A and B start their new jobs on the same day. A’s schedule is 3 working days followed by 1 rest day, while B’s schedule is 7 working days followed by 3 rest days. On how many of their first 1000 days do both have rest-days on the same day?

This is a question which requires you to observe the pattern. 

For A, there are three working days and one rest day. Hence, A has a rest day on every fourth day (4, 8, 12, 16, 20, 24......1000).

For B, there are seven working days and three rest days after that. Hence, B has three rest days in the span of ten days (8/9/10, 18/19/20, 28/29/30, 38/39/40, 48/49/50..... 998/999/1000). 

From the pattern, it is clearly visible that A and B will have exactly one common day in every three-day rest period of B. This means, there will be 100 common days of rest for both A and B in the first 1000 days.