A clock loses 20 minutes every real hour. If it shows 12:00 noon now, how many minutes slow will it be when it later shows 2:00 p.m.?

Introduction / Context:
This question deals with a faulty clock that does not keep correct time. Instead of advancing 60 minutes for every real hour that passes, it shows 20 minutes less each hour. The puzzle asks how many minutes the clock will be behind the correct time when it reads 2:00 p.m. after starting at 12:00 noon. Such questions test understanding of relative speed in terms of timekeeping, similar to problems about gaining or losing minutes per day on a watch.

Given Data / Assumptions:

- The faulty clock loses 20 minutes for each real hour that passes. - At the starting moment, both the real time and the clock reading are 12:00 noon. - We want to know the real time when the faulty clock shows 2:00 p.m. - One real hour equals 60 real minutes. - The clock is assumed to lose time at a constant rate.

Concept / Approach:
When a clock loses a fixed amount of time each real hour, its displayed time advances slower than real time. Here, for every 60 real minutes, the clock advances only 40 minutes, because it loses 20 minutes. This means the clock speed is 40 minutes of display per 60 minutes of real time, or two thirds of the normal rate. The relationship between displayed time and real time can be expressed as a simple proportionality, allowing us to calculate how much real time has passed when the clock shows a given display time, such as 2 hours after noon on its own face.

Step-by-Step Solution:
Step 1: In 60 real minutes, the clock advances only 40 display minutes. Step 2: Therefore, the ratio of real time to displayed time is 60 to 40, which simplifies to 3 to 2. Step 3: Let the displayed time interval from noon to the later reading be 2 hours, that is 120 display minutes. Step 4: Using the ratio, the corresponding real time interval is (3 / 2) * 120 = 180 real minutes. Step 5: Starting from 12:00 noon, 180 real minutes corresponds to 3 hours later, which is 3:00 p.m. Step 6: When the clock reads 2:00 p.m., the true time is 3:00 p.m. Step 7: Thus, the clock is 1 hour, or 60 minutes, slow.

Verification / Alternative check:
We can compute hour by hour. After 1 real hour, the correct time is 1:00 p.m. but the faulty clock shows only 12:40 p.m., so it is 20 minutes slow. After 2 real hours, the correct time is 2:00 p.m. but the faulty clock has advanced only 80 minutes, so it shows 1:20 p.m., now 40 minutes slow. After 3 real hours, the correct time is 3:00 p.m. while the faulty clock has advanced 120 display minutes from noon and now shows 2:00 p.m., which is 60 minutes behind real time. This step by step view confirms the proportional method result.

Why Other Options Are Wrong:
A loss of 36 or 48 minutes would correspond to smaller slowdowns than the actual 20 minutes per hour loss and do not match the proportional relationship of 3 real minutes to 2 display minutes over a two hour displayed period. A loss of 35 or 40 minutes similarly fails to align with the exact calculations. Only a loss of 60 minutes matches both the proportional reasoning and the direct hour by hour simulation.

Common Pitfalls:
A common mistake is to assume that after two hours on the display, only two real hours have passed, which ignores the slower running of the clock. Others subtract 20 minutes only once instead of for each real hour, leading to an underestimate of the total error. Carefully distinguishing between real time and displayed time and using either the ratio method or a consistent stepwise approach prevents these errors.

Final Answer:
When the faulty clock shows 2:00 p.m., it is 60 minutes slow compared to the correct time.