PUMaC 2012 · 团队赛 · 第 1 题

PUMaC 2012 — Team Round — Problem 1

(2 digits) For some reason, people in math problems like to paint houses. Alice can paint a house in one hour. Bob can paint a house in six hours. If they work together, it takes them seven hours to paint a house. You might be thinking “What? That’s not right!” but I did not make a mistake. When Alice and Bob work together, they get distracted very easily and simultaneously send text messages to each other. When they are texting, they are not getting any work done. When they are not texting, they are painting at their normal speeds (as if they were working alone). Carl, the owner of the house decides to check up on their work. He randomly picks a time during the seven hours. The probability that they are texting during that time can be written as r/s , where r and s are integers and gcd( r, s ) = 1. What is r + s ? √ √

解析

Problem: (2 digits) For some reason, people in math problems like to paint houses. Alice can paint a house in one hour. Bob can paint a house in six hours. If they work together, it takes them seven hours to paint a house. You might be thinking “What? That’s not right!” but I did not make a mistake. When Alice and Bob work together, they get distracted very easily and simultaneously send text messages to each other. When they are texting, they are not getting any work done. When they are not texting, they are painting at their normal speeds (as if they were working alone). Carl, the owner of the house decides to check up on their work. He randomly picks a time during the seven hours. The probability that they are texting during that time can be written as r/s , where r and s are integers and gcd( r, s ) = 1. What is r + s ? Answer: 92 Solution: Alice’s painting speed is 1 house per hour and Bob’s is 1 / 6 house per hour. Thus, if they don’t text, it would take them 1 / (1 + 1 / 6) = 6 / 7 of an hour to paint the house. The probability is time spent texting 7 − 6 / 7 43 P = = = total time 7 49 so the answer is 43 + 49 = 92 . Author: Alan 8 √ √