AMC 8 2004 Problem 22

AMC8 2004 年第 22 题

比与比例★★★☆☆

At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is

\frac25

. What fraction of the people in the room are married men?

在某个聚会上，只有单身女性，以及已婚男性和他们的妻子。若随机选择一个女性，则她是单身的概率为

\displaystyle \frac{2}{5}

。那么这个房间的所有人中，已婚男性占了多少比例？

A. $\frac13$
B. $\frac38$
C. $\frac25$
D. $\displaystyle \frac{5}{12}$
E. $\frac35$

正确答案 · Correct Answer

解析 · Solution

Solution 1

Assume arbitrarily (and WLOG) there are 5 women in the room, of which

5 \cdot \frac25 = 2

are single and 5-2=3 are married. Each married woman came with her husband, so there are 3 married men in the room as well for a total of 5+3=8 people. The fraction of the people that are married men is

\frac38

. Let single women be

S

, married wives to be

W

, and men to be

M

. As such, the number of men is equal to married woman since each husband has his own wife or W=M. We are asked to find the probability of men in the total group or

\displaystyle \frac{M}{S+W+M}

. We are also given that out of all the women,

\displaystyle \frac{2}{5}

are single or

\displaystyle \frac{S}{S+W}

\displaystyle \frac{2}{5}

. Rewrite this as

\displaystyle S= \frac{2}{5}(S+W)

\displaystyle S= \frac{2S}{5} + \frac{2W}{5}

, isolating

S

gives us

\displaystyle \frac{3S}{5}

\displaystyle \frac{2W}{5}

\displaystyle S = \frac{2M}{3}

(because W=M ). Now, substituting gives us

\displaystyle \frac{M}{\frac{2M}{3}+M+M}= \frac{M}{\frac{2M}{3}+ \frac{6M}{3}}= \frac{M}{\frac{8M}{3}}=\frac38

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