Jack and Jill decide to go for a run. Jack says he will run alternately north and west. He will go 4 minutes north, 6 minutes west, 8 minutes north, 10 minutes west, 12 minutes north, 4 minutes west, 6 minutes north and so on. Jill says she will run alternately west and north. She will go 12 minutes west, 10 minutes north, 8 minutes west, 6 minutes north, 4 minutes west, 12 minutes north, 10 minutes west and so on. They will set off at the same time and run at exactly the same speed. How long will it be after they first set off that their paths will cross again?

To solve the problem, we need to analyze the running patterns of both Jack and Jill step by step. ### Jack's Running Pattern: - Jack runs alternately north and west. - His running times are as follows: - 4 minutes north - 6 minutes west - 8 minutes north - 10 minutes west - 12 minutes north - 4 minutes west - 6 minutes north - ... We can observe that Jack's running times for north and west are increasing in a specific pattern. The north running times are 4, 8, 12, ... (increasing by 4 minutes each time), and the west running times are 6, 10, 4, 6, ... (alternating). ### Jill's Running Pattern: - Jill runs alternately west and north. - Her running times are as follows: - 12 minutes west - 10 minutes north - 8 minutes west - 6 minutes north - 4 minutes west - 12 minutes north - 10 minutes west - ... Jill's running times for west and north are also increasing in a specific pattern. The west running times are 12, 8, 4, ... (decreasing by 4 minutes each time), and the north running times are 10, 6, 12, 10, ... (alternating). ### Finding the Time Until Their Paths Cross Again: To find when their paths will cross again, we need to determine the total time taken for each runner after each segment of their run and see when they will be at the same coordinates. 1. **Calculate Jack's total time after each segment:** - 4 min north → 4 min - 6 min west → 10 min - 8 min north → 18 min - 10 min west → 28 min - 12 min north → 40 min - 4 min west → 44 min - 6 min north → 50 min - ... 2. **Calculate Jill's total time after each segment:** - 12 min west → 12 min - 10 min north → 22 min - 8 min west → 30 min - 6 min north → 36 min - 4 min west → 40 min - 12 min north → 52 min - 10 min west → 62 min - ... Next, we will find the first common time in both sequences after they start running. ### Finding the Common Time: We will check the total times calculated for both Jack and Jill until we find a match. Let's calculate the total times for both Jack and Jill until we find the first common time. - Jack's times: 4, 10, 18, 28, 40, 44, 50, ... - Jill's times: 12, 22, 30, 36, 40, 52, 62, ... The first common time is 40 minutes. ### Conclusion: Jack and Jill's paths will cross again after **40 minutes**.