How much time does a watch lose in one day if its hands coincide every 64 minutes instead of at the correct interval?

Introduction / Context:
This question concerns a faulty watch whose hands coincide more frequently than they should. On a correct clock, the time between two consecutive coincidences of the hour and minute hands is a fixed value. If a watch shows coincidence earlier than that, it is running slow in terms of real time. The goal is to determine how much time the watch loses in one real day. This tests precise use of the formula for coincidence interval and proportional reasoning.

Given Data / Assumptions:

• On a correct clock, the hands coincide at a standard interval.
• On this faulty watch, the hands coincide every 64 minutes as shown by the watch.
• The watch is assumed to be uniformly slow.
• We want the amount of time the watch loses in one real day of 24 hours.

Concept / Approach:
For a correct clock, the interval between two coincidences is: T_correct = 720 / 11 minutes, which is approximately 65 5/11 minutes. For the faulty watch, let k be the number of real minutes corresponding to 1 minute shown by the watch. Then both hands slow equally, so their relative angular speed in terms of shown minutes is scaled by k, and the interval between coincidences in shown minutes is: T_shown = T_correct / k. We are told T_shown is 64 minutes, so we can solve for k and then find how much real time corresponds to 24 shown hours.

Step-by-Step Solution:
Step 1: Standard coincidence interval for a correct clock. T_correct = 720 / 11 minutes. Step 2: Relation between real time and shown time for the faulty watch. Let 1 shown minute represent k real minutes. Then for the faulty watch, the relative speed is slower by factor k, so the coincidence interval in shown minutes is: T_shown = T_correct / k. Step 3: Use the given coincidence interval. T_shown = 64 minutes, so: 64 = (720 / 11) / k. k = (720 / 11) / 64 = 720 / (11 * 64). k = 720 / 704 = 90 / 88 = 45 / 44. So 1 shown minute equals 45 / 44 real minutes, which is more than 1, meaning the watch is slow. Step 4: Find how much time is lost in one day. In 24 hours shown by the watch, real time elapsed = 24 * k hours = 24 * 45 / 44 hours. Real time = 1080 / 44 hours = 24 12/22 hours = 24 6/11 hours. Extra real time beyond 24 hours = 6/11 hour. Convert 6/11 hour to minutes: 6/11 * 60 = 360 / 11 = 32 8/11 minutes. Since in 24 hours of true time the watch shows only 24 hours, it has lost 32 8/11 minutes in a day.

Verification / Alternative check:
The ratio k = 45 / 44 means that in 44 minutes shown, real time is 45 minutes. So the watch loses 1 minute every 45 minutes. In 24 real hours (1440 minutes), loss = 1440 / 45 = 32 minutes exactly plus a fraction which matches 32 8/11 minutes. This alternative way supports the same result.

Why Other Options Are Wrong:
Options b, c, and d (33 8/11, 34 8/11, 35 8/11) give larger daily losses that do not match the derived ratio from the coincidence interval. These values would correspond to different faulty intervals, not the given 64 minutes between coincidences.

Common Pitfalls:
A common mistake is to assume that if the hands coincide more often, the watch is fast, whereas here it is actually slow in real time. Another error is to forget that both hands slow at the same rate, so the relative angular speed decreases and the formula must be adjusted via the factor k for shown minutes. Also, mishandling fractions like 720 / 11 often leads to approximate answers instead of an exact one.

Final Answer:
The watch loses 32 8/11 minutes in one day.