Problem:
On Monday, Millie puts a quart of seeds, 25% of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only 25% of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
Answer Choices:
A. Tuesday
B. Wednesday
C. Thursday
D. Friday
E. Saturday
Solution:
On Monday, day 1, the birds find 41 quart of millet in the feeder. On Tuesday they find
41+43⋅41
quarts of millet. On Wednesday, day 3, they find
41+43⋅41+(43)2⋅41
quarts of millet. The number of quarts of millet they find on day n is
41+43⋅41+(43)2⋅41+⋯+(43)n−1⋅41=1−43(41)(1−(43)n)=1−(43)n
The birds always find 43 quart of other seeds, so more than half the seeds are millet if 1−(43)n>43, that is, when (43)n<41. Because (43)4=25681>41 and (43)5=1024243<41, this will first occur on day 5 which is (D)Friday.
The problems on this page are the property of the MAA's American Mathematics Competitions