For an inaccurate clock, the minute and hour hands are observed to coincide every 65 minutes instead of the usual interval of 65 5/11 minutes.\nHow much time (in minutes) does this clock gain in one full day?

Introduction / Context:
Clock-aptitude questions involving coinciding hands test understanding of relative angular speed and timekeeping accuracy. In a correct clock, the hour and minute hands coincide at fixed time intervals. Here, an inaccurate clock has its hands coinciding every 65 minutes instead of the correct interval, and you must determine how much the clock gains in one day compared with true time.

Given Data / Assumptions:

In a correct clock, successive coincidences of the hands occur every 65 5/11 minutes, which is 720/11 minutes.

In the inaccurate clock, successive coincidences occur every 65 real minutes.

The gear ratio between hour and minute hands is unchanged; only the running rate of the clock differs.

We seek the daily time gain (how many minutes fast per day the faulty clock becomes).

Concept / Approach:
The dial interval between coincidences (in minutes shown on that clock) is fixed at 720/11 for any correctly geared 12-hour clock. If the faulty hands coincide every 65 real minutes, the clock is running faster than normal. Let r be the rate of the clock: the number of dial minutes it shows per real minute. Over one coincidence interval, the clock shows 720/11 minutes on its face, but only 65 real minutes have elapsed. This allows us to find r, then compute how much time the clock gains over 24 hours of real time.

Step-by-Step Solution:
Step 1: Express the relation between dial minutes and real minutes. Let r = dial minutes / real minute. For one coincidence interval: dial minutes shown = 720/11, real minutes = 65. Thus r = (720/11) / 65 = 720 / (11 * 65) = 720 / 715. Step 2: Find how many dial minutes the clock shows in one real day. Real minutes in a day = 1440. Dial minutes shown = r * 1440 = (720 / 715) * 1440. Step 3: Compute the daily gain. Correct clock should show 1440 minutes in a day. Faulty clock shows (720 / 715) * 1440 minutes. Daily gain = (720 / 715) * 1440 - 1440. Factor 1440: gain = 1440 * (720/715 - 1) = 1440 * (5 / 715). Simplify: gain = (1440 * 5) / 715 = 7200 / 715 = 1440 / 143 minutes.

Verification / Alternative check:
1440 / 143 is approximately 10.07 minutes. Over a day, the faulty clock reads about 1450.07 minutes while actual time is 1440 minutes, so it is about 10 minutes fast. That level of error agrees with the small difference between the correct coincidence interval (about 65.4545 minutes) and the observed 65 minutes; a gain of around 0.45 minutes per coincidence interval, repeated many times, leads to roughly a 10-minute daily gain.

Why Other Options Are Wrong:
184/13 minutes is about 14.15 minutes, 1425/18 minutes is over 79 minutes, and 541/9 minutes is roughly 60 minutes. All of these are far too large compared with the small difference in coincidence interval and would require the clock to be much more inaccurate than given. Only 1440/143 minutes matches the calculated gain from the relative-speed analysis.

Common Pitfalls:
A frequent misunderstanding is to treat 65 minutes as the dial interval instead of real minutes, leading to the incorrect conclusion that the clock is losing time. Another mistake is forgetting that the coincidence interval on the dial is always 720/11 minutes for any correctly geared clock, regardless of its speed. Careful handling of the ratio between dial time and real time, and then scaling over 24 hours, is crucial to avoid these traps.

Final Answer:
The inaccurate clock gains 1440/143 minutes per day compared with correct time.