Clocks and Calendars

From 6 a.m. to 9 p.m. is a 15-hour period (6 am to 6 pm = 12 hours, plus 3 more hours to 9 pm). If the minute hand gains 2 minutes every hour, total gain = 15 hours × 2 minutes/hour = 30 minutes. Option A is correct. Option B (25), C (28), and D (34) are miscalculations. The key is correctly counting the hours and multiplying by the gain rate.

From noon to 3:45 p.m. is 3 hours 45 minutes = 3.75 hours. The hour hand moves 360° in 12 hours, so it moves 30° per hour. In 3.75 hours, it turns 3.75 × 30° = 112.5°. The question asks for the angle turned by the hour hand starting from noon.

February 24 is Sunday. The Fridays in February 2024 (leap year) would be: Feb 2 (1st Friday), Feb 9 (2nd), Feb 16 (3rd), Feb 23 (4th), Feb 29 (5th). Wait, if Feb 24 is Sunday, then Feb 23 is Saturday, Feb 22 is Friday... Let me recount: If Feb 24 is Sunday, then the Friday of that week is Feb 21. Counting back: Feb 21 (4th Friday), Feb 14 (3rd), Feb 7 (2nd), Jan 31 (1st). So Feb 28 is Thursday, Feb 29 is Friday - this would be the 5th Friday of February.

At 5:45, the hour hand is at 5.75 hours from 12 = 5.75 × 30 = 172.5 degrees. The minute hand at 45 minutes is at 45 × 6 = 270 degrees. The angle between them is |270 - 172.5| = 97.5 degrees. Formula: |30H - 5.5M| = |150 - 247.5| = 97.5.

The hour and minute hands coincide 11 times every 12 hours, approximately every 65 minutes. Between 11 am and 1 pm (a 2-hour window), they coincide only once at exactly 12:00 noon. The next coincidence occurs around 1:05 pm, which is outside the given range.

From Monday 8 AM to Tuesday 10 PM = 38 hours. Clock goes from 2 min fast to 4 min slow = change of -6 min (lost 6 minutes). Rate of loss per hour = 6/38 = 3/19 min per hour. The clock is losing time at 3/19 min per hour.

At 1:20, the minute hand points at 4 (120° from 12). The hour hand is 1/3 of the way from 1 to 2 (30+10=40° from 12). The angle between them is |120-40|=80°. This calculation accounts for the hour hand moving as minutes progress.

At 4 o'clock, the hour hand is at 120° and minute hand at 0°. The angle between them is 120°. We need this to become 60°, meaning we need the angle to reduce by 60°. The relative speed between hour and minute hands is 5.5° per minute. Time = 60/5.5 = 120/11 = 10 10/11 minutes. But wait, the angle can be 60° in two ways: when hands are 60° apart approaching or separating. The first 60° occurs at 32 8/11 minutes past 4. Calculation: (120-60)/5.5 = 60/5.5 = 120/11 = 10 10/11 minutes for 60° separation from 120° starting point... but that would give 60° not 60°... Let me recalculate. For angle θ between hands at time t past 4: |30×4 - 5.5t| = 60. |120 - 5.5t| = 60. Case 1: 120 - 5.5t = 60 → 5.5t = 60 → t = 60/5.5 = 120/11 = 10 10/11. Case 2: 5.5t - 120 = 60 → 5.5t = 180 → t = 180/5.5 = 360/11 = 32 8/11. The answer is 4:32 8/11, which is option A.

At 8:30, minute hand at 6 (180° from 12). Hour hand at 8.5 (8 + 30/60) × 30° = 255° from 12. Difference = |255-180| = 75°. The hour hand has moved halfway between 8 and 9 (15° from 8), making it 75° from the minute hand.

In 12 hours, clock hands are in a straight line 22 times (11 times at 180° opposite, 11 times overlapped at 0°). In 84 hours = 7 × 12 hours, so 7 × 22 = 154 times. The hands form a straight line when they are either exactly opposite (180° apart) or exactly overlapping (0° apart), happening 11 times each in 12 hours.

At 10:20, the hour hand moves 0.5° per minute. Position from 12 o'clock: 10×30° + 20×0.5° = 300° + 10° = 310°. The minute hand moves 6° per minute. Position: 20×6° = 120°. Angle difference = |310° - 120°| = 190°. The supplementary (smaller) angle is 360° - 190° = 170°.

Using the formula for when hands coincide: time = (5h/11) past h o'clock. For h=9: (5×9/11) = 45/11 = 4 1/11 minutes past 9. So time is 9:49 1/11.

Clock hands align every 65 5/11 minutes. In 12 hours, they align 11 times (not 12 because they skip one overlap at 11-12). In 24 hours, they align 22 times. Straight line includes both overlapping (0 degrees) and opposite (180 degrees) positions, happening 22 times each, totaling 44 straight line occurrences.

At 8:20, hour hand at 250° (8×30 + 20×0.5), minute hand at 120° (20×6). Angle = 250-120 = 130°. Minute hand gains 5.5° per minute on hour hand. To get to 90°, need to cover 40° at 5.5°/min: t = 40/5.5 = 7/11 minutes. Time = 8:20 + 7/11 min = 8:27 3/11.

At 8:38, the hour hand is at (8 + 38/60) × 30° = 240° + 19° = 259° from 12 o'clock. The minute hand is at 38 × 6° = 228°. The angle between them is |259° - 228°| = 31°. Option B is correct.