Between 9 and 10 o'clock, at what time will the hands of a watch be together?

Problem restatement
Find the exact time between 9 and 10 when the hour and minute hands coincide.

Given data

• Hour-hand position at time t minutes after 9: 30×9 + 0.5t = 270 + 0.5t (degrees).
• Minute-hand position: 6t (degrees).

Concept/Approach
At coincidence: positions equal. Solve 270 + 0.5t = 6t.

Step-by-Step calculation
270 + 0.5t = 6t 270 = 5.5t = (11/2)t t = 270 × (2/11) = 540/11 = 49 1/11 minutes

Verification/Alternative
Time = 9:49 1/11, i.e., 9:49 and 5 5/11 seconds (since 1/11 min = 60/11 s ≈ 5.4545 s).

Common pitfalls

• Using 65 5/11 minutes after 9:00 (that gives the next overlap overall, not within each hour).

Final Answer
9:49 1/11