Please help. How do I solve the problem?

Question

At a school with 100 students, 32 were taking Arabic, 36 Bulgarian, and 30 Chinese. 19 students take only Arabic, 21 take only Bulgarian, and 14 take only Chinese. In addition, 10 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?

Answer 1

Let:

A = students taking Arabic

B = students taking Bulgarian

C = students taking Chinese

We are given:

A = 32

B = 36

C = 30

Arabic only = 19

Bulgarian only = 21

Chinese only = 14

10 students take both Arabic and Bulgarian (this includes those who also take Chinese).

Let x be the number of students taking all three languages.

Then:

Arabic and Bulgarian only = 10 − x

Using the Arabic total:

19 + (10−x) + (Arabic and Chinese only) + x = 32

So, Arabic and Chinese only = 3

Using the Bulgarian total:

21 + (10−x) + (Bulgarian and Chinese only) + x = 36

So, Bulgarian and Chinese only = 5.

Now use the Chinese total:

14 + 3 + 5 + x = 30

22 + x = 30

x = 8

​So, 8 students take all three languages.

Now find the total taking at least one language:

Arabic only: 19

Bulgarian only: 21

Chinese only: 14

Arabic & Bulgarian only: 10 − 8 = 2

Arabic & Chinese only: 3

Bulgarian & Chinese only: 5

All three: 8

Total: 19 + 21 + 14 + 2 + 3 + 5 + 8 = 72

Therefore, the number taking none of the three languages is

100 − 72 = 28

Final Answers:

Students taking all three languages: 8

Students taking none of the three languages: 28