# Need Help With Simple Mathematical Sets Difference (Intersection) Problem

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?

• Have you tried drawing a Venn diagram? How many had nausea but not headache? – saulspatz Mar 26 '18 at 19:36
• 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. – Pedro _PR Mar 26 '18 at 19:40
• Can you find how many distinct volunteers had either headaches or nausea or both from what you've already done? – K B Dave Mar 26 '18 at 19:41
• @Pedro_PR If you fill out the numbers in the Venn diagram carefully, you'll see what's missing. – saulspatz Mar 26 '18 at 19:46

headache and no nausea = 12 - 4

nausea and no heaqdache = 8 - 4

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

HINT There are four mutually exclusive categories of volunteers:

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.)