The question is from the topic relative Speed. This is relatively a straightforward question and only tests the basic concepts. Time Speed and Distance is a favorite in CAT Exam and appears more often than expected in the CAT Quantitative Aptitude section in the CAT Exam
Question 25: Two persons A and B start moving from point P towards Q which are 1400 Km apart. Speed of A is 50 Km/hr and that of B is 20 Km/hr. How far is A from Q when he meets B for the 22nd time?
Let's consider the Distance between P and Q, which is 1400km as one lap.
The Speeds of A and B are in the ratio 5 : 2. That means by the time A coimpletes 5 laps, B completes just 2 laps.
Both of them start moving from P, so after 5 laps(odd number of laps) A will be at Q, and B will be at P(even number of laps).
Seeing it from the perspective of A, he would have met P once evevy lap that he has made except the first lap since they are moving in the same dirction in the first lap.
So A and B would have met 5-1=4 times.
Let's consider another round of 5 laps. A runs 5 more laps and B runs 2 more laps
This time A starts from Q and after 5 laps reaches P, while in the same time B starts from P and ends at P.
This time they would have met exactly 5 times
Consolidating everything, we can say that A runs 10 rounds and B runs 4 rounds in the same time, reaching their original positions, meeting each other exactly 9 times
So cyclicity kicks in here....
22 can be written as 9 + 9 + 4.
After every 9 meets, A and B reach their original positions, that is both of them are at P. So, after 18 meets they both remain at P, For 4 more meets A must take 5 laps and B 2 laps, A being at Q and B at P at the end.
So at the end of this, A is at Q and B is at P.
Which means their last meet happened while they were travelling away from each other, which implies that they in the process together covered one lap(1400 km) after the last meet.
Which mean sA covered 5/7 th of 1400km after the last meet, ehich is the same as the distance between the last meeting point and Q.
So distance A and Q when he meets B for the 22nd time = 5/7 th of 1400km = 1000km
The question is " How far is A from Q when he meets B for the 22nd time? "
Choice A is the correct answer.
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