10 kmph
Distance = 36 km
Time = 36/10^{ }= 3.6 hours
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