Step-by-step Solution:

Quick Solution

To solve this, we need to find the relationship between the time measured by the faulty watch and the correct time. We can then use this relationship to find out how much correct time has actually passed when the watch shows 7:20 pm.

1. Calculate the Time Elapsed on the Faulty Watch

• The watch was set correctly at 9:00 AM.
• It later shows 7:20 PM on the same evening.
• The total time that has passed on this faulty watch is 10 hours and 20 minutes.
• Converting to minutes: (10 × 60) + 20 = 620 minutes (faulty time).

2. Find the Relationship Between Faulty and Correct Time

• The watch gains 10 seconds in 5 minutes.
• This means that for every 5 minutes (or 300 seconds) of correct time, the faulty watch measures 5 minutes and 10 seconds (or 310 seconds).
• So, 300 seconds of Correct Time = 310 seconds of Faulty Time.
• The ratio to find the correct time from the faulty time is:
  Correct Time = (300/310) × Faulty Time = (30/31) × Faulty Time.

3. Calculate the Actual (Correct) Time Passed

- We know 620 minutes have passed on the faulty watch. Let's find the actual correct time that this corresponds to:
- Correct Time = (30/31) × 620 minutes
- Correct Time = 30 × (620 / 31)
- Correct Time = 30 × 20 = 600 minutes.

600 minutes is equal to exactly 10 hours.

4. Determine the Correct Final Time

- The actual, correct time that has passed is 10 hours from when the watch was set.
- Start Time: 9:00 AM.
- Correct Time = 9:00 AM + 10 hours = 7:00 PM.

Final Answer: The correct time is 7.00 pm.