Difference between revisions of "2012 AMC 8 Problems/Problem 19"

Latest revision as of 23:30, 17 February 2024

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 1 (Trial and Error)

$6$ are blue and green - $b+g=6$

$8$ are red and blue - $r+b=8$

$4$ are red and green- $r+g=4$

We can do trial and error. Let's make blue $5$ . That makes green $1$ and red $3$ because $6-5=1$ and $8-5=3$ . To check this, let's plug $1$ and $3$ into $r+g=4$ , which works. Now count the number of marbles - $5+3+1=9$ . So the answer is $\boxed{\textbf{(C)}\ 9}.$

Solution 2

We already knew the facts: $6$ are blue and green, meaning $b+g=6$ ; $8$ are red and blue, meaning $r+b=8$ ; $4$ are red and green, meaning $r+g=4$ . Then we need to add these three equations: $b+g+r+b+r+g=2(r+g+b)=6+8+4=18$ . It gives us all of the marbles are $r+g+b = 18/2 = 9$ . So the answer is $\boxed{\textbf{(C)}\ 9}$ . ~LarryFlora

Solution 3 (Venn Diagrams)

We may draw three Venn diagrams to represent these three cases, respectively.

Let the amount of all the marbles be $x$ , meaning $R+G+B = x$ .

The Venn diagrams give us the equation: $x = (x-6)+(x-8)+(x-4)$ . So $x = 3x-18$ , $x = 18/2 =9$ . Thus, the answer is $\boxed{\textbf{(C)}\ 9}$ . ~LarryFlora

Solution 4 (Answer Choices)

Since we know all but $8$ marbles in the jar are green, the jar must have at least $9$ marbles. Then we can just start from $C$ and keep going. If there are $9$ marbles total, there are $3$ red marbles $(9-6)$ , $1$ green marble $(9-8)$ , and $5$ blue marbles $(9-4)$ . Since we assumed there were $9$ marbles and $3+1+5=9$ , the answer is $\boxed{\textbf{(C)}\ 9}$ .

Solution 5 (Algebra)

Let $x$ be the number of total marbles. There are $x – 6$ red marbles, $x – 8$ green marbles, and $x – 4$ blue marbles. We can create an equation: $(x – 6)+(x – 8)+(x – 4)=x$ Solving, we get $x=9$ , which means the total number of marbles is $\boxed{\textbf{(C)}\ 9}$ . -J.L.L (Feel free to edit)

Solution 6

Let $x$ be the number of total marbles, $r$ be the number of red marbles, $g$ be the number of green marbles, and $b$ be the number of blue marbles. Then we have $x - r = 6$ , $x - g = 8$ , $x - b = 4$ , and $r + g + b = x$ . Adding the first three equations together, we get $3x - r - g - b = 18$ or $3x - (r + g + b) = 18$ . Substituting in the fourth equation, we have $3x - x = 18$ $\implies$ $\boxed{\textbf{(C)}\ 9}$ .

~cxsmi

Video Solution

https://youtu.be/mMph7QH1kX0 Soo, DRMS, NM

https://youtu.be/-p5qv7DftrU ~savannahsolver

Video Solution by OmegaLearn

https://youtu.be/TkZvMa30Juo?t=1316

~pi_is_3.14

@@ Line 4: / Line 4: @@
 <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 1==
+==Solution 1 (Trial and Error)==
 <math>6</math> are blue and green - <math>b+g=6</math>
@@ Line 29: / Line 29: @@
 The Venn diagrams give us the equation: <math>x = (x-6)+(x-8)+(x-4)</math>.
 So <math>x = 3x-18</math>, <math>x = 18/2 =9</math>.
-Thus, the answer is <math>\boxed{\textbf{(C)}\ 9}</math>.    ---LarryFlora
+Thus, the answer is <math>\boxed{\textbf{(C)}\ 9}</math>.
+~LarryFlora
 ==Solution 4 (Answer Choices)==
@@ Line 42: / Line 43: @@
 ==Solution 6==
-Let <math>x</math> be the number of total marbles, <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 red marbles. Then we have <math>x - r = 6</math>, <math>x - g = 8</math>, <math>x - b = 4</math>, and <math>r + g + b = x</math>. Adding the first three equations together, we get <math>3x - r - g - b = 18</math> or <math>3x - (r + g + b) = 18</math>. Substituting in the fourth equation, we have <math>3x - x = 18</math> <math>\implies</math> <math>\boxed{\textbf{(C)}\ 9}</math>.
+Let <math>x</math> be the number of total marbles, <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. Then we have <math>x - r = 6</math>, <math>x - g = 8</math>, <math>x - b = 4</math>, and <math>r + g + b = x</math>. Adding the first three equations together, we get <math>3x - r - g - b = 18</math> or <math>3x - (r + g + b) = 18</math>. Substituting in the fourth equation, we have <math>3x - x = 18</math> <math>\implies</math> <math>\boxed{\textbf{(C)}\ 9}</math>.
-~cxsmi
+~[https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi]
 ==Video Solution==

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