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Question

Difficulty: HardClocks and Calendars

A mechanical clock is synchronized to the correct standard time at exactly 00:00 (midnight) on February 15th of a leap year. This particular clock is known to gain exactly 44 minutes every 2424 hours of true time.

What is the acute angle (in degrees) between the hour and minute hands of this faulty clock at exactly 12:00 noon (true time) on March 2nd of the same leap year?

Answer: 3 degrees

Answer

The acute angle between the hands is 3 degrees.
By determining the exact time elapsed across the leap year boundary (16.516.5 days), the total time gained by the clock is 16.5×4=6616.5 \times 4 = 66 minutes. Adding this to the true time of 12:00 yields a faulty clock time of 13:06 (or 1:06). At 1:06, the minute hand is at 36∘36^\circ (6×6∘6 \times 6^\circ) and the hour hand has moved past the 1 o'clock mark (30∘30^\circ) by an additional 3∘3^\circ ((6/60)×30∘(6/60) \times 30^\circ). The difference between 36∘36^\circ and 33∘33^\circ is 3∘3^\circ.

Step-by-Step Solution

1
Calculate the total true time elapsed in days.
The total elapsed time is 16.5 days.
From February 15th 00:00 to March 1st 00:00 in a leap year (29 days in February) is exactly 15 days. From March 1st to March 2nd 00:00 is 1 day. From 00:00 to 12:00 noon is 0.5 days. Total: 15 + 1 + 0.5 = 16.5 days.
2
Calculate the total time gained by the faulty clock.
The clock gains a total of 66 minutes.
The clock gains 4 minutes per 24 hours (1 day). For 16.5 days, the total gain is 16.5 * 4 = 66 minutes.
3
Determine the time shown on the faulty clock.
The faulty clock shows 13:06 (or 1:06 PM).
The true time is 12:00 noon. Adding the gained 66 minutes (1 hour and 6 minutes) gives 13:06.
4
Calculate the precise position of the minute and hour hands at 1:06.
Minute hand is at 36 degrees; Hour hand is at 33 degrees.
The minute hand moves 6 degrees per minute: 6 * 6 = 36 degrees from the 12 o'clock position. The hour hand moves 30 degrees per hour plus 0.5 degrees per minute: (1 * 30) + (6 * 0.5) = 30 + 3 = 33 degrees.
5
Find the acute angle between the two hands.
The acute angle is 3 degrees.
The difference between the two positions is |36 - 33| = 3 degrees.

Key Concept

Calculating elapsed time across leap year month boundaries combined with continuous clock drift and geometric clock angle formulas.
Estimated Time:2m 30s
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Topics

Clocks and Calendars

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