Class of 100 students, 10 who speak German, 20 who speak Italian, 30 who speak Spanish, 8 who speak both Italian and Spanish, 3 speak all 3 languages **How many people in the class speak none of the 3 languages?** 10 speak German, minus the 3 who speak all languages is **7**. 20 speak Italian, minus the 3 who speak all languages and the 8 who speak both Italian and Spanish so **9**. 20 speak Spanish, minus the 3 who speak all languages and the 8 who speak both Italian and Spanish so **19**. Add back in the 3 and 8 people who spoke multiple languages = **11** So the amount of people who don't speak any of the 3 languages should be: **100 - 7 - 9 - 19 + 11 = 76** Have I gotten the correct answer or have I miscounted somewhere?

Look at this self explanatory diagram:

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So the answer is $100-7-22-12=59$

If you start drawing this diagramm, it is clear which number comes in which area. You could start writing all numbers you know first.

But then you count to many. For example if you write a 10 into (ger) and also a 3 into the intersection of (ger), (esp) and (it) then you count 3 to many. The 3 in the intersection are also in (ger), so the 10 becomes a 7. And so on.

Also every student who speaks three languages also speaks two, and so on.

> 20 speak Italian, minus the 3 who speak all languages and the 8 who speak both Italian and Spanish so **9**.

Here you miscounted.

If you subtract 8, you subtract 3 to many, because you already subtracted three students who speak every language, but these students are also among the 8 who speak two languages.

> 20 speak Spanish, minus the 3 who speak all languages and the 8 who speak both Italian and Spanish so 19.

Same mistake.