There are 31 students in a class.The only languages available for the class to study are French and Spanish.17 students study French.15 students study Spanish.6 students study neither French nor Spanish. Using a Venn diagram, or otherwise, work out how many students study only one language.
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We know that 10 of the 50 students study neither French nor Spanish. Of the remaining 40 students we know that 31 take French and 17 take Spanish. Since 17 plus 31 is 48 and 48 is more than 40, there MUST be overlap, that is, it is not possible that the two groups are mutually exclusive.
Now, if all 17 who study Spanish also study French then 9 students are unaccounted for. And some number of students are unaccounted for for any number down to 9 studying both languages since if 9 study both languages that leaves 8 who study only Spanish and 31 plus 8 is only 39, 1 short of 40. That is, 1 student of the 40 is not accounted for. On the other hand we know that it cannot be 7 (or fewer) students who study both French and Spanish, since that would mean there were 31 minus 7 = 24 who study only French and 17 minus 7 = 10 who study only Spanish and 7 who study both languages, and 24 plus 10 plus 7 = 41, which is more students than we have in the group. The lower the number of overlaps the higher the number necessarily in the group. But we already know that the group (after the 10 who study neither Spanish nor French) who study Spanish or French or both is 40, no more and no less. Therefore the answer is 8, since 31 minus 8 = 23 and 17 minus 8 = 9 and 23 + 9 + 8 = 40.
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