A watch which gains uniformly is 2 min slow at 10 am on Wednesday and is `2^(1/2)` min fast at 1 pm the following day When was it correct?

Options

A. 10:30 pm Wednesday
B. 9 :15 pm Wednesday
C. 10 :45 pm Wednesday
D. 10 pm wednesday

Answer

To solve the problem step by step, we need to analyze the information given about the watch's timekeeping over a specified period.

### Step 1: Determine the time interval
The watch is 2 minutes slow at 10 AM on Wednesday and 2.5 minutes fast at 1 PM on Thursday. We need to calculate the total time interval from 10 AM Wednesday to 1 PM Thursday.

- From 10 AM Wednesday to 10 AM Thursday is 24 hours.
- From 10 AM Thursday to 1 PM Thursday is an additional 3 hours.

So, the total time interval is:
\[ 24 \text{ hours} + 3 \text{ hours} = 27 \text{ hours} \]

**Hint:** Calculate the total time between the two given times by breaking it down into manageable parts.

### Step 2: Calculate the total gain of the watch
The watch was 2 minutes slow and then became 2.5 minutes fast. The total change in time is:
\[ \text{Total gain} = 2 \text{ minutes (slow)} + 2.5 \text{ minutes (fast)} = 4.5 \text{ minutes} \]

**Hint:** To find the total gain, add the absolute values of the slow and fast times.

### Step 3: Understand the gain per hour
Since the watch gained 4.5 minutes over 27 hours, we can find out how much it gains per hour:
\[ \text{Gain per hour} = \frac{4.5 \text{ minutes}}{27 \text{ hours}} = \frac{1}{6} \text{ minutes per hour} \]

**Hint:** Divide the total gain by the total time to find the gain per hour.

### Step 4: Calculate when the watch was correct
The watch will be correct when it has gained enough time to offset the 2 minutes it was slow. We need to find out how long it takes to gain 2 minutes:
\[ \text{Time to gain 2 minutes} = \frac{2 \text{ minutes}}{\frac{1}{6} \text{ minutes per hour}} = 2 \times 6 = 12 \text{ hours} \]

**Hint:** To find the time taken to gain a specific amount, divide that amount by the gain per hour.

### Step 5: Determine the correct time
Starting from 10 AM Wednesday, if we add 12 hours, we get:
\[ 10 \text{ AM} + 12 \text{ hours} = 10 \text{ PM Wednesday} \]

Thus, the watch was correct at **10 PM on Wednesday**.

**Hint:** Add the calculated time to the starting time to find when the watch was correct.

### Final Answer
The watch was correct at **10 PM on Wednesday**.