A watch gains 5 seconds per minute and was set right at 6 AM. What would be the time shown on the watch when the correct time is 2 PM?
2.40 PM
This problem involves calculating how much time a faulty watch gains over a specific period and determining the time it will show when the correct time is known.
Understanding the Problem: Watch Gain Rate
The question states that the watch gains 5 seconds every minute. This means the watch runs faster than a standard clock.
- Gain rate: 5 seconds per minute.
- Starting time (set right): 6 AM.
- Ending time (correct time): 2 PM.
Calculating the Total Time Duration
First, we need to find the total duration for which the watch has been running since it was set correctly.
The time duration is from 6 AM to 2 PM.
- From 6 AM to 12 PM (noon) is 6 hours.
- From 12 PM to 2 PM is 2 hours.
- Total duration = 6 hours + 2 hours = 8 hours.
Converting Duration to Minutes for Gain Calculation
Since the gain is given per minute, we need to convert the total duration into minutes.
Total duration in minutes = Total hours × Minutes per hour
Total duration in minutes = \(8 \text{ hours} \times 60 \text{ minutes/hour}\)
Total duration in minutes = \(480 \text{ minutes}\)
Calculating the Total Time Gained by the Watch
The watch gains 5 seconds for every minute it runs. The total duration is 480 minutes.
Total gain = Gain per minute × Total minutes
Total gain = \(5 \text{ seconds/minute} \times 480 \text{ minutes}\)
Total gain = \(2400 \text{ seconds}\)
Converting Total Gain from Seconds to Minutes
It is easier to add the gain to the correct time if we convert the total gain from seconds into minutes.
Total gain in minutes = Total gain in seconds ÷ Seconds per minute
Total gain in minutes = \(2400 \text{ seconds} \div 60 \text{ seconds/minute}\)
Total gain in minutes = \(40 \text{ minutes}\)
Determining the Time Shown on the Watch
The watch started correctly at 6 AM. It has been running for 8 hours and has gained 40 minutes during this time.
When the correct time is 2 PM, the time shown on the fast watch will be the correct time plus the total time gained.
Time shown on watch = Correct time + Total gain
Correct time = 2 PM
Total gain = 40 minutes
Time shown on watch = 2 PM + 40 minutes
Time shown on watch = 2:40 PM
| Metric | Calculation | Result |
|---|---|---|
| Time Duration | 6 AM to 2 PM | 8 hours |
| Duration in Minutes | 8 hours × 60 minutes/hour | 480 minutes |
| Gain per Minute | Given | 5 seconds |
| Total Gain in Seconds | 480 minutes × 5 seconds/minute | 2400 seconds |
| Total Gain in Minutes | 2400 seconds ÷ 60 seconds/minute | 40 minutes |
| Time on Watch | 2 PM + 40 minutes | 2:40 PM |
Therefore, when the correct time is 2 PM, the watch that gains 5 seconds per minute will show 2:40 PM.
Revision Table: Watch Time Problems
| Concept | Description | Formula/Approach |
|---|---|---|
| Watch Gain/Loss Rate | How much time a watch gains or loses in a specific interval (e.g., per minute, per hour). | Rate = Gain or Loss / Time interval |
| Total Duration | The total time elapsed from when the watch was set correct until the target time. | End Time - Start Time |
| Total Gain/Loss | The total amount of time the watch gains or loses over the total duration. | Total Gain/Loss = Rate per unit time × Total duration in those units |
| Time Shown on Fast Watch | Correct Time + Total Gain | Correct Time + Total Gain |
| Time Shown on Slow Watch | Correct Time - Total Loss | Correct Time - Total Loss |
Additional Information: Relative Speed in Time Problems
Watch problems can sometimes be thought of in terms of relative speed. A watch that gains time is like moving faster than a standard clock. A watch that loses time is like moving slower.
- A standard clock completes one full circle (12 hours or 720 minutes) in 720 real minutes.
- A watch gaining time completes 720 "real" minutes faster than a standard clock. For example, if it gains 1 minute per hour (60 minutes), it gains 12 minutes in 12 hours. So, it would complete its 12-hour cycle in less than 12 real hours.
- In this specific problem, the watch gains 5 seconds per minute. This means in 1 real minute (60 seconds), the fast watch measures 65 seconds.
- Ratio of fast watch time to real time = 65 seconds / 60 seconds = 65/60.
- Over 8 real hours (480 minutes), the fast watch measures: \(480 \text{ minutes} \times \frac{65 \text{ seconds}}{60 \text{ seconds}}\) which is incorrect application of the ratio as the gain is per minute. The total time measured by the fast watch is 480 real minutes PLUS the gain.
- Correct approach using gain: Calculate total duration, calculate total gain based on that duration, add gain to the correct time. This is the method used in the step-by-step solution above, which is typically simpler for these types of problems.
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