Clock angle — Between 2 PM and 3 PM, at what exact time is the angle between the minute and hour hands equal to 100 degrees? Answer as 2 : mm (with fractional minutes if needed).

Introduction / Context:
We need the time after 2:00 when the hands subtend 100 degrees. This uses the relative speed method.

Given Data / Assumptions:

• At 2:00, hour angle = 60 degrees.
• Let m be minutes after 2:00.
• Hour angle = 60 + 0.5*m; minute angle = 6*m.

Concept / Approach:
Set |(60 + 0.5m) − 6m| = 100. Solve for m in [0, 60].

Step-by-Step Solution:
|60 − 5.5m| = 100.Case 1: 60 − 5.5m = 100 → −5.5m = 40 → m = −40/5.5 (invalid).Case 2: 5.5m − 60 = 100 → 5.5m = 160 → m = 160/5.5 = 320/11 ≈ 29.091.Hence time = 2 : 320/11.

Verification / Alternative check:
Plugging m = 320/11 gives hour angle = 60 + 160/11, minute angle = 1920/11; difference = 100.

Why Other Options Are Wrong:
46/11, 126/11, and 186/11 do not satisfy the equation; only 320/11 works within the hour.

Common Pitfalls:
Forgetting to take absolute value or accepting a negative minute solution.

Final Answer:
2 : 320/11, not listed; therefore “None of these.”