2012 AMC 8 Problems/Problem 19

Problem

In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar?

$\textbf{(A)}\hspace{.05in}6\qquad\textbf{(B)}\hspace{.05in}8\qquad\textbf{(C)}\hspace{.05in}9\qquad\textbf{(D)}\hspace{.05in}10\qquad\textbf{(E)}\hspace{.05in}12$

Solution

Let $r$ be the number of red marbles, $g$ be the number of green marbles, and $b$ be the number of blue marbles.

We have three equations:

$g + b = 6$

$r + b = 8$

$r + g = 4$

Now we use some algebraic manipulation.

We add all the equations to obtain a fourth equation:

$2r + 2g + 2b = 18$

Now divide by $2$ on both sides to find the total number of marbles:

$r + g + b = 9$ . The total number of marbles in the jar is $\boxed{\textbf{(C)}\ 9}$ .

2012 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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2012 AMC 8 Problems/Problem 19

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