Stefania’s family is looking for a new apartment and Stefania is hoping to get a dog after they move. Of the \(65\) apartments she has found online, \(40\) are close to a park, \(28\) allow dogs, and \(8\) are not close to a park and do not allow dogs. How many of the apartments are close to a park and allow dogs?
Solution 1
Let \(n\) be the number of apartments that are close to a park and allow dogs. Since \(40\) apartments are close to a park and \(n\) apartments are close to a park and allow dogs, then \(40-n\) apartments are close to a park but do not allow dogs. Similarly, since \(28\) apartments allow dogs and \(n\) apartments are close to a park and allow dogs, then \(28-n\) apartments allow dogs but are not close to a park. We also know that \(8\) apartments are not close to a park and do not allow dogs. This information is summarized in the following Venn diagram.
Since there are \(65\) apartments in total, then \[\begin{aligned} 65 &= (40-n)+n+(28-n)+8\\ 65 &= 76-n\\ n &= 76-65=11 \end{aligned}\] Therefore, \(11\) of the apartments are close to a park and allow dogs.
Solution 2
Since \(8\) of the \(65\) apartments are not close to a park and do not allow dogs, we know that \(65-8 = 57\) apartments must be close to a park or allow dogs, or both. We know that the \(40\) apartments close to a park will include the apartments that are close to a park and also allow dogs. Similarly, the \(28\) apartments that allow dogs will include the apartments that are close to a park and also allow dogs. Thus, if we consider \(40+28 = 68\), we will be double counting the apartments that are close to a park and also allow dogs. Since there are \(57\) apartments that are close to a park or allow dogs, or both, we have double-counted \(68-57 = 11\) apartments. Thus, there must be \(11\) apartments that are both close to a park and allow dogs.