I have two watches with a 12 hour cycle. One of them gains one minute a day and the other loses $1 \frac{1}{2}$ minutes per day. If I set them both at the correct time, how long will it be before they again tell the correct time together?

### Concept & Faulty Clocks & Cycles For a standard 12-hour watch to show the *correct* time again after drifting, it must gain or lose exactly 12 hours (720 minutes). To find when two independent watches show the correct time simultaneously, find the Least Common Multiple (LCM) of their individual correction cycles. ### Step-by-Step Solution * Given: 12-hour cycle clocks. Watch 1 gains 1 min/day. Watch 2 loses 1.5 min/day. * To show the correct time again, a watch must accumulate a drift of a full 12 hours (which is $12 \times 60 = 720$ minutes). * Time for Watch 1 to gain 720 minutes: $720 / 1 \text{ min/day} = 720 \text{ days}$. * Time for Watch 2 to lose 720 minutes: $720 / 1.5 \text{ min/day} = 480 \text{ days}$. * To show correct time together, we need the Least Common Multiple (LCM) of their individual cycles (720 days and 480 days). * LCM of 720 and 480: $720 = 240 \times 3$ $480 = 240 \times 2$ $\text{LCM} = 240 \times 3 \times 2 = 1440$. ### Exam Strategy & Shortcut Determine how many days it takes for each watch to drift by 720 minutes. Watch A takes 720 days. Watch B takes 480 days. Find the LCM of (720, 480) immediately, which is 1440. ### Common Pitfall Calculating when they show the *same* time rather than the *correct* time together. If they just needed to show the same time, you would find when their relative drift is 12 hours. ### Final Answer Therefore, the correct answer is **1440 days**.