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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:

enter image description here

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
up vote 4 down vote accepted

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 $103+67-49=121$.

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