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