# Percentages word problem

There has been an outbreak of bedbugs at a college. 30% of students have the pests in their beds. 40% of the infested beds are in the dorms and 60% are in apartments. Of the uninfested beds, 20% are in the dorms and 80% are in apartments. What percentage of the students live in the dorms?

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A tabular approach is convenient. To start with you’re given one percentage involving the entire student body:

$$\begin{array}{r|cc|c} &\text{Pests}&\text{No Pests}&\text{Total}\\ \hline \text{Dorm}&\\ \text{Apt.}&\\ \hline \text{Total}&30\% \end{array}$$

Clearly the other $100-30=70$% of the students have uninfested beds, you can immediately expand the table:

$$\begin{array}{r|cc|c} &\text{Pests}&\text{No Pests}&\text{Total}\\ \hline \text{Dorm}&\\ \text{Apt.}&\\ \hline \text{Total}&30\%&70\%&100\% \end{array}$$

Now you’re told that $40$% of the infested beds are in the dorms; that’s $30$% or $40$%, or $12$%.

$$\begin{array}{r|cc|c} &\text{Pests}&\text{No Pests}&\text{Total}\\ \hline \text{Dorm}&12\%\\ \text{Apt.}&\\ \hline \text{Total}&30\%&70\%&100\% \end{array}$$

By similar reasoning you can fill in the rest of the first two columns of the table, and once you have those, you can easily get the last column. Which entry in the table contains the answer?

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