Maximizing Multilingualism: Solving Venn Diagram Problems in High School

[thaneshsan]

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?

 

[HOI]

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"?

 

[thaneshsan]

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 ;)

 

[mrtwhs]

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.