Question

At a school with 100​ students, 34 were taking​ Arabic, 32 ​Bulgarian, and 33 Chinese. 20 students take only​ Arabic, 17 take only​ Bulgarian, and 17 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?

Answer 1

Let A, B, and C be the number of students taking Arabic, Bulgarian, and Chinese, respectively. From the information given, we have:

A = 34, B = 32, C = 33
A-only = 20, B-only = 17, C-only = 17
A&B = 11
A&B&C = ?
None of these three languages = ?

To find the number of students taking all three languages, we can use the formula:

A + B + C - (A&B) - (A&C) - (B&C) + (A&B&C) = total

Substituting the values given, we get:

34 + 32 + 33 - 11 - (A&C) - (B&C) + (A&B&C) = 100
88 - (A&C) - (B&C) + (A&B&C) = 100
(A&C) + (B&C) - (A&B&C) = 12

We don't have enough information to solve for (A&C), (B&C), or (A&B&C) individually, but we do know that the sum of the three quantities is less than or equal to the smallest of the three values A, B, and C (which is 32). Therefore, we have:

(A&C) + (B&C) + (A&B&C) ≤ 32

To find the number of students taking none of these three languages, we can subtract the total number of students taking at least one language from the total number of students:

none = 100 - (A-only) - (B-only) - (C-only) - (A&B) - (A&C) - (B&C) + (A&B&C)
none = 100 - 20 - 17 - 17 - 11 - (A&C) - (B&C) + (A&B&C)

Substituting the previous equation, we have:

none = 100 - 20 - 17 - 17 - 11 - 12
none = 23

Therefore, there are 12 students taking all three languages, and 23 students taking none of these three languages