Understanding the Clock Problem

This question involves a clock that is not perfectly accurate; it gains time at a constant rate. We are given the starting time when it was correct, the rate at which it gains time, and the target time we need to find what the clock shows.

The key steps to solve this kind of problem are:

1. Calculate the total duration between the initial correct time and the target time.
2. Calculate the total amount of time the clock has gained during this duration based on its gaining rate.
3. Add the total gained time to the target time to find what the faulty clock shows.

Calculating the Elapsed Time Duration

The clock was set correctly at 4:00 AM on Thursday.

We want to find the time shown at 8:30 PM on Friday night.

• From Thursday 4:00 AM to Friday 4:00 AM is exactly 24 hours.
• From Friday 4:00 AM to Friday 8:30 PM:
  • Friday 4:00 AM to 12:00 PM (noon) = 8 hours
  • Friday 12:00 PM to 8:00 PM = 8 hours
  • Friday 8:00 PM to 8:30 PM = 30 minutes

So, the total time from Friday 4:00 AM to 8:30 PM is $8 \text{ hours} + 8 \text{ hours} + 30 \text{ minutes} = 16 \text{ hours } 30 \text{ minutes}$.

Total elapsed time from Thursday 4:00 AM to Friday 8:30 PM is:

$24 \text{ hours } + 16 \text{ hours } 30 \text{ minutes} = 40 \text{ hours } 30 \text{ minutes}$.

To make calculations easier, we can convert 30 minutes into hours: $30 \text{ minutes} = \frac{30}{60} \text{ hours} = 0.5 \text{ hours}$.

Total elapsed time is $40.5$ hours.

Determining the Total Time Gain

The clock gains 20 seconds in every 3 hours.

We need to find the total gain in 40.5 hours.

Number of 3-hour intervals in 40.5 hours = $\frac{40.5 \text{ hours}}{3 \text{ hours/interval}}$

Number of intervals = $13.5$ intervals.

Since the clock gains 20 seconds per 3-hour interval, the total gain is:

Total gain = Number of intervals $\times$ Gain per interval

Total gain = $13.5 \times 20 \text{ seconds}$

Total gain = $270 \text{ seconds}$.

Converting Total Gain to Minutes and Seconds

We know that 60 seconds make 1 minute.

To convert 270 seconds into minutes and seconds, we divide 270 by 60:

$270 \div 60 = 4$ with a remainder of $30$.

So, 270 seconds is equal to 4 minutes and 30 seconds.

Calculating the Final Time Shown

The correct time at the end of the elapsed duration is 8:30 PM on Friday night.

The clock has gained 4 minutes and 30 seconds over this period.

The time shown on the clock will be the correct time plus the gain.

Time shown = Correct time + Total gain

Correct time = 8:30 PM

Total gain = 4 minutes 30 seconds

Time shown = 8 hours 30 minutes 0 seconds PM + 0 hours 4 minutes 30 seconds

Time shown = 8 hours 34 minutes 30 seconds PM.

Comparing with Options

Let's look at the given options:

• 8 hours 34 minutes PM
• 9 hours 34 minutes PM
• 8 hours 34 minutes 30 seconds PM
• 8 hours 30 minutes 30 seconds PM

Our calculated time is 8 hours 34 minutes 30 seconds PM, which matches one of the options.

Revision Table: Clock Gain Calculation

Additional Information: Types of Clock Problems

Clock problems in competitive exams often fall into a few categories:

• Angle between hands: Calculating the angle between the hour and minute hand at a specific time.
• Clock gaining/losing time: Problems like this one, where a clock runs faster or slower than a standard clock.
• Mirror image time: Finding the time shown in a mirror if a clock shows a certain time.
• Clocks striking bells: Calculating the time taken or number of strikes.

For clock gaining/losing problems, the key is always to determine the total elapsed time and then calculate the total gain or loss based on the given rate. The final time shown on the faulty clock is the correct time plus the total gain, or the correct time minus the total loss.