Quant Boosters  Rajesh Balasubramanian  Set 4

Q28) Two trains left from two stations P and Q towards station Q and station P respectively. 3 hours after they met, they were 675 Km apart. First train arrived at its destination 16 hours after their meeting and the second train arrived at its destination 25 hours after their meeting. How long did it take the first train to make the whole trip?
a) 18h
b) 36h
c) 25h
d) 48h

Total distance travelled by both the trains before meeting = D. This distance will be covered in the proportion of their speeds. Clarify with the diagram.
3 hours after meeting distance travelled by A = SA x 3 and by B = SB x 3
Therefore, 3 (SA + SB) = 675 =) SA + SB = 225.
Now the remaining distance to be covered by first train is DSB/(SA + SB).
Therefore, time taken = DSB/(SA+SB)SA = 16  (1)
Similarly, DSA/(SA+SB)SB = 25  (2)
Dividing equation 1 by 2
SA^2/SB^2 = 25/16 => SA/SB = 5/4.
Therefore, SA + 4/5 ∗ SA = 225 => SA = 125 Km/hr and SB = 100 Km/hrFrom equation 2, D/SA = 16 ∗ (SA + SB)/SB = 16 ∗ 225/100 = 36 h which is the time taken for the first train to complete the journey.

Q29) Arjun travels from A to B, a distance of 200 Km at the speed of 40 Km/hr. At the same time, Rakesh starts from point C at a speed of 20 Km/hr along a roadwhich is perpendicular to AB. Find the time in which Arjun & Rakesh will be closer to each other?
a) 1.5 h
b) 3.33 h
c) 5 h
d) 4 h

Distance between the two at any time t, D = sqrt((200−40t)^2 + (20t)^2)
One way to calculate is to find the minima using d/dt (D) = 0
=) 1/(2 sqrt(Doesn't matter)) x 2 x (200  40t) x 40 + (2 x 20t) x 20) = 0
=) 2 x (200  40t) x 40 = 40t x 20
=) 400  80t = 20t
=) t = 4 hours.Alternatively, you can find the right answer by putting the options in the equation.

Q30) Station X of length 900 meters has two station masters A and B. But as the station is not a busy one, they are mostly jobless and decide to conduct an experiment. They stand at either end of the station and decide to note the exact time when trains cross the stationmasters. They synchronize their watches and proceed to either end of the station. Two trains P and Q go past the station (neither train stops here), and after having taken down their readings, the station masters sit down to have a chat
A: Train P entered the station at exactly 8:00:00
B: Train Q entered the station at exactly 8:00:10 (10 seconds past 8)
A: The last carriage of train P crossed me by at 8:00:20, and precisely two seconds after this, the engines of the two trains went past each other. (Engines are at the front of the train)
B: The last carriage of train Q crossed me 22 seconds after the engine of P went past me.
A: After the last carriage of train P crossed by me, it took 35 seconds for the engine of train Q to cross me.
B: I got bored and I came here.Q1. What is the length of train Q?
Q2. At what time do the rear ends of the two trains cross each other?
Q3. How far from station master A do the rear ends of the two trains cross each other?
Q4. You are told that the two trains enter the station at the same times mentioned and the length of the two trains are unchanged. Furthermore, train P continues to travel at the same speed (as computed above). At what minimum speed should train Q travel such that the rear ends of the two trains cross each other at a point within the length of the platform?

Now, let us try to jot down the points in a slightly different format and see if we can make any inferences. Let us assume speed of train P = p, speed of train Q = q, length of train P = L, length of train Q = M
Take statements 1 and 3
Train P crosses the stationmaster A entirely in 20 seconds (enters station at 8:00:00 and last carriage passes at 8:00:20) => Length of train = 20p => L = 20p
Statement 5 tells us that engine of train Q crosses station master A at 8:00:55. Train Q enters the station at 8:00:10, so the train takes 45 seconds to cross the entire station. Or, it takes 45 seconds to travel 900 meters => Speed to train Q, q = 20m/s.
Statement 3 tells us that the two trains cross each other at 8:00:22. This implies train P has traveled for 22 seconds since entering the station and train Q has traveled for 12 seconds since entering the station before they cross each other. The cumulative distance traveled by the two trains should be equal to the length of the station = 900m.
=> 900 = 22p + 12q
q = 20m/s => p = 30m/s.
So, train P_ takes 900/p = 30 seconds to cross the station. So, engine of train P will cross stationmaster B at 8:00:30.
Statement 4 states that the last carriage of Q went past 22 seconds after the engine of P went by. Or, the last carriage of Q went by at 8:00:52.
Engine of train Q went by at 8:00:10, last carriage went by at 8:00:52, or train Q took 42 seconds to cross station master B. Train Q travels at 20m/s. => Length of train Q = 840m
Now, to the questions
Q1) What is the length of train Q?
Train Q is 840m long
Q2) At what time do the rear ends of the two trains cross each other?
The engines cross each other at 8:00:22. The relative speed of the two trains = 20+30 = 50m/s. The relative distance traveled by two trains from the time the engines cross each other to the times the rear ends cross each other = Sum of the two lengths = 600 + 840 =1440. Time taken = 1440/50 = 28.8 seconds past 8:00:22, or at 8:00:50.8 seconds.
Q3) How far from station master A do the rear ends of the two trains cross each other?
At 8:00:50.8, P would have traveled 50.8 * 30m/s post entering the station. Or, train P would have traveled 1524m. The rear end of train P would be at a point 1524600m = 924m from stationmaster A (or 24 meters from stationmaster B and outside the station)
Q4) You are told that the two trains enter the station at the same times mentioned and the length of the two trains are unchanged. Furthermore, train P continues to travel at the same speed (as computed above). At what minimum speed should train Q travel such that the rear ends of the two trains cross each other at a point within the length of the platform?
Rear end of train P crosses the station completely at 8:00:50. (Train P takes 30 seconds to travel the station and 20 seconds to travel a distance equal to its length). Train Q should have traveled 840m by this time. => Train Q should travel 840 within 40 seconds.
Or, minimum speed of Q = 840/40 = 21m/s.

we can do this another way too:
The time in which antony covers 20 m, amar will cover 20 + 12  14 = 18 m
=> (20/antony) = (18/amar)
=> amar = (18/20) antony = (9/10) antony
Choice a .

As C is at some fixed distance say x from B
shouldnt the distance at time t include x also?
D= sqrt((20040t)^2+ (x20t)^2)???

In every meet their combined distance travelled is 100 m.
Let there be "a" meets.
=>[(100a) * 5/(5+3)]=300
Which gives a=4.8Though it cant be decimal.
But What is wrong with this approach????
As there is a similar question in gyan roomarithmetic shortcuts...and i hv applied the same approach here..

@rajesh_balasubramanian Hello. I was just looking for an online CAT coaching and stumbled upon this. The answer seems to be 7. And i don't think 10 can be a value for B. Eg: Take the Track length as 300m. A will complete it in 50secs. And B will take 75 secs to overlap A for the first time(300/4). I can email you my solution if you'd like to give it a look.
Thank you.