A faulty watch has its hour and minute hands coinciding every 64 minutes instead of the usual interval. How many minutes does this watch lose in one full day of 24 hours?

Introduction / Context:
Clock problems that involve faulty watches often use the fact that, in a correct clock, the hour and minute hands coincide at fixed regular intervals. If the watch is slow or fast, this coincidence interval changes. By comparing the incorrect interval with the correct one, we can find how much time the watch gains or loses in a full day. This question asks you to determine how many minutes a faulty watch loses in twenty four hours when its hands coincide too frequently compared with a perfect clock.

Given Data / Assumptions:

• In a correct clock, the hour and minute hands coincide every 65 5/11 minutes.
• In the faulty watch, the hands coincide every 64 minutes, as read on that watch.
• We assume that the fault affects the overall speed of the watch uniformly throughout the day.
• We must find the total loss of time in one real day of 24 hours, that is, 1440 real minutes.

Concept / Approach:
In a perfect clock, the relative speed between the minute hand and the hour hand is 5.5 degrees per minute, and they need to cover 360 degrees relative to each other to coincide. This gives the standard interval of 360 / 5.5 = 65 5/11 minutes between coincidences. If a faulty watch shows coincidences every 64 minutes on its own dial, that means that in 64 minutes of indicated time, the amount of real time is actually equal to one true coincidence interval. The ratio between shown time and real time can be used to find the daily loss. Once that rate is known, we simply scale it up to 24 hours.

Step-by-Step Solution:
Step 1: Correct interval between coincidences for a good clock is 65 5/11 minutes. Write this as an improper fraction: 65 5/11 = 720/11 minutes. Step 2: For the faulty watch, its hands coincide at 64 minutes of indicated time. In that same period, real time must be 720/11 minutes, because the true relative motion is unchanged. Step 3: Therefore, in 720/11 real minutes, the watch shows only 64 minutes. The rate of the faulty watch compared with real time is: rate = shown time / real time = 64 / (720/11) = (64 * 11) / 720 = 704 / 720 = 44 / 45. Step 4: In one real day, there are 1440 minutes. The faulty watch will show: shown time = (44 / 45) * 1440 = 44 * 32 = 1408 minutes. Step 5: The loss per day is real time minus shown time: loss = 1440 - 1408 = 32 minutes. Step 6: Thus the watch loses 32 minutes in one full day.

Verification / Alternative check:
You can also view the error per coincidence interval. The correct interval is 65 5/11 minutes, while the watch indicates only 64 minutes. So it loses 65 5/11 - 64 = 1 5/11 minutes per coincidence interval. The number of coincidence intervals in a full day for a correct clock is 1440 / (65 5/11) = 1440 / (720/11) = 22. Multiplying 1 5/11 minutes of loss per interval by 22 intervals gives (16/11) * 22 = 32 minutes, exactly the same result as before.

Why Other Options Are Wrong:
30 minutes: This is close but underestimates the accumulated loss and does not match the exact ratio calculation.
34 3/11 minutes: This would correspond to a slightly larger error per interval than we actually have.
36 7/9 minutes: This is significantly higher than the computed 32 minutes and does not arise from any consistent use of the coincidence interval.
28 minutes: This underestimates the loss and cannot be obtained from the correct fractions.

Common Pitfalls:
Students sometimes mix up whether the watch is gaining or losing time by comparing 64 minutes and 65 5/11 minutes in the wrong order. Another common error is to forget that 65 5/11 must be converted to a single fraction, which can lead to arithmetic slips. Some learners try to compare hours instead of using the coincidence interval, which complicates the problem unnecessarily. Carefully setting up the ratio shown time over real time and then scaling to a full day is the safest way to avoid mistakes.

Final Answer:
The faulty watch loses 32 minutes in one day of 24 hours.