AMC8 2025

AMC8 2025 · Q17

AMC8 2025 · Q17. It mainly tests Fractions, Probability (basic).

In the land of Markovia, there are three cities: $A$, $B$, and $C$. There are 100 people who live in $A$, 120 who live in $B$, and 160 who live in $C$. Everyone works in one of the three cities, and a person may work in the same city where they live. In the figure below, an arrow pointing from one city to another is labeled with the fraction of people living in the first city who work in the second city. (For example, $\frac{1}{4}$ of the people who live in $A$ work in $B$.) How many people work in $A$?

在马尔科维亚，有三个城市：$A$、$B$和$C$。$A$有100人居住，$B$有120人，$C$有160人。每人都在一个城市工作，可以在居住的城市工作。下图中，从一个城市指向另一个城市的箭头标有居住在第一个城市而在第二个城市工作的人的 fraction。（例如，居住在$A$的$\frac{1}{4}$的人在$B$工作。）有多少人在$A$工作？

(A) \ 55 \ 55

(B) \ 60 \ 60

(D) \ 115 \ 115

(E) \ 160 \ 160

Answer

Correct choice: (D)

正确答案：（D）

Solution

There are $100 \cdot (\frac{1}{4} + \frac{1}{5}) = 100 \cdot \frac{9}{20} = 45$ people who do not work in city $A$ that live in city $A$, meaning that $100 - 45 = 55$ people who live in city $A$ work in city $A$. There are $\frac{1}{3} \cdot 120 = 40$ people who live in city $B$ and work in $A$, as well as $\frac{1}{8} \cdot 160 = 20$ people who live in city $C$ that work in city $A$. Therefore, the answer is $55 + 40 + 20 = \boxed{\textbf{(D)}\ 115}$.

居住在$A$但不在$A$工作的人有$100 \cdot (\frac{1}{4} + \frac{1}{5}) = 100 \cdot \frac{9}{20} = 45$人，因此居住在$A$并在$A$工作的人是$100 - 45 = 55$人。居住在$B$并在$A$工作的人有$\frac{1}{3} \cdot 120 = 40$人，居住在$C$并在$A$工作的人有$\frac{1}{8} \cdot 160 = 20$人。因此，总人数是$55 + 40 + 20 = \boxed{\textbf{(D)}\ 115}$。

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