If the two incorrect watches are set at 12 : 00 noon at correct time, when will both the watches show the correct time for the first time given that the first watch gains 1 min in 1 hour and second watch loses 4 min in 2 hours :

Options

A. 6 pm, 25 days later
B. 12 : 00 noon, 30 days later
C. 12 noon, 15 days later
D. 6 am 45 days later

Answer

To solve the problem, we need to determine when both watches will show the correct time for the first time after being set at 12:00 noon.

### Step-by-Step Solution:

1. **Understanding the Watches' Behavior**:
- The first watch **gains** 1 minute every hour.
- The second watch **loses** 4 minutes every 2 hours, which means it loses 2 minutes every hour.

2. **Calculating the Time for the First Watch**:
- In 1 hour, the first watch gains 1 minute.
- In 24 hours (1 day), the first watch will gain:
\[ 24 \text{ hours} \times 1 \text{ minute/hour} = 24 \text{ minutes} \]
- Therefore, in 30 days (30 x 24 hours), the total gain will be:
\[ 30 \text{ days} \times 24 \text{ minutes/day} = 720 \text{ minutes} \]
- 720 minutes is equivalent to:
\[ \frac{720 \text{ minutes}}{60 \text{ minutes/hour}} = 12 \text{ hours} \]
- So, after 30 days, the first watch will show:
\[ 12:00 \text{ noon} + 12 \text{ hours} = 12:00 \text{ midnight} \]

3. **Calculating the Time for the Second Watch**:
- The second watch loses 2 minutes every hour.
- In 24 hours (1 day), the second watch will lose:
\[ 24 \text{ hours} \times 2 \text{ minutes/hour} = 48 \text{ minutes} \]
- Therefore, in 15 days (15 x 24 hours), the total loss will be:
\[ 15 \text{ days} \times 48 \text{ minutes/day} = 720 \text{ minutes} \]
- 720 minutes is also equivalent to:
\[ \frac{720 \text{ minutes}}{60 \text{ minutes/hour}} = 12 \text{ hours} \]
- So, after 15 days, the second watch will show:
\[ 12:00 \text{ noon} - 12 \text{ hours} = 12:00 \text{ midnight} \]

4. **Finding the Least Common Multiple (LCM)**:
- The first watch shows the correct time after 30 days, and the second watch shows the correct time after 15 days.
- We need to find the LCM of 30 and 15:
- The multiples of 30 are: 30, 60, 90, ...
- The multiples of 15 are: 15, 30, 45, 60, 75, 90, ...
- The LCM is 30.

5. **Conclusion**:
- Therefore, both watches will show the correct time for the first time after **30 days** at **12:00 noon**.

### Final Answer:
Both watches will show the correct time for the first time at **12:00 noon, 30 days later**.