Between 7 and 8 o'clock, at what time will the clock hands be in a straight line but not together (i.e., 180° apart)?

Problem restatement
Find the time between 7 and 8 when the hands are 180° apart.

Given data

• At t minutes after 7, hour-hand angle = 210° + 0.5t°.
• Minute-hand angle = 6t°.
• Required: |difference| = 180°.

Concept/Approach
Solve |6t − (210 + 0.5t)| = 180 for t; discard any t outside 0–60.

Step-by-Step calculation
6t − 210 − 0.5t = ±180Case 1: 5.5t = 390 → t = 70 10⁄11 (invalid > 60)Case 2: 5.5t = 30 → t = 5 5⁄11Time = 7 : 05 5⁄11

Verification/Alternative
Relative speed = 5.5°/min; using a diagram or quick check confirms the second solution is within the hour.

Common pitfalls
Accepting the extraneous solution t > 60 minutes; mixing up 0° (together) with 180° (opposite).

Final Answer
7:05 5⁄11