2009 AMC 10B Problem 14

Question

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?

Tuesday

Wednesday

Thursday

Friday

Saturday

Answer: D

Concepts:geometric sequence recursion inequality

Difficulty rating: 1660

Small hint:

Each day the standing millet is multiplied by

\dfrac34,

then

\dfrac14

quart is added

Big hint:

Other seeds always total

\dfrac34

quart; find when millet first exceeds that

Solution:

Each quart adds

\dfrac14

quart of millet, and the birds leave

\dfrac34

of the standing millet. On day

n

the millet present is

\begin{gathered} \dfrac14\left(1+\dfrac34+\cdots+\Big(\dfrac34\Big)^{n-1}\right) \\ = 1-\Big(\dfrac34\Big)^{n}. \end{gathered}

The other seeds always total

\dfrac34

quart. Millet exceeds half when

1-\Big(\dfrac34\Big)^n\gt\dfrac34,

i.e.

\Big(\dfrac34\Big)^n\lt\dfrac14.

Since

\Big(\dfrac34\Big)^4=\dfrac{81}{256}\gt\dfrac14

but

\Big(\dfrac34\Big)^5=\dfrac{243}{1024}\lt\dfrac14,

this first happens on day

5,

which is Friday.

Thus, the correct answer is D.