Understanding Clock Hand Coincidence

The question asks us to determine the number of times the hour hand and the minute hand of a clock coincide within a period of one week. To solve this, we first need to understand how often the hands coincide in a shorter period, like 12 hours or 24 hours.

Clock Hands Relative Movement

The minute hand and the hour hand of a clock move at different speeds. This difference in speed causes them to overlap or coincide at regular intervals.

• The minute hand completes a full circle (360 degrees) in 60 minutes. Its speed is $\frac{360}{60} = 6$ degrees per minute.
• The hour hand completes a full circle (360 degrees) in 12 hours (720 minutes). Its speed is $\frac{360}{720} = 0.5$ degrees per minute.

The minute hand is faster than the hour hand. The difference in their speeds is the relative speed at which the minute hand gains on the hour hand.

Relative speed = Speed of minute hand - Speed of hour hand = $6 - 0.5 = 5.5$ degrees per minute.

How Often Hands Coincide in 12 Hours?

For the hands to coincide, the minute hand must gain 360 degrees on the hour hand (or 0 degrees difference between them, relative to their position at coincidence). Since the minute hand gains 5.5 degrees every minute, it takes $\frac{360}{5.5} = \frac{3600}{55} = \frac{720}{11}$ minutes for them to coincide after they were previously together.

This time interval is approximately 65.45 minutes. Starting from 12:00, the hands coincide at:

• 12:00
• Approximately 1:05
• Approximately 2:10
• ...and so on.

In a 12-hour period, the hands coincide 11 times, not 12. This is because the coincidence that would typically occur between 11 o'clock and 12 o'clock happens exactly at 12 o'clock, which is the start of the next 12-hour cycle. So, over the interval from, say, 12 AM to 12 PM, the hands coincide at 12:00 AM, and then 10 more times throughout the morning/afternoon, finally coinciding again at 12:00 PM. This makes a total of 11 coincidences in the 12-hour period.

How Often Hands Coincide in 24 Hours (One Day)?

A day consists of two 12-hour periods. Since the hands coincide 11 times in each 12-hour period, the total number of coincidences in 24 hours is:

$11 \text{ coincidences/12 hours} \times 2 \text{ (12-hour periods/day)} = 22 \text{ coincidences/day}$.

So, the hands of a clock coincide 22 times in one day.

Calculating Total Coincidences in a Week

A week has 7 days. To find the total number of times the hands coincide in a week, we multiply the number of daily coincidences by the number of days in a week:

Total coincidences in a week = Coincidences per day $\times$ Number of days in a week

Total coincidences in a week = $22 \times 7$

Let's calculate the product:

$22 \times 7 = 154$

Therefore, the hands of the clock will coincide with each other 154 times in a week.

Revision Table: Clock Hand Coincidence Frequency

Additional Information: Why 11 Times in 12 Hours?

The key reason the hands coincide 11 times in 12 hours instead of 12 is the relative movement. The minute hand has to "catch up" to the hour hand from behind. If they coincide at a certain time, they won't coincide again until the minute hand has made almost a full extra lap compared to the hour hand. This takes slightly longer than one hour (about 65.45 minutes). Over a 12-hour period, this slightly longer interval means that one coincidence 'slot' is missed. The coincidence at 12:00 serves as both the end of one 12-hour cycle and the beginning of the next.

Another way to think about it: If they coincided exactly every hour, they would coincide 12 times. But since the minute hand takes more than an hour to catch up again, the twelfth catch-up in a 12-hour period doesn't happen within the period itself but exactly at the end, which is the start point, making it count only once for the entire 12-hour interval from the beginning to the end point.