1993 AHSME Problems/Problem 18

Problem

Al and Barb start their new jobs on the same day. Al's schedule is 3 work-days followed by 1 rest-day. Barb's schedule is 7 work-days followed by 3 rest-days. On how many of their first 1000 days do both have rest-days on the same day?

Solution

We note that Al and Barb's work schedules clearly form a cycle, with Al's work cycle being 4 days long and Barb's work schedule being 10 days long. Since , we know that Al's and Barb's work cycles coincide, and thus collectively repeat, every 20 days. As a result, we only need to figure out how many coinciding rest days they have in the first 20 days, and multiply that number by to determine the total number of coinciding rest days. These first 20 days are written out below:

Day 1: Al works; Barb works Day 2: Al works; Barb works Day 3: Al works; Barb works Day 4: Al rests; Barb works Day 5: Al works; Barb works Day 6: Al works; Barb works Day 7: Al works; Barb works Day 8: Al rests; Barb rests Day 9: Al works; Barb rests Day 10: Al works; Barb rests Day 11: Al works; Barb works Day 12: Al rests; Barb works Day 13: Al works; Barb works Day 14: Al works; Barb works Day 15: Al works; Barb works Day 16: Al rests; Barb works Day 17: Al works; Barb works Day 18: Al works; Barb rests Day 19: Al works; Barb rests Day 20: Al rests; Barb rests

In this 20-day cycle, Al and Barb's rest days coincide twice: on day 8 and day 20. As a result (as discussed above), the answer is . ~cw357

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.