I'm not sure if I'm correct? Can someone check?

Question

At a school with 100 students, 32 were taking Arabic, 40 Bulgarian, and 36 Chinese. 20 students take only Arabic, 27 take only Bulgarian, and 26 take only Chinese. In addition, 11 are taking both Arabic and Bulgarian, some of whom also take Chinese. How many students are taking all three languages? None of these three languages?

 

*students are taking all three languages.....9

 

*students are taking none of the three languages......29

Answer 1

Create a three-way Venn Diagram using A, B, C for the first letter of each language. Add the values that we know, which is only A (A=20), only B (B=27), and only C (C=26). There are four areas here where they cross over, so there are four unkowns:

• People taking exactly Arabic & Bulgarian (AB)
• People taking exactly Bulgarian & Chinese (BC)
• People taking exactly Arabic & Chinese (AC)
• People taking all three (ABC)

Now, see how anyone who is taking Arabic can also be taking another language, or even all three. We can create three equations, one for each language:

1. 20+AB+AC+ABC=32
2. 27+AB+BC+ABC=40
3. 26+BC+AC+ABC=36

Since we have four unkowns, we will need four equations to solve for any of the values. Luckily, they also tell us that there are 11 students taking Arabic & Bulgarian, but some of them also take Chinese. Thus we have:

4. AB+ABC=11

 

Start with that last equation since there are less variables. Pick one variable to solve for:

AB = 11-ABC

Plugging into equation 1:

20+11-ABC+AC+ABC=32

31+AC=32

AC=1

Plugging into equation 3:

26+BC+1+ABC=36

27+BC+ABC=36

BC=9-ABC

Plugging all we know into equation 2:

27+ 11-ABC+9-ABC+ABC=40

47-ABC=40

ABC=7

Use this to solve for AB & BC:

AB=4 & BC=2

 

So there are 7 students taking all languages. To find out how many are taking no languages, take 100 and subract from it your A, B, C, AB, BC, AC & ABC because none of these values overlap students. You should get 13.