Q.7 Rohit goes to a restaurant for lunch at about 1 PM.
When he enters the restaurant, he notices that the hour and minute hands
on the wall clock are exactly coinciding. After about an hour, when he
leaves the restaurant, he notices that the clock hands are again exactly
coinciding. How much time (in minutes) did Rohit spend at the restaurant?

Rohit enters around 1 PM when clock hands coincide and leaves about an hour later at the next coincidence. The exact time spent is 65 ⁵/₁₁ minutes, matching option (C).

Clock Mechanics Basics

Minute hand moves at 6° per minute (360°/60). Hour hand moves at 0.5° per minute (30°/60). Relative speed is 5.5° per minute, so hands coincide every 360°/5.5° = 720/11 ≈ 65 ⁵/₁₁ minutes.

First Coincidence After 1 PM

At H:00, hour hand at 30H°. Let m minutes pass: hour position = 30H + 0.5m, minute position = 6m. Set equal: 6m = 30×1 + 0.5m → 5.5m = 30 → m = 30/5.5 = 60/11 ≈ 5 ⁵/₁₁ minutes past 1.

Time to Next Coincidence

Hands coincide 11 times in 12 hours, interval 12/11 hours = 720/11 minutes. Rohit enters at first (60/11 min past 1), leaves at second: 2×720/11 – 60/11 = (1440/11 – 60/11) = 1380/11 = 125 ⁵/₁₁ min past 1, but spent time is exactly 720/11 = 65 ⁵/₁₁ min.

Why Options Are Wrong