MHB Maximizing Multilingualism: Solving Venn Diagram Problems in High School

AI Thread Summary
The discussion revolves around calculating the number of students taking all three languages—Spanish, French, and German—at a high school with 103 foreign language students. It is established that 29 students take at least two languages, leading to the equation a + b + c + d = 29, where 'd' represents students taking all three languages. The total counts for each language include overlaps, resulting in expressions for students taking only one language. However, when these expressions are summed and equated to the total of 103 students, the calculation yields d = -4, indicating a discrepancy in the provided numbers. The participants conclude that there may be an error in the initial data given for the language enrollments.
thaneshsan
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There are a total of 103 foreign language students in a high school where they offer Spanish,
French, and German. There are 29 students who take at least 2 languages at once. If there
are 40 Spanish students, 42 French students, and 46 German students, how many students
take all three languages at once?
 
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Draw three overlapping circles and label them "F" (for "French"), "G" (for "German"), and "S" (for "Spanish").

You are told "There are 29 students who take at least 2 languages at once." So the total number of students who would fit into the overlaps of those circles is 29. You are not told how many take, say, "French and German but not Spanish" or "all three languages" so enter "a" where "S" and only "F" overlap, "b" where only "G" and "F" overlap, "c" where only "S" and "G" overlap, and "d" where all three circles overlap. We must have a+ b+ c+ d= 29.

You are told that "there are 40 Spanish students" but that includes the "a" students who take Spanish and French, the "c" students who take Spanish and German, and the "d" students who take all three language. There are 40- a- c- d students who take Spanish only.

Similarly, you are told that there are "42 French students" so there are 42- a- b- d students who take French only.

And you are told that there are "46 German students" so there are 46- b- c- d students who take German only.

So in the 7 areas where those three circles overlap, we have "40- a- c- d", "42- a- b- d", "45- b- c- d", "a", "b", "c", and "d" where, now, each student is counted only once. Add those together and set it equal to 103 since we are told that is the number of foreign language students.

You are asked, "how many students take all three languages at once?". That is what we called "d". Are you able to find "d"?
 
I do understand the concept but it'll be easy for me to visualize it. Can you insert an image of the venn diagram? Thank you for the help ;)
 
All of Country Boys's expressions for the seven regions are correct. If you add them up and set them equal to 103, you get that d = -4. I think there is something wrong with the numbers you gave.
 
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