At what time between 4:00 PM and 5:00 PM will the hour hand and the minute hand of a clock...
Nvidia technical mcq question, verified with a worked answer. Free to practise - no sign-up.
At what time between 4:00 PM and 5:00 PM will the hour hand and the minute hand of a clock be exactly $10^\circ$ apart?
Show answer & explanation
The relative speed between the minute hand (6°/min) and the hour hand (0.5°/min) is 5.5° per minute. To find when they are 10° apart, we calculate the time required for the minute hand to either trail the hour hand by 10° or lead it by 10° starting from the 4:00 position.
Step-by-step Derivation:
Step 1: Determine the initial position at 4:00 PM. The hour hand is at 4 * 30° = 120° from the 12 o'clock mark. The minute hand is at 0°.
Step 2: Let 'm' be the number of minutes past 4:00. The position of the minute hand is 6m and the position of the hour hand is 120 + 0.5m.
Step 3: Set up the equation for the absolute difference in angles: |6m - (120 + 0.5m)| = 10.
Step 4: Solve Case 1 (Minute hand is behind the hour hand): 120 + 0.5m - 6m = 10 => 5.5m = 110 => m = 110 / 5.5 = 20 minutes. This corresponds to 4:20 PM.
Step 5: Solve Case 2 (Minute hand has passed the hour hand): 6m - (120 + 0.5m) = 10 => 5.5m = 130 => m = 130 / 5.5 = 260 / 11 = 23 7/11 minutes. This corresponds to 4:23 7/11 PM.
Step 6: Both values fall between 4:00 and 5:00, matching Option A.