AMC12 2009 B

AMC12 2009 B · Q11

AMC12 2009 B · Q11. It mainly tests Sequences & recursion (algebra), Interest / growth (simple).

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?

周一，米莉向一个鸟食器中放入一夸脱种子，其中 $25\%$ 是小米。在接下来的每一天，她都会再加入一夸脱相同配比的种子，并且不取出任何剩下的种子。每天鸟只吃掉食器中小米的 $25\%$，但会吃掉所有其他种子。问在哪一天，米莉刚放入种子后，鸟会发现食器中超过一半的种子是小米？

(A) Tuesday 星期二

(B) Wednesday 星期三

(D) Friday 星期五

(E) Saturday 星期六

Answer

Correct choice: (D)

正确答案：（D）

Solution

On Monday, day 1, the birds find $\frac 14$ quart of millet in the feeder. On Tuesday they find \[\frac 14 + \frac 34 \cdot \frac 14\] quarts of millet. On Wednesday, day 3, they find \[\frac 14 + \frac 34 \cdot \frac 14 + \left(\frac34\right)^2 \cdot \frac 14\] quarts of millet. The number of quarts of millet they find on day $n$ is \[\frac 14 + \frac 34 \cdot \frac 14 + \left(\frac34\right)^2 \cdot \frac 14 + \cdots + \left(\frac 34\right)^{n-1} \cdot \frac 14 = \frac {(\frac 14)(1 - (\frac 34)^n)}{1 - \frac 34} = 1 - \left(\frac 34\right)^n .\] The birds always find $\frac 34$ quart of other seeds, so more than half the seeds are millet if $1 - \left(\frac 34\right)^n > \frac 34$, that is, when $\left(\frac 34\right)^n < \frac 14$. Because $\left(\frac 34\right)^4 = \frac {81}{256} > \frac 14$ and $\left(\frac 34\right)^5 = \frac {243}{1024} < \frac 14$, this will first occur on day $5$ which is $\boxed {\text{Friday}}$. The answer is $\mathrm{(D)}$.

周一（第 1 天），鸟在食器中发现 $\frac 14$ 夸脱小米。周二它们发现 \[\frac 14 + \frac 34 \cdot \frac 14\] 夸脱小米。周三（第 3 天）它们发现 \[\frac 14 + \frac 34 \cdot \frac 14 + \left(\frac34\right)^2 \cdot \frac 14\] 夸脱小米。第 $n$ 天它们发现的小米夸脱数为 \[\frac 14 + \frac 34 \cdot \frac 14 + \left(\frac34\right)^2 \cdot \frac 14 + \cdots + \left(\frac 34\right)^{n-1} \cdot \frac 14 = \frac {(\frac 14)(1 - (\frac 34)^n)}{1 - \frac 34} = 1 - \left(\frac 34\right)^n .\] 鸟总是会发现 $\frac 34$ 夸脱的其他种子，因此当 $1 - \left(\frac 34\right)^n > \frac 34$ 时，小米占种子的一半以上，也就是当 $\left(\frac 34\right)^n < \frac 14$。因为 $\left(\frac 34\right)^4 = \frac {81}{256} > \frac 14$ 且 $\left(\frac 34\right)^5 = \frac {243}{1024} < \frac 14$，这第一次发生在第 5 天，即 $\boxed {\text{Friday}}$。答案是 $\mathrm{(D)}$。

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