This problem involves two main steps: first, determining the actual time from a mirror reflection, and second, calculating the current time based on a given time difference.

Mirror Time Reflection: Deciphering the Wall Clock

When observing a wall clock in a mirror, the reflected time is horizontally inverted. To find the actual time from a mirror image, we use a simple rule. If the mirror image shows a time, say \(M_h:M_m\), the actual time \(A_h:A_m\) can be found using the formula:

\(\text{Actual Time} = 11:60 - \text{Mirror Time}\)

In this question, the reflection of the wall clock seemed to show 1:30.

Let's apply the formula to find the actual time that was shown two and a quarter hours back:

• Mirror Time = 1:30
• Actual Time (two and a quarter hours back) = 11:60 - 1:30
• Actual Time (two and a quarter hours back) = 10:30

So, the actual time on the wall clock, two and a quarter hours back, was 10:30. This is the precise time the clock displayed at that moment, not the reflected time.

Current Time Calculation: Finding the Actual Time

The problem states that the observation was made two and a quarter hours back. This means we need to add two and a quarter hours to the time we just found (10:30) to get the actual current time shown by the clock.

First, let's convert "two and a quarter hours" into hours and minutes:

• Two hours and a quarter = 2 hours and 15 minutes (\(\frac{1}{4}\) hour = 15 minutes).

Now, let's add this duration to the actual time from earlier:

• Time two and a quarter hours back = 10:30
• Duration passed = 2 hours 15 minutes

Adding the hours:

\(10 \text{ hours} + 2 \text{ hours} = 12 \text{ hours}\)

Adding the minutes:

\(30 \text{ minutes} + 15 \text{ minutes} = 45 \text{ minutes}\)

Therefore, the actual current time shown by the clock is 12:45. This is the final actual time after accounting for the time elapsed since the mirror observation.

Final Current Time: Summarizing the Clock's Display

To summarize the steps for finding the actual current time:

1. Determined the actual time from the mirror reflection of 1:30, which was 10:30. This corrects the inverted view from the mirror.
2. Added the duration of two and a quarter hours (2 hours and 15 minutes) to 10:30. This advances the time to the current time.
3. The resulting actual current time shown by the clock is 12:45.

This method accurately finds the actual current time by properly interpreting the mirror reflection and accounting for the time difference.