Between 3 and 4 o'clock, after how many minutes will the two hands coincide?

Problem restatementFind the exact minutes past 3 when the hour and minute hands overlap.

Given data

• At t minutes after 3, hour-hand angle = 90° + 0.5t°.
• Minute-hand angle = 6t°.
• Coincide → angles equal.

Concept/ApproachSet 6t = 90 + 0.5t and solve for t.

Step-by-Step calculation6t = 90 + 0.5t5.5t = 90 → t = 90 ÷ 5.5 = 16 4⁄11 minutes

Verification/AlternativeGeneral formula: minutes past h o’clock for overlap = (60h) ÷ 11; for h = 3, t = 180÷11 = 16 4⁄11.

Common pitfallsUsing 60/11 blindly (that gives overlap after 12 o’clock), not adjusting for h.

Final Answer16 4⁄11 minutes