In overlapping sets, people or items will be categorized by their “membership” or “non-membership” in either of two groups. For example, workers in a factory could be salaried or non-salaried. They could also work in an Operations role, or not work in an Operations role. These problems can be represented by a simple Venn diagram, as shown:
The two key points to note are the following:
1. The workers will always fall into one of four groups:
Therefore, there are four unknowns in this type of problem, generally (although the question itself may only require that you work with two or three of them).
2. The problem will often give you total amounts for the groups (salaried, and in Operations), and you will have to use logic to figure out whichever unknown the question is asking about.
The various sections can be labeled as follows:
As you can see, c = a − e; d = b − e, and Total = a + b − e + f.
Alternatively, Total = c + d + e + f.
Here’s an example:
At Factory X, there are 400 total workers. Of these workers, 240 are salaried, and 220 work in Operations. If at least 100 of the workers are non-salaried and do not work in Operations, what’s the minimum number of workers who both are salaried and work in Operations?
Graphically, this looks like:
Mathematically, you can use Total = a + b − e + f.
Thus, at least 160 workers are salaried and work in operations.
Of 320 consumers, 200 eat strawberries and 300 eat oranges. If all 320 eat at least one of the fruits, how many eat both?