AMC12 2007 B

AMC12 2007 B · Q10

AMC12 2007 B · Q10. It mainly tests Systems of equations, Percent.

Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\%$ of the group are girls. How many girls were initially in the group?

一些男孩和女孩在为班级中国之旅筹款而洗车。最初小组的 $40\%$ 是女孩。不久之后，两个女孩离开，两个男孩到来，然后小组的 $30\%$ 是女孩。最初小组中有多少女孩？

(A) 4 4

(B) 6 6

(D) 10 10

(E) 12 12

Answer

Correct choice: (C)

正确答案：（C）

Solution

If we let $p$ be the number of people initially in the group, then $0.4p$ is the number of girls. If two girls leave and two boys arrive, the number of people in the group is still $p$, but the number of girls is $0.4p-2$. Since only $30\%$ of the group are girls, \begin{align*} \frac{0.4p-2}{p}&=\frac{3}{10}\\ 4p-20&=3p\\ p&=20\end{align*} The number of girls initially in the group is $0.4p=0.4(20)=\boxed{\mathrm{(C) \ } 8}$

设最初小组人数为 $p$，则女孩人数为 $0.4p$。两个女孩离开、两个男孩到来后，小组人数仍为 $p$，但女孩人数变为 $0.4p-2$。由于此时只有 $30\%$ 是女孩， \begin{align*} \frac{0.4p-2}{p}&=\frac{3}{10}\\ 4p-20&=3p\\ p&=20\end{align*} 因此最初女孩人数为 $0.4p=0.4(20)=\boxed{\mathrm{(C) \ } 8}$

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