M and N start at 5:00 a.m. from the same point on a circular track and walk in opposite directions. M walks at 5 rounds per hour and N at 2 rounds per hour. How many times will they cross each other before 6:30 a.m. on the same day?

Question

Accepted Answer

Let the length of the circular track be 1 unit (since only rounds matter).

M's speed = 5 rounds/hour
N's speed = 2 rounds/hour

They walk in opposite directions, so their relative speed = 5 + 2 = 7 rounds/hour.

Time interval: From 5:00 a.m. to 6:30 a.m. = 1.5 hours

Total distance (in rounds) covered by their relative speed in 1.5 hours = 7 × 1.5 = 10.5 rounds

Each time their combined distance equals a whole number of rounds, they meet (cross each other).

Number of times they cross each other = Number of whole rounds completed by their relative speed before 6:30 a.m.

So, in 10.5 rounds, they cross each other at the end of 1st, 2nd, ..., 10th round (i.e., after 1, 2, ..., 10 rounds).

Therefore, they cross each other 10 times before completing 10.5 rounds. But, since they start together at 0, the first meeting is at the starting point (which is not counted as a crossing after starting), so we count only the crossings after they start.

But in such problems, the number of meetings is equal to the integer part of the total rounds covered by their relative speed in the given time.

So, Number of crossings = Integer part of 10.5 = 10.

However, since they start together, the first meeting is at the starting point (at 0 time), which is not to be counted as a crossing after starting. So, the number of times they cross each other after starting = 10.

But let's check the standard approach:

- They meet every 1/7 hours (since relative speed is 7 rounds/hour, so 1 round = 1/7 hour).
- Number of meetings in 1.5 hours = 1.5 / (1/7) = 10.5
- So, they meet at 1/7, 2/7, ..., 10/7, 11/7, ..., up to 1.5 hours.
- The number of meetings before 1.5 hours = Integer part of 10.5 = 10.

Therefore, the correct answer is 10.

But let's check the options: 10, 3, 5, 7. Option 1 is 10.

So, the correct answer is 10.

But the answer key says 5. Let's check if the question is asking about 'crossings' or 'overtakings'.

Since they are walking in opposite directions, every time their combined distance is a multiple of the track's length, they cross each other.

So, the number of crossings = Integer part of (relative speed × time) = Integer part of (7 × 1.5) = 10.

Therefore, the correct answer is 10.

But since the answer key says 5, perhaps the question is asking about the number of times they cross each other excluding the starting point and only counting when they meet at the starting point. But that's not what the question says.

Therefore, the correct answer is 10.

Answer

Answer

Answer