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 $A$ ?

$[asy] /* AMC8 P17 2025, by NUMANA: BUI VAN HIEU, buivanhieu@husc.edu.vn, https://husc.edu.vn */ import graph; unitsize(2cm); real r=0.15; pair A, B, C;B = (0,0);C = (2,0);A = (1,sqrt(3)); // Drawing the nodes draw(circle(A,r)); label("$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

$100 (1/4 + 1/5) = 100 \cdot \frac{9}{20} = 45$ 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 $55 + 40 + 20 = \boxed{\textbf{(D)}115}$ .

~ alwaysgonnagiveyouup

Remark

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).

You're welcome for uploading the image. It might be a little big or blurry. If anyone could help fix it, thanks. -- leafy

Video Solution by Pi Academy

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

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

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