A colck loses 2 minutes in an hour and another clock gains 2 minutes in every 2 hours. Both these clocks are set correctly at a certain time on Sunday and both the clocks stop simultaneously on the next day with the time shown being 9 am and 10:06 am. What is the correct time at which they stopped?

Options

A. 9:54 am
B. 9:44 pm
C. 9:54 pm
D. 9:44 am

Answer

To solve the problem, we need to determine the correct time at which both clocks stopped. Let's break down the solution step by step.

### Step 1: Determine the Time Lost by Each Clock
- **Clock 1** loses 2 minutes every hour. Therefore, in 1 hour, it shows 58 minutes.
- **Clock 2** gains 2 minutes every 2 hours, which means it gains 1 minute every hour. Therefore, in 1 hour, it shows 61 minutes.

### Step 2: Calculate the Relative Speed of the Clocks
- The relative speed of Clock 2 with respect to Clock 1 can be calculated as:
\[ \text{Relative Speed} = \text{Speed of Clock 2} - \text{Speed of Clock 1} = 61 \text{ minutes/hour} - 58 \text{ minutes/hour} = 3 \text{ minutes/hour} \]

### Step 3: Calculate the Time Difference Between the Two Clocks
- The time shown by Clock 1 when it stopped is 9:00 AM.
- The time shown by Clock 2 when it stopped is 10:06 AM.
- The difference in time shown by the two clocks is:
\[ 10:06 - 9:00 = 1 \text{ hour and } 6 \text{ minutes} = 66 \text{ minutes} \]

### Step 4: Calculate the Time Taken for the Difference
- To find out how long it took for this difference to occur, we can use the relative speed:
\[ \text{Time taken} = \frac{\text{Time Difference}}{\text{Relative Speed}} = \frac{66 \text{ minutes}}{3 \text{ minutes/hour}} = 22 \text{ hours} \]

### Step 5: Calculate the Actual Time Passed
- During this 22 hours, we need to find out how much actual time has passed according to Clock 1:
- Since Clock 1 loses time, in 22 hours, it would show:
\[ \text{Time shown by Clock 1} = 22 \text{ hours} \times \frac{58}{60} = 21.33 \text{ hours} \approx 21 \text{ hours and } 20 \text{ minutes} \]

### Step 6: Convert the Time Shown by Clock 1 to Minutes
- Convert 21 hours and 20 minutes to minutes:
\[ 21 \text{ hours} = 21 \times 60 = 1260 \text{ minutes} \]
\[ \text{Total minutes} = 1260 + 20 = 1280 \text{ minutes} \]

### Step 7: Calculate the Actual Time Passed
- The actual time passed in minutes is:
\[ \text{Actual Time} = 22 \text{ hours} \times 60 = 1320 \text{ minutes} \]

### Step 8: Find the Time Lost
- The difference between the actual time and the time shown by Clock 1 is:
\[ 1320 \text{ minutes} - 1280 \text{ minutes} = 40 \text{ minutes} \]

### Step 9: Determine the Correct Time
- Since Clock 1 shows 9:00 AM, we need to add the time lost:
\[ 9:00 \text{ AM} + 40 \text{ minutes} = 9:40 \text{ AM} \]

### Final Answer
The correct time at which both clocks stopped is **9:40 AM**.

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