Clock angles — Between 5:00 and 5:30, at what exact time are the hands 5 minute-spaces apart (i.e., 30 degrees apart)?

Introduction / Context:
Clock problems convert time to angles. One “minute space” on the dial is 6 degrees. “5 minutes apart” therefore means a 30-degree separation between hour and minute hands. We solve for the time t minutes after 5:00 when the angle difference equals 30 degrees.

Given Data / Assumptions:

• Time window: between 5:00 and 5:30.
• Minute hand speed: 6 degrees per minute.
• Hour hand speed: 0.5 degrees per minute; at 5:00 its angle is 150 degrees.
• Required separation: 30 degrees.

Concept / Approach:
Let t be minutes after 5:00. Minute angle = 6t. Hour angle = 150 + 0.5t. We want |6t − (150 + 0.5t)| = 30. Solve the linear equations for t and select the solution in (0, 30).

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Final Answer:
240/11 min past 5