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I'm having trouble with a seemingly simple problem in Math and I need some help with it. The problem states:

A hospital is doing a treatment research on 50 volunteers. Of those, some have had a few reactions: 12 have had headaches, 8 felt nausea and 4 had headaches and nausea. How many volunteers had headaches and not nausea? How many volunteers didn't feel neither (headache and nausea simultaneously)?

It's not a homework as I'm not in school anymore but studying by myself to try my country's equivalent of the SAT to get into college.
So, I know that my Universe (U) in this case is 50. I know the sets (let's say A and B) are A={12}, B={8} and A∩B={4}, so the number of volunteers that had only headaches exclusively is 8 (12 - 4 people that also had nausea) but i can't figure out the number of volunteers that haven't had neither.
I've tried putting together a simple equation to find the number: 12 volunteers that felt headache + 8 that felt nausea + x people that felt neither = 50 that will amount to x=30 volunteers but the book (without explaining why) says the correct answer is 34.
I've tried breaking the problem down by its inquiries, writing them one by one (which I won't do here to not extend the question) and it seems to be a simple thing, really, and that what I've done is correct.
Is the book wrong (unlikely)? Is my solution correct? How should I go about when solving this problem?

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    $\begingroup$ Have you tried drawing a Venn diagram? How many had nausea but not headache? $\endgroup$ – saulspatz Mar 26 '18 at 19:36
  • $\begingroup$ Yes but it's just like the numbers I've stated on the question, the diagram represents the same thing. It doesn't give me any information that I don't know. $\endgroup$ – Pedro _PR Mar 26 '18 at 19:40
  • $\begingroup$ Can you find how many distinct volunteers had either headaches or nausea or both from what you've already done? $\endgroup$ – K B Dave Mar 26 '18 at 19:41
  • $\begingroup$ @Pedro_PR If you fill out the numbers in the Venn diagram carefully, you'll see what's missing. $\endgroup$ – saulspatz Mar 26 '18 at 19:46
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enter image description here

headache and no nausea = 12 - 4

nausea and no heaqdache = 8 - 4

neither = 50 - (12 + 8 - 4) = 34

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HINT There are four mutually exclusive categories of volunteers:

Those who had no reaction.

Those who had headache, but not nausea.

Those who had nausea, but not headache.

Those who had both reactions.

The sum of the numbers in these groups must add up to $50.$ Are you certain you've counted each group correctly? (No, you haven't, actually, although your calculation of those with headache but not nausea is correct.)

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  • $\begingroup$ Thank you. From your answer and the other one I see where I was wrong and how. $\endgroup$ – Pedro _PR Mar 26 '18 at 20:10
  • $\begingroup$ Glad to help you. Good luck on the exam. $\endgroup$ – saulspatz Mar 26 '18 at 20:20

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