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 $A?$

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The number of people who live in $A$ and work in $A$ is $$100 - 100 \times \frac{1}{4} - 100 \times \frac{1}{5}$$ which is $$100 - 25 - 20 = 55.$$ The number of people who live in $B$ and work in $A$ is $$120 \times \frac{1}{3} = 40$$ The number of people who live in $C$ and work in $A$ is $$160 \times \frac{1}{8} = 20$$ So, the answer is $$55 + 40 + 20 = 115,$$ which is choice D.