AMC 8 2024 Problem 9
AMC8 2024 年第 9 题
比与比例★★☆☆☆
All of the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?
Maria收藏有红色、绿色、蓝色三种颜色的大理石，其中红色大理石的数量是绿色大理石数量的一半，蓝色大理石数量是绿色大理石数量的两倍。请问Maria收藏的大理石总数可能是多少？
- A.24
- B.25
- C.26
- D.27
- E.28
正确答案 · Correct Answer
E
解析 · Solution
解法1
由于玛丽亚的红色弹珠数量是绿色的一半，我们可以设红色弹珠的数量为，绿色弹珠的数量为。由于她的绿色弹珠数量是蓝色的一半，我们可以设蓝色弹珠的数量为。加起来，我们得到：颗弹珠。因此弹珠总数必须是7的倍数，因为是整数，所以唯一可能的答案是 。 小修改
假设Maria有颗绿色弹珠和颗总弹珠。那么她有颗红色弹珠和颗蓝色弹珠。总体而言，Maria有
颗弹珠，所以因此必须是7的倍数。答案选项中唯一的7的倍数是 。 (countmath1) and anabel.disher
这个解法与……类似。解决这个问题最简单有效的方法是列方程。我们有以下方程：2r=g 和 2g=b。我们本质上想求出Maria收藏的弹珠总数，即用方程表示为 (r+g+b)。转换该方程并代入值，我们得到 r+2r+4r=7r。现在，我们知道总数是7的倍数，因此答案就是 。
Solution 1
Since Maria has half as many red marbles as green, we can call the number of red marbles , and the number of green marbles . Since she has half as many green marbles as blue, we can call the number of blue marbles . Adding them up, we have: marbles. The number of marbles therefore must be a multiple of 7, as represents an integer, so the only possible answer is. Minor Edit
Suppose Maria has green marbles and total marbles. She then has red marbles and blue marbles. Altogether, Maria has
marbles, so so must be a multiple of 7. The only multiple of 7 in the answer choices is. (countmath1) and anabel.disher
This solution is similar to. The most simplest and effective way to solve this problem is equations. We have the following equations: 2r=g and 2g=b We essentialy want to find out the total number of mables in Maria's collection or in equation terms (r+g+b). Converting that equation and substituting the values we get r+2r+4r=7r Now, we know the total is a multiple of 7 leaving our answer as.
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