Difference between revisions of "2012 AMC 8 Problems/Problem 19"
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| + | ==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? | 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? | ||
<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== | + | ==Solution 1== |
| − | + | 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 and it does work. Now count the number of marbles- 5+3+1=9. So 9 (C) is the answer. | |
| − | + | ==Solution 2== | |
| − | We add all the | + | We already knew the facts: <math>6</math> are blue and green, meaning <math>b+g=6</math>; <math>8</math> are red and blue, meaning <math>r+b=8</math>; <math>4</math> are red and green, meaning <math>r+g=4</math>. Then we need to add these three equations: <math>b+g+r+b+r+g=2(r+g+b)=6+8+4=19</math>. It gives us all of the marbles are <math>r+g+b = 19/2 = 9</math>. So the answer is <math>\boxed{\textbf{(C)}\ 9}</math>. ---LarryFlora |
| − | + | ==Solution 3 Venn Diagrams== | |
| + | We may draw three Venn diagrams to represent these three cases, respectively. | ||
| − | + | [[File: Screen_Shot_2021-08-29_at_9.14.51_AM.png]] | |
| − | <math> | + | Let the amount of all the marbles is <math>x</math>, meaning <math>R+G+B = x</math>. |
| + | |||
| + | 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 | ||
| + | |||
| + | ==Solution 4 Venn Diagrams== | ||
| + | We may draw three Venn diagrams to represent these three cases, respectively. | ||
| + | |||
| + | [[File: Screen_Shot_2021-08-29_at_9.14.51_AM.png]] | ||
| + | |||
| + | Let the amount of all the marbles is <math>x</math>, meaning <math>R+G+B = x</math>. | ||
| + | |||
| + | Adding the three Venn diagrams, it gives us the equation: <math>x+18 = 3x</math>. | ||
| + | So <math>2x = 18</math>, <math>x = 18/2 =9</math>. | ||
| + | Thus, the answer is <math>\boxed{\textbf{(C)}\ 9}</math>. ---LarryFlora | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2012|num-b=18|num-a=20}} | {{AMC8 box|year=2012|num-b=18|num-a=20}} | ||
| + | {{MAA Notice}} | ||
Revision as of 13:55, 3 September 2021
Contents
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?
Solution 1
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 and it does work. Now count the number of marbles- 5+3+1=9. So 9 (C) is the answer.
Solution 2
We already knew the facts: 
are blue and green, meaning 
; 
are red and blue, meaning 
; 
are red and green, meaning 
. Then we need to add these three equations: 
. It gives us all of the marbles are 
. So the answer is 
. ---LarryFlora
Solution 3 Venn Diagrams
We may draw three Venn diagrams to represent these three cases, respectively.
Let the amount of all the marbles is 
, meaning 
.
The Venn diagrams give us the equation: 
.
So 
, 
.
Thus, the answer is . ---LarryFlora
Solution 4 Venn Diagrams
We may draw three Venn diagrams to represent these three cases, respectively.
Let the amount of all the marbles is , meaning .
Adding the three Venn diagrams, it gives us the equation: 
.
So 
, .
Thus, the answer is . ---LarryFlora
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
