Clock Hands Coincidence Explained

This question asks us to determine how many times the hour hand and the minute hand of a standard analog clock are in the exact same position (coincide) during a period of half a day. A half day is equivalent to 12 hours.

Understanding Clock Hand Movement

To figure this out, let's consider the speeds of the two hands:

• The minute hand completes a full circle (360 degrees) in 60 minutes. Its speed is:

  Speed_minute = \frac{360^{\circ}}{60 \text{ min}} = 6^{\circ} \text{ per minute}

• The hour hand completes a full circle (360 degrees) in 12 hours (which is $12 \times 60 = 720$ minutes). Its speed is:

  Speed_hour = \frac{360^{\circ}}{720 \text{ min}} = 0.5^{\circ} \text{ per minute}

Calculating Relative Speed

For the hands to coincide, the faster minute hand needs to "catch up" to the slower hour hand. We look at their relative speed:

Relative Speed = Speed_minute - Speed_hour

Relative Speed = $6^{\circ} \text{ per minute} - 0.5^{\circ} \text{ per minute} = 5.5^{\circ} \text{ per minute}

Or, in fractional form: Relative Speed = $\frac{11}{2}^{\circ} \text{ per minute}$

Determining Coincidence Frequency

The hands coincide when the minute hand gains a full 360 degrees on the hour hand. Let's calculate the time it takes for this to happen:

Time between coincidences = \frac{360^{\circ}}{\text{Relative Speed}}

Time between coincidences = \frac{360^{\circ}}{5.5^{\circ} \text{ per minute}} = \frac{360}{\frac{11}{2}} \text{ minutes}

Time between coincidences = \frac{720}{11} \text{ minutes}

This means the hands coincide approximately every $\frac{720}{11}$ minutes (about 65.45 minutes).

Coincidences in Half a Day (12 Hours)

Now, we need to find out how many times this happens in 12 hours.

Total time = 12 hours = $12 \times 60 = 720$ minutes.

Number of coincidences = \frac{\text{Total time}}{\text{Time between coincidences}}

Number of coincidences = \frac{720 \text{ minutes}}{\frac{720}{11} \text{ minutes}}

Number of coincidences = $720 \times \frac{11}{720} = 11$

Alternatively, think about a 12-hour period. The hands start together at 12:00. They then coincide roughly once every hour (e.g., around 1:05, 2:11, etc.). However, the coincidence that would happen around 11:xx actually occurs exactly at 12:00, completing the cycle. So, in a 12-hour duration, there are exactly 11 instances where the hands coincide.

Conclusion

The hour hand and minute hand of a clock coincide 11 times in a 12-hour period (half a day).