2009 AMC 12B Problem 11

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: 1610

Small hint:

Each morning

\tfrac{3}{4}

of the previous millet remains and

\tfrac{1}{4}

quart of new millet is added, while the other seeds always total

\tfrac{3}{4}

quart

Big hint:

After

n

days the millet is

1 - \left(\tfrac{3}{4}\right)^n

quart; require this to exceed

\tfrac{3}{4}

Solution:

Each day the birds leave

\dfrac{3}{4}

of the millet and Millie adds

\dfrac{1}{4}

quart of new millet, so after

n

days the millet is

\begin{aligned} &\dfrac{1}{4}\left(1 + \dfrac{3}{4} + \cdots + \left(\dfrac{3}{4}\right)^{n-1}\right) \\ &= 1 - \left(\dfrac{3}{4}\right)^n \end{aligned}

quart.

The non-millet seeds always total

\dfrac{3}{4}

quart, so millet exceeds half when

1 - \left(\dfrac{3}{4}\right)^n \gt \dfrac{3}{4},

that is,

\left(\dfrac{3}{4}\right)^n \lt \dfrac{1}{4}.

Since

\left(\dfrac{3}{4}\right)^4 = \dfrac{81}{256} \gt \dfrac{1}{4}

and

\left(\dfrac{3}{4}\right)^5 = \dfrac{243}{1024} \lt \dfrac{1}{4},

this first happens on day

5,

which is Friday.

Thus, the correct answer is D.