2003 AMC 10B Problems/Problem 12

Problem

Al, Betty, and Clare split $$$$ $1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year, they have a total of $$$$ $1500$ dollars. Betty and Clare have both doubled their money, whereas Al has managed to lose $$$$ $100$ dollars. What was Al's original portion?

$\textbf{(A) }$ $250 \qquad\textbf{(B) }$ $350 \qquad\textbf{(C) }$ $400\qquad\textbf{(D) }$ $450\qquad\textbf{(E) }$ $500$

Solution

For this problem, we will have to write a three-variable equation, but not necessarily solve it. Let $a, b,$ and $c$ represent the original portions of Al, Betty, and Clare, respectively. At the end of one year, they each have $a-100, 2b,$ and $2c$ . From this, we can write two equations.

$a+b+c=1000$

$a-100+2b+2c=1500$

Since all we need to find is $a,$ substitute $b+c$ in terms of $a$ and solve for $a.$

$b+c=1000-a$ $\begin{align*} a-100+2(b+c)&=1500\\ a+2(1000-a)&=1600\\ a+2000-2a&=1600\\ a&=400\end{align*}$

Al's original portion was $\boxed{\mathrm{(C) \ } 400}$ .

2003 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2003 AMC 10B Problems/Problem 12

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