Clock problem — Between 3:00 and 4:00, at what exact time are the hour and minute hands together?

Introduction / Context:
Many clock problems ask for the exact instant when the hour and minute hands coincide. Between 3:00 and 4:00, the minute hand must catch up the fractional advance of the hour hand after 3:00. This tests the standard relative speed technique for clock angles.

Given Data / Assumptions:

• Hour hand speed = 0.5 degrees per minute.
• Minute hand speed = 6 degrees per minute.
• At 3:00, the hour hand is at 90 degrees from 12.

Concept / Approach:
When hands are together, their angular positions are equal. Let t be minutes after 3:00. Then hour angle = 90 + 0.5*t and minute angle = 6*t. Set them equal and solve for t.

Step-by-Step Solution:
At 3:00, gap = 90 degrees.Relative speed = 6 - 0.5 = 5.5 degrees per minute.Time to close gap t = 90 / 5.5 = 180 / 11 = 16 4/11 minutes.Thus the hands coincide at 16 4/11 minutes past 3.

Verification / Alternative check:
In each hour, there is one coincidence (except around 11–1 boundary). The value 16 4/11 is the standard result for the interval after 3:00, matching known formula t = (60*H)/11 for hour H = 3.

Why Other Options Are Wrong:
“15 7/11 past 4” refers to the window 4–5, not 3–4. “16 2/11 past 2” refers to 2–3. “None” is invalid because a valid time exists at 16 4/11 after 3.

Common Pitfalls:
Using degrees per hour instead of per minute; forgetting that the hour hand moves during those minutes; mixing the 2–3 and 4–5 intervals with 3–4; rounding too early.

Final Answer:
16 4/11 minutes past 3