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?
$\text{(A) } 48\quad \text{(B) } 50\quad \text{(C) } 72\quad \text{(D) } 75\quad \text{(E) } 100$
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 $\text{LCM}\left(4,10\right)=20$, 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 $\frac{1000}{20}=50$ 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 $2\cdot50=\fbox{100\text{ (E)}}$.    ~cw357