# Question related to a Venn diagram

The problem description is this:

In a survey of $190$ manufacturing companies, $103$ hired operators, $67$ hired technicians, and $49$ hired both operators and technicians, as illustrated in the Venn diagram below:

How many companies surveyed have hired at least one category of employees: operators and technicians?

First of all what I have tried is to understand how much operators and technicians were without having both professions, so I have added $103 + 67 = 170$; then $170 + 49 = 219$, so finally $219 - 190 = 29$. So now what should I do? If we note that $29$ are both, then we can see that only operators are $103 - 29 = 74$ and only technicans are $67 - 29 = 38$, is it right? But what about question itself? If I add both results, I get $74 + 38 = 112$, but in answer there is $121$ as correct answer, what is worng?

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The $190$ companies are simply all companies; even those that didn't hire any employees. So for your question, this number is irrelevant. – TMM Oct 22 '12 at 8:46

The companies that hired both operators and technicians are counted twice when you add those that hired operators and those that hired technicians. So the correct answer is $103+67-49=121$.