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2012 AMC 8 Problems/Problem 19: Difference between revisions

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<math> \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 </math>
<math> \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 </math>
==Solution==
Let <math> r </math> be the number of red marbles, <math> g </math> be the number of green marbles, and <math> b </math> be the number of blue marbles.
We have three equations:
<math> g + b = 6 </math>
<math> r + b = 8 </math>
<math> r + g = 4 </math>
Now we use some algebraic manipulation.
We add all the equations to obtain a fourth equation:
<math> 2r + 2g + 2b = 18 </math>
Now divide by <math> 2 </math> on both sides to find the total number of marbles:
<math> r + g + b = 9 </math>. The total number of marbles in the jar is <math> \boxed{\textbf{(C)}\ 9} </math>.


==See Also==
==See Also==
{{AMC8 box|year=2012|num-b=18|num-a=20}}
{{AMC8 box|year=2012|num-b=18|num-a=20}}

Revision as of 11:47, 24 November 2012

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 $\boxed{\textbf{(C)}\ 9}$.

See Also

2012 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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
All AJHSME/AMC 8 Problems and Solutions
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