1993 AMC 12 Problem 18

Question

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?

48

50

72

75

100

The combined schedule repeats every

\operatorname{lcm}(4,10)

days

Within one

20

-day cycle, compare Al’s rest days with Barb’s

The combined pattern repeats every

20

days. Al rests on days

4,

8,

12,

16,

20,

while Barb rests on days

8,

9,

10,

18,

19,

20.

Their common rest days are

8

and

20,

two per cycle. There are

\frac{1000}{20}=50

cycles, giving

2\cdot50=100

common rest days. Thus the correct answer is E.

Problem 18 in other years