# 2007 AMC 10B Problems/Problem 14

The following problem is from both the 2007 AMC 12B #10 and 2007 AMC 10B #14, so both problems redirect to this page.

## Problem

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?

$\textbf{(A) } 4 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 10 \qquad\textbf{(E) } 12$

## 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 $0.4p=0.4(20)=\boxed{\mathrm{(C) \ } 8}$

## Alternate Solution

There are the same number of total people before and after, but the number of girls has dropped by two or $10\%$ of the total. $\frac{2}{0.1}=20$, and $40\%\cdot20=8$, so the answer is $\mathrm{(C)}$.

## Yet Another Alternate Solution

Let $x$ be the number of people initially in the group and $g$ the number of girls. $\frac{2}{5}x = g$, so $x = \frac{5}{2}g$. Also, the problem states $\frac{3}{10}x = g-2$. Substituting $x$ in terms of $g$ into the second equation yields that $g = \boxed{ 8\ \mathrm{(C)}}$.

~mobius247