Clock problem — between 9:00 PM and 10:00 PM, at what exact time do the minute hand and the hour hand coincide (overlap)? Answer in the form 9 : mm (fractional minutes allowed).

Introduction / Context:
Coincidence of hands occurs when both hands are at the same angle from 12. We compute the exact minute t after 9:00 PM when this happens.

Given Data / Assumptions:

• Minute hand speed = 6 degrees/minute.
• Hour hand speed = 0.5 degrees/minute.
• At 9:00 the hour hand is at 270 degrees; the minute hand is at 0 degrees.

Concept / Approach:
Set angles equal: 6t = 270 + 0.5t.

Step-by-Step Solution:
6t = 270 + 0.5t5.5t = 270t = 270 / 5.5 = 540/11 ≈ 49 1/11 minutesTherefore time = 9 : 540/11.

Verification / Alternative check:
Substitute t = 540/11 back: hour angle = 270 + 0.5*(540/11) = 270 + 270/11; minute angle = 6*(540/11) = 3240/11; both equal mod 360.

Why Other Options Are Wrong:
491/11, 481/11, and 441/11 are not equal to 540/11; using them will not equalize the two angles.

Common Pitfalls:
Confusing 49 1/11 with 491/11 (text concatenation error) or simplifying to the nearest integer minute.

Final Answer:
9 : 540/11 is correct, not present; hence “None of these.”