Imagine that your watch was correct at noon, but then it began to lose 30 minutes each hour. It now shows 4 p.m. but it stopped 5 hours ago. What is the correct time now?

### Concept & Faulty Clocks When a clock loses time consistently, it operates at a slower rate compared to real time. We can establish a ratio of "Clock Time" to "Real Time" to determine how much actual time has passed. $$ \text{Real Time Elapsed} = \frac{\text{Clock Time Elapsed}}{\text{Clock Speed Rate}} $$ ### Step-by-Step Solution * Given: The watch loses 30 minutes every hour. This means for every 1 hour (60 mins) of real time, the watch advances only 30 minutes. * Clock Speed Rate: The watch runs at half speed ($0.5$ hours on watch = $1$ hour real time). * Clock Time Elapsed: From noon (12:00) to 4 p.m. is 4 hours shown on the watch. * Real Time Elapsed: To show 4 hours, the real time passed must be $4 \times 2 = 8$ hours. * The watch stopped at 8 p.m. real time. * It stopped 5 hours ago. * Current Real Time: 8 p.m. + 5 hours = 1 a.m. ### Exam Strategy & Shortcut Think of it logically: If it loses half an hour every hour, it takes 2 real hours to show 1 hour of progress. 4 hours of progress on the face = 8 hours of real time. 12:00 PM + 8 hours + 5 hours = 1:00 AM. ### Common Pitfall Adding the 5 hours to the 4 p.m. shown on the broken clock (getting 9 p.m.) without converting the 4 hours of "broken clock time" into actual elapsed real time first. ### Final Answer Therefore, the correct answer is **1 a.m.**.