After 9 o’clock and before 10 o’clock, at what time are the hour and minute hands in opposite directions (180° apart)?

Introduction / Context:
“Opposite directions” means the angle between hands is 180°. Use the relative speed 5.5 degrees/minute and the equation for angle difference.

Given Data / Assumptions:

• Between 9 and 10 → hour hand is past 9.
• At 9 : 00, hour-hand angle = 270°.

Concept / Approach:
For t minutes after 9:00, minute-hand angle = 6t; hour-hand angle = 270 + 0.5t. Require |(270 + 0.5t) − 6t| = 180.

Step-by-Step Solution:
|270 − 5.5t| = 180 → 270 − 5.5t = 180 (valid solution in the first hour).5.5t = 90 → t = 90 / 5.5 = 16 4/11 minutes.Therefore the time is 9 : 16 4/11.

Verification / Alternative check:
Other algebraic branch gives a time beyond 10:00; only this one lies in (9,10).

Why Other Options Are Wrong:
15, 16, 17 1/11 are close but not exact; precise fraction is required.

Common Pitfalls:
Using 6t − (270 + 0.5t) = 180 without absolute value can produce the later invalid time.

Final Answer:
16 4/11 min past 9