Question

The minute-hand and second-hand of a clock cross each other ............... times between 09:15:00 AM and 09:45:00 AM on a day.

• A. 30
• B. 15
• C. 29
• D. 31

Hint:

For questions about the second hand and minute hand crossing, remember that they cross approximately once per minute. For a time interval of N minutes, the answer will almost always be N or N-1. Check the boundaries to be sure, but in most cases, it's simply the number of minutes in the duration.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to determine how many times the second-hand overtakes or "crosses" the minute-hand within a specific 30-minute time frame.
Step 2: Key Formula or Approach:
The second-hand moves much faster than the minute-hand. The second-hand completes a full 360-degree rotation every 60 seconds (1 minute). In that same minute, the minute-hand moves only a small amount. Therefore, the second-hand will lap (cross) the minute-hand approximately once every minute.
Step 3: Detailed Explanation:
1. The time interval is from 09:15:00 AM to 09:45:00 AM.
2. The duration of this interval is exactly 30 minutes.
3. Let's analyze one minute, for example, from 09:15:00 to 09:16:00. At 09:15:00, the minute hand is exactly on the '3' and the second hand is on the '12'. As the second hand sweeps around the clock face, it will inevitably cross the position of the minute hand before the minute is over.
4. This event—the second hand crossing the minute hand—happens once in every single minute interval.
5. The minute intervals in our time frame are:
- 09:15:xx (the first crossing happens in this minute)
- 09:16:xx (the second crossing)
- 09:17:xx (the third crossing)
- ... and so on, up to ...
- 09:44:xx (the final crossing within the interval)
6. The crossing at 09:45:xx would happen after our interval ends at 09:45:00.
7. To find the total number of crossings, we just need to count the number of minutes from 15 to 44, inclusive.
8. The number of minutes is \(44 - 15 + 1 = 29 + 1 = 30\).
Step 4: Final Answer:
The minute-hand and second-hand will cross each other 30 times.
Step 5: Why This is Correct:
Since the second hand laps the minute hand once every minute, a 30-minute duration will contain exactly 30 such lapping events. The start and end points of the interval do not coincide with a crossing, so we simply count the number of minutes in the interval.