Clock problem — between 9:00 PM and 10:00 PM, at what exact time are the minute hand and the hour hand opposite to each other (i.e., separated by 180 degrees)? Provide the answer in the form 9 : mm (with fractional minutes allowed).

Introduction / Context:
Questions on clock angles test relative speed and angle formulas between the hour and minute hands. Here we need the time between 9 PM and 10 PM when the hands are opposite (180 degrees apart).

Given Data / Assumptions:

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

Concept / Approach:
Let t be minutes after 9:00. Angle(hour) = 270 + 0.5*t. Angle(minute) = 6*t. Opposite means the absolute difference equals 180 degrees.

Step-by-Step Solution:
|6t - (270 + 0.5t)| = 180Case 1: 6t - 270 - 0.5t = 180 → 5.5t = 450 → t = 900/11 ≈ 81.818 (not within 0–60 minutes)Case 2: 270 + 0.5t - 6t = 180 → 5.5t = 90 → t = 180/11 ≈ 16.364 (valid)Therefore time = 9 : 180/11.

Verification / Alternative check:
At t = 180/11, hour angle = 270 + 0.5*(180/11) = 270 + 90/11 = 2970/11. Minute angle = 6*(180/11) = 1080/11. Difference = 1890/11 = 180 degrees.

Why Other Options Are Wrong:
9:154/11, 9:164/11, 9:124/11 are all different from 9:180/11 and do not yield 180 degrees.

Common Pitfalls:
Picking the extraneous solution t = 900/11 (>60 minutes) or rounding fractional minutes to neat decimals.

Final Answer:
9 : 180/11 is correct, which is not listed; hence “None of these.”