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 $A$ ?

$[asy] import graph; unitsize(2cm); real r=0.15; pair A, B, C;B = (0,0);C = (2,0);A = (1,sqrt(3)); // Drawing the nodes draw(circle(A,r)); label("$A$", A); draw(circle(B,r)); label("$B$", B); draw(circle(C,r)); label("$C$", C); guide AB=A+r*dir(-135)..{down}B+r*dir(90), BA=B+r*dir(60)..{up}A+r*dir(-105), BC=B+r*dir(0)..(1,-0.2)..C+r*dir(180), CB=C+r*dir(150)..(1,0.3)..B+r*dir(30), CA=C+r*dir(90){up}..A+r*dir(-45), AC=A+r*dir(-75){down}..C+r*dir(120); draw(AB,L=Label("$1/4$", MidPoint, W),Arrow(HookHead)); draw(BA,L=Label("$1/3$", MidPoint, W),Arrow(HookHead)); draw(BC,L=Label("$1/6$", MidPoint, S),Arrow(HookHead)); draw(CB,L=Label("$1/10$", MidPoint, S),Arrow(HookHead)); draw(CA,L=Label("$1/8$", MidPoint, E),Arrow(HookHead)); draw(AC,L=Label("$1/5$", MidPoint, E),Arrow(HookHead)); [/asy]$

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

Solution 1+2

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 $55 + 40 + 20 = \boxed{\textbf{(D)}\ 115}$ .

~ alwaysgonnagiveyouup

We could also make an equation. Let's denote the number of people that live in city $A$ as $a$ , $B$ as $b$ , and $C$ as $c$ . If $x$ denotes the number of people working in city $A$ , then $x = \frac{1}{8}c + \frac{1}{3}b + a - \frac{1}{4}a - \frac{1}{5}a$ . This is because $\frac{1}{8}$ of the people from City $C$ and $\frac{1}{3}$ of the people from City $B$ work in city $A$ as shown in the image. We also know that $\frac{9}{20}$ of the people living in $A$ work in other cities. We are already given the values of variables $a$ , $b$ , and $c$ as $100$ , $120$ , and $160$ respectively. Plug the values it into the main equation like this: $x = 160 \cdot \frac{1}{8} + 120 \cdot \frac{1}{3} + 100 - 100 \cdot \frac{1}{4} - 100 \cdot \frac{1}{5}$ . We solve for it and get our answer $\boxed{\textbf{(D)}\ 115}$ .

~sushipie2025

Remark

Main article: Markov Chains

This model is known as the Markov Chain, a type of stochastic process that models systems where the next state depends only on the current state, not on the sequence of events that preceded it. This is known as the Markov property (memoryless property).

Video Solution by Pi Academy

https://youtu.be/Iv_a3Rz725w?si=E0SI_h1XT8msWgkK

Video Solution(Quick, fast, easy!)

https://youtu.be/fdG7EDW_7xk

~MC

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/VP7g-s8akMY?si=fV-dPbMPVzWTkSV3&t=2020 ~hsnacademy

Video Solution by Thinking Feet

https://youtu.be/PKMpTS6b988

2025 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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