The time a watch loses per day, if its two hands coincide every 64 minutes (in mins.) is

To solve the problem, we first need to determine how often the hands of a normal watch coincide and then compare it to the given time.

### Step 1: Calculate the time for hands to coincide on a normal watch
In a normal watch, the hour and minute hands coincide 11 times in 12 hours (because 12 hours = 720 minutes). The time interval between each coincidence is:

\[
\text{Time interval} = \frac{720 \text{ minutes}}{11} = 65 \frac{5}{11} \text{ minutes}
\]
This simplifies to approximately \( 65.4545 \) minutes.

### Step 2: Compare with the given interval
The problem states that the hands coincide every 64 minutes. Let's compare this with the normal interval:

\[
\text{Difference} = 65.4545 - 64 = 1.4545 \text{ minutes}
\]

### Step 3: Calculate the total time lost per day
To find the total time lost per day (24 hours = 1440 minutes):

The number of coincidences in a normal 24-hour period is \( 22 \times 65.4545 = 1440 \text{ minutes} \). For the given watch, where coincidences happen every 64 minutes:

\[
\text{Total time for the given watch} = 22 \times 64 = 1408 \text{ minutes}
\]

The difference in time for 24 hours:

\[
\text{Time lost per day} = 1440 - 1408 = 32 \text{ minutes}
\]

So, the watch loses **32 minutes** per day.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How would the calculation change if the watch's hands coincided every 65 minutes?
2. What would be the time lost in 48 hours under the same conditions?
3. How can we calculate the time gain per day if the watch's hands coincided every 66 minutes?
4. Can you derive a formula to calculate the time lost or gained if hands coincide every \( x \) minutes?
5. What happens if the watch gains time instead of losing time? How would the hands' coincidence change?

**Tip:** Understanding the relationship between the hour and minute hands helps in solving more complex problems related to time measurement.

Explore how to calculate the time lost per day by a watch where its hands coincide every 64 minutes. Understand the relationship between the hour and minute hands and learn the mathematical process to determine the time discrepancy. Suitable for students in Grades 6-8.

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