Exact time of coincidence between 3 PM and 4 PM — At what time between 3 and 4 do the hour and minute hands coincide?

Introduction / Context:
To find the exact time of overlap between H and H+1 hours, use the standard formula for minutes past H:00 when the hands coincide: t = (60H) / 11. This arises from equating the angular positions of the hour and minute hands and solving for t.

Given Data / Assumptions:

• Hour window: between 3:00 and 4:00.
• Formula for coincidence: t = (60H)/11 minutes past H.

Concept / Approach:
Plug H = 3 into the formula and then convert the fractional minutes into a mixed number of minutes and seconds if desired. Compare the exact fraction with the options provided.

Step-by-Step Solution:
t = (60 × 3) / 11 = 180/11 minutes = 16 4/11 minutes.Time = 3 : 16 4/11.

Verification / Alternative check:
General derivation: set minute-hand angle 6t equal to hour-hand angle 30H + 0.5t; solving 6t = 30H + 0.5t yields t = 60H/11.

Why Other Options Are Wrong:
3 : 164/11 (≈ 14 10/11), 3 : 154/11 (14), 3 : 174/11 (≈ 15 9/11) are not equal to 16 4/11. Hence “None of these” is correct.

Common Pitfalls:
Misreading “164/11” as “16 4/11”; note that 164/11 = 14 10/11, not 16 4/11.

Final Answer:
None of these (correct time = 3 : 16 4/11)