2025 AMC 8 Problems/Problem 17

Problem

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 $?

$\textbf{(A)}\ 55\qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 115\qquad \textbf{(E)}\ 160$

Solution 1

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

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2025 AMC 8 Problems/Problem 17

Problem

Solution 1