A clock is set right at 8:00 AM in the morning. The minute hand and the hour hand of the clock come together after every 65 minutes. What is the actual time when the clock shows 8:00 PM, approximated to the nearest minutes?

In a standard, accurate clock, the minute and hour hands coincide every 65 5/11 minutes (approximately 65.45 minutes). However, this specific clock's hands meet every 65 minutes, meaning it is running faster than a normal clock. The clock is gaining time because it completes the coincidence cycle in less time than required. To find the actual time when the clock shows 8:00 PM (12 hours or 720 minutes of clock time), we use the ratio of correct time to clock time: (65 5/11) / 65. Total actual minutes = 720 * (720/11) / 65 = 720 * (144/143) ≈ 725.03 minutes. Since 720 minutes is exactly 12 hours, the actual time elapsed is approximately 725 minutes, which is 12 hours and 5 minutes. However, because the clock is fast, when it shows 8:00 PM, the actual time is approximately 7:55 PM. Re-calculating the loss/gain: the clock gains 5/11 minutes every 65 minutes. In 720 clock minutes, it gains (5/11 / 65) * 720 ≈ 5.03 minutes. Thus, the actual time is 8:00 PM minus ~5 minutes, which is 7:55 PM. Wait, if the clock is fast, the actual time is behind the clock time. 8:00 PM minus 5 minutes is 7:55 PM. But standard aptitude logic for 'clock shows 8 PM' with a fast clock means actual time is 7:55 PM. Nearest option is 7:55 PM.