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 $A$ ?
A. 55
B. 60
C. 85
D. 115
E. 160

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Answer (D): Of the 100 people who live in $A$ , $\frac{1}{4} \cdot 100 = 25$ work in $B$ , and $\frac{1}{5} \cdot 100 = 20$ work in $C$ . This means $100 - 25 - 20 = 55$ people who live in $A$ also work there.

In addition, $\frac{1}{3} \cdot 120 = 40$ people who live in $B$ work in $A$ and $\frac{1}{8} \cdot 160 = 20$ people who live in $C$ work in $A$ . Therefore the total number of people who work in $A$ is $55 + 40 + 20 = 115$ .

Answer

D. 115

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