Two motorists Anil and Sunil are practising with two different sports cars: Ferrari and Maclaren, on the circular racing track, for the car racing tournament to be held next month. Both Anil and Sunil start from the same point on the circular track. Anil completes one round of the track in 1 minute and Sunil takes 2 minutes to complete for a round. While Anil maintains same speed for all the rounds, Sunil halves his speed after the completion of each round. How many times Anil and Sunil will meet between the 6th round and 9th round of Sunil (6th and 9th rounds are excluded and the starting point is excluded)? Assume that the speed of Sunil remains steady throughout each round and changes only after the completion of that round.

The ratio of initial speeds of Anil and Sunil is 2:1 (Since speed is inversely proportional to time )

So in the time Sunil covers 'x' distance, Anil will cover '2x' distance.

Thus, at the start of the second round, they will meet for the first time time(Since Anil will complete 2 rounds while Sunil will complete his round).

Now, since the speed of Sunil is halved, the ratio will become 4:1.

At the starting point of the third round, they will meet for $$1+3=4^{th}$$ time(Since Anil will complete 4 rounds while Sunil will complete his round).

Now we can observe a pattern that at the starting point of every $$n^{th}$$ round, they are meeting at the $$\Sigma\ \left(2^k-1\right)^{th}time$$, where k ranges from 1 to (n-1).

Thus, at the start of the 7^th round, they will meet for the 120^th time, and at the starting point of the 9^th round, they will meet for the 502^th time.

The $$n^{th}$$ time they will meet between the 6^th and 9^th rounds (excluding the 6^th and 9^th rounds) are $$120^{st}$$, $$122^{nd}$$, $$123^{rd}$$....... $$501^{st}$$, $$502^{nd}$$.

So, the total meeting points will be 383 points.

Since the starting point of the $$7^{th}$$ round is excluded,

Total meets = 383 - 1 (Starting of 7^thround) = 382 meets


Alternate explanation:

Since Time = $$\frac{Distance}{Speed}$$ therefore as Sunil halves his speed the time required to complete the round will double therefore time required for Round 1 = 2 mins

Round 2 = 4 mins

Round 3 = 8 mins

Round 4 = 16 mins

Round 5 = 32 mins

Round 6 = 64 mins

Round 7 = 128 mins

Round 8 = 256 mins

Round 9 = 512 mins

The time required by Anil for Round 1 to Round 9 = 1 mins each

Thus, in the first two minutes, Sunil finished his 1st round, at the same time Anil finished 2 rounds.

When Sunil finished his 6th round, at the same time Anil finished his =2+4+8+16+32+64 rounds

.

Now for the 7th round of Sunil, he will take 128 mins.

Thus Anil will run 128 rounds.

Thus both of them will meet 128-1 =127 times

.

Now for the 8th round of Sunil, he will take 256 mins.

Thus Anil will run 256 rounds.

Thus both of them will meet 256-1 =255 times

= 255 + 127

= 382

Therefore answer is option 'C'