Exact time when hands are opposite between 3 PM and 4 PM — At what time between 3 and 4 are the hour and minute hands 180° apart?

Introduction / Context:
Opposite positions (180°) occur once between consecutive hour marks. The formula for the minutes past H:00 when the hands are opposite is t = (60H + 30) / 11. This follows from setting the absolute angular difference to 180° and solving for t.

Given Data / Assumptions:

• Hour window: 3:00–4:00.
• Opposite condition: |6t − (30H + 0.5t)| = 180.
• Derived formula: t = (60H + 30)/11.

Concept / Approach:
Apply H = 3, compute the exact fraction in minutes, and compare against the options.

Step-by-Step Solution:
t = (60 × 3 + 30) / 11 = 210/11 minutes = 19 1/11 minutes.Time = 3 : 19 1/11.

Verification / Alternative check:
Plug t back into angles: minute = 6t, hour = 90 + 0.5t; difference evaluates to 180°.

Why Other Options Are Wrong:
3 : 471/11, 3 : 481/11, 3 : 491/11 correspond to ≈ 42 10/11, 43 8/11, 44 7/11 minutes, none equal to 19 1/11. Hence “None of these” is correct.

Common Pitfalls:
Using the coincidence formula (60H/11) instead of the opposite formula (60H+30)/11.

Final Answer:
None of these (correct time = 3 : 19 1/11)