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2009 AMC 10B Problem 14

2009 AMC 10B Problem 14AMC 10 Step by Step

Name: Date: September 13, 2026

Sequences & Series · difficulty 3 of 5 · about 5 min · position P11-15

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?

A)
$\text{Tuesday}$
B)
$\text{Wednesday}$
C)
$\text{Thursday}$
D)
$\text{Friday}$
E)
$\text{Saturday}$

Answer

Key insight

Non-millet seeds are always exactly 3/4 quart after a refill, while millet obeys m -> 3m/4 + 1/4, so the day it first exceeds 3/4 quart is day 5.

Solution

Measure in quarts. Each fresh quart contains $\frac14$ millet and $\frac34$ other seeds. The birds eat all the other seeds every day, so just after a refill the non-millet amount is always exactly $\frac34$ . Millet, on the other hand, survives at $75\%$ and is topped up by $\frac14$ .

Let $m_n$ be the millet just after the refill on day $n$ (Monday is day $1$ ). Then $m_1 = \frac14$ and $m_{n} = \frac34 m_{n-1} + \frac14$ :
$m_2 = \tfrac{7}{16}, \quad m_3 = \tfrac{37}{64}, \quad m_4 = \tfrac{175}{256}, \quad m_5 = \tfrac{781}{1024}.$

Millet exceeds half the feeder exactly when $m_n > \frac34$ (the amount of other seeds). Now $\frac{175}{256} < \frac{192}{256} = \frac34$ , while $\frac{781}{1024} > \frac{768}{1024} = \frac34$ . So day $5$ is the first such day: Friday.

The answer is $\boxed{\textbf{(D)}\ \text{Friday}}$ .

Why this works

Separating the two seed types reveals that one quantity is constant and the other follows a simple recursion. In closed form, $m_n = 1 - \left(\frac34\right)^n$ , so the condition is $\left(\frac34\right)^n < \frac14$ , first true at $n = 5$ . A "more than half" comparison is easiest as "part exceeds the rest," which avoids computing totals.

The trap

Comparing millet to the total of the fresh quart only, or forgetting that the birds eat all non-millet seeds so the 3/4 quart never accumulates.

Common mistakes

Comparing millet to the total of the fresh quart only, or forgetting that the birds eat all non-millet seeds so the 3/4 quart never accumulates.
Miscounting days: Monday is day 1, so day 5 is Friday, not Saturday.

Techniques

Compute small cases, spot the pattern, generalize · Define states/recurrence and iterate