1190.32 – The Phone Booth

You are on a scavenger hunt, following clues here and there around the city. At this moment your watch says 1:30 pm, but at the moment it is 20 minutes slow. Not only that, it is losing 10 minutes every hour after a bus ran over it last week. You must be in the phone booth at 6th Avenue and 44th street at exactly 4:00 to receive a call with at your next clue. If you don't reset your watch between now and then, at what time by your watch should you be at the phone booth?

Solution

This is messy. Let's see: right now it is really 1:50. That is 2 hours and 10 minutes before 4:00, or $2 \dfrac{1}{6}$ hours, or $\dfrac{13}{6}$ hours. In that time your watch will lose $\dfrac{13}{6} \times 10$ minutes more, or $21 \dfrac{2}{3}$ minutes.

This means that the real time of 4:00 your watch will be $20 + 21 \dfrac{2}{3}$ slow, i.e. 41 minutes and 40 seconds slow. So you should show up that much early by your watch, namely 3:18:20. 3:18:20 + 41:40 = 4:00. Whew. You need to get your watch fixed, or replace it.