Question: Arjun and Bhim can run a full around a circular track in 4 minutes and 7 minutes respectively. If they start simultaneously, after how much time will they meet together at a point diametrically opposite form the starting point?

1. 3 minutes
2. 5.5 minutes
3. 11 minutes
4. 28 minutes
5. Never

Answer:
Solution with Explanation:
Approach Solution (1):

Time taken by A = 4 minutes and that by B = 7 minutes
To avoid complicated calculations, we can assume the length of the track as 28 meters (LCM of 4 and 7)
[Note: You may also take it as 28k meters]

Thus, speeds of A and B = 7 meters/min and 4 meters/min
Same direction:
Speed ratio = 7 : 4 (lowest form)
Thus, the number of meeting points = 7 – 4 = 3
Each gap (length) between points = \(\frac{28}{3}\)meters
Thus, the meetings are: \(\frac{28}{3}\) meters from starting point;\(\frac{56}{11}\)meters from starting point and the starting point itself

Opposite direction:
Speed ratio = 7 : 4 (lowest from)
Thus, the number of meetings points = 7 + 4 = 11
Each gap (length) between meeting points = \(\frac{28}{11}\)meters
Thus, the meetings are: \(\frac{28}{11}\) meters from starting point, \(\frac{56}{11}\) meters from starting point and so on.
Thus, there would never be any meeting diagonally opposite to the starting point

Correct Option: E

Approach Solution (2):

Let they meet at a point diametrically opposite. For that, let A cover x laps and B cover y laps.
We have already assumed the lap length as 28 meters
Thus, equating the time taken by A and B, we have:
\(\frac{(28x+14)}{7} = \frac{(28y+14)}{4}\)
\(\Rightarrow 4x+2 = 7y+3.5\)
\(\Rightarrow 4x-7y = 1.5\)

However, x and y are integers, hence the above is not possible

Correct Option: E

Approach Solution (3):

Ratio of speed of Bhim and Arjun = 7:4

a.If the length of circular track = 28 m, the speeds of Bhim and Arjun are 7 and 4 m/ min
The time when they are together for the first time will be when Bhim (the faster one) has taken one round more than Arjun.
Therefore, if time when they meet is ‘t’ then
7t - 4t = 28. which means t = 28/3 min

b.They will meet at the starting place the first time at a time which is the LCM of the times each one of them takes to reach the starting place
Therefore, LCM of 4, 7 is 28 min.

c.Diametrically opposite point is at a circular distance of 14 m.
Bhim reaches this point in 14/7 = 2 min and Arjun reaches this point in 14 / 4 = 3.5 min.
Bhim reaches this point in the 2nd min, 2+4 = 6 min, 6+4 = 10th min... so on. Arjun reaches after 3.5 min, 10.5 min, 17.5 min ...so on.
The time after the start when Bhim reaches the point is a natural number, Whereas the time when Arjun reaches this point will always be a non-natural number. So they will never meet.

Correct Option: E

“Arjun and Bhim can run a full around a circular track in 4 minutes and 7 minutes respectively. If they start simultaneously, after how much time will they meet together at a point diametrically opposite form the starting point?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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