At a school with 100 students, 35 take French, 32 take German, and 30 take Spanish. Twenty students take only French, 20 take only German, and 14 take only Spanish. In addition, 7 students are taking both French and German, some of whom also take Spanish. How many students are taking all 3 languages? How many are taking none of these 3 languages?

Question

I have drawn a Venn Diagram with 3 circles and put in the 20, 20, and 14, but I don't know what to do with the 7 since some of the 7 is also in the intersection of all 3. When I know that, I think I can solve for the missing regions in the Venn diagram.

Answer 1

We don't know how many take all three, which is our first main question.

So in your Venn diagram, place x in the intersection of all three circles.
Now look at the intersection of the F and G circles
We are told that this is 7 but x are already counted, ("some of whom also take Spanish")
So place 7-x in the region of ONLY F and G

Now look at the F circle , so far we have
20 + x + 7-x = 27
We are told that 35 take F
So place 8 in the region ONLY F and S
Do the same for circle G, placing 5 in the G and S only circle.
We can now find x from the S circle
14+8+5+x = 30
x = 3

So 3 students study all three languages

remember that some students don't take any of the three languages.
Check:
Sum of all entries in our circles = 74
So 100-74 or 26 don't take any language course

Answer 2

IN A CLASS OF 55 STUDENTS, 35 TAKE ENGLISH,40 TAKE FRENCH AND 5 TAKE OTHER LANGUAGES. DETERMINE HOW MANY STUDENTS TAKE BOTH LANGUAGES

Answer 3

25