2003 AMC 10B Problems/Problem 12

Problem

Al, Betty, and Clare split $\textdollar 1000$ 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 $\textdollar 1500$ dollars. Betty and Clare have both doubled their money, whereas Al has managed to lose $\textdollar100$ dollars. What was Al's original portion?

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

Solution 1

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.

$a+b+c=1000 \implies 2a+2b+2c=2000$ $(a-100)+2b+2c=1500 \implies a+2b+2c=1600$

Since all we need to find is $a,$ subtract the second equation from the first equation to get $a=400.$ Al's original portion was $\boxed{\textbf{(C)}\ \textdollar 400}$ .

Solution 2

Suppose Betty and Clare $1000-x$ dollars and Al $x$ dollars originally. Then, $(x-100)+2(1000-x)=1500 \implies x=400$ , so Al's original portion was $\boxed{\textbf{(C)}\ \textdollar 400}$ .

~Mathkiddie

Solution 3

If Al had not lost $\textdollar 100$ , the total amount would be $\textdollar 1600$ . So, Betty and Clare collectively gained $\textdollar 600$ and as they have doubled their collective fortune, they must have $\textdollar 600$ with them at the beginning. This leaves $\boxed{\textbf{(C)}\ \textdollar 400}$ for Al. ~Anshulb

Solution 4

Let $a, b,$ and $c$ represent the original portions of Al, Betty, and Clare, respectively. We can write some equations: $a+b+c = 1000$ $2b+2c+a-100 = 1500 \Longrightarrow 2b+2c+a=1600$ Subtracting the first equation from the second we get $b+c = 600$ . So, $a = 1000-600 = \boxed{\textbf{(C)}\ \textdollar 400}$

-jb2015007

2003 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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All AMC 10 Problems and Solutions

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