AMC 8 2012 Problem 19
AMC8 2012 年第 19 题
方程与不等式★★★☆☆
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?
一个罐子里装有红绿蓝三种颜色的玻璃球，除了 6 个之外的其他玻璃球都是红色的，除了 8 个之外的其他玻璃球都是绿色的，除了 4 个之外的其他玻璃球都是蓝色的。那么罐子里有多少个玻璃球？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
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 . So the answer is 9. We may draw three Venn diagrams to represent these three cases, respectively. Let the amount of all the marbles be , meaning R+G+B = x. The Venn diagrams give us the equation: x = (x-6)+(x-8)+(x-4). So x = 3x-18, . Thus, the answer is 9.
-similar to solution 3 but you can skip Venn Diagrams Let x equal total marbles Then: red = x-6 Green = x-8 Blue = x-4 Then x= r + g+ b So (x-4)+ (x-8) + (x-4) = x Simplifying, you get x=9 so 9 marbles, answer choice C 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 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 9. Let 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 9. -J.L.L (Feel free to edit)
Let be the number of total marbles, be the number of red marbles, be the number of green marbles, and 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 .
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