Hands swapped (hour/minute) between given intervals: A man left between 3–4 and returned between 8–9. He found the hands had exactly interchanged positions. He returned at what time?

Introduction / Context:
“Hands interchanged” means the minute hand's earlier minute-mark becomes the hour hand's later position and vice-versa. Using the standard positions (minute = M; hour = 5H + M/12, in minute-marks), we can write two linear equations for the unknown minute values at the two times and solve exactly.

Given Data / Assumptions:

• First time: between 3 and 4 ⇒ hour = 3, minute m1.
• Second time: between 8 and 9 ⇒ hour = 8, minute m2.
• Interchange: m1 = 5×8 + m2/12 and m2 = 5×3 + m1/12.

Concept / Approach:
Write the two equations and solve simultaneously. Fractions with denominator 13 or 143 commonly emerge in such “swap” problems due to the 12-based hour-hand progress.

Step-by-Step Solution:

Verification / Alternative check:
66/143 ≈ 0.4615 ≈ 6/13. Hence 8:18 6/13 is the standard exact form found in classic solutions.

Why Other Options Are Wrong:
Other minute values do not satisfy both interchange equations simultaneously.

Common Pitfalls:
Using degrees instead of minute-mark equations for swaps; rounding too early and missing the exact fractional form.

Final Answer:
18 6/13 minutes past 8