AMC 8 2017 Problem 9
AMC8 2017 年第 9 题
比与比例★★☆☆☆
All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Marcy could have?
Marcy 的玻璃球只有蓝色，红色，绿色和黄色这 4 种颜色。她的玻璃缘的三分之一是蓝色，四分之一是红色，6 个玻璃球是绿色，那么黄色玻璃球的最小数目是多少个？
- A.1
- B.2
- C.3
- D.4
- E.5
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
The 6 green marbles and yellow marbles form of the total marbles. Now, suppose the total number of marbles is . We know the number of yellow marbles is and a positive integer. Therefore, 12 must divide . Trying the smallest multiples of 12 for , we see that when x = 12, we get there are -1 yellow marbles, which is impossible. However when x = 24, there are yellow marbles, which must be the smallest possible.
Since of the marbles are blue and are red, it is clear that the total number of marbles must be divisible by 12. If there are 12 marbles, then 4 are blue, 3 are red, and 6 are green, meaning that there are -1 yellow marbles. This is impossible. Trying the next multiple of 12, 24, we find that 8 are green, 6 are red, and 6 are green, meaning that the minimum number of yellow marbles is 4.
Assume the total number of marbles is . Since of her marbles are blue the total number of blue marbles she has is . Similarly she has red marbles. Saying that the number of yellow marbles is y we get this equation . Simplifying it we get which becomes . Multiplying on both sides we get . Because is a natural number 12(6+y) has to be divisible by 5. This means that 6+y has to be divisible by 5. The value that satisfies that and is in the answer choices in 4.
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