A company that specializes in language lists the following information concernin

Question

A company that specializes in language lists the following information concerning its English-speaking employees: 23 employees speak German; 25 speak French; 33 speak Spanish; 45 speak Spanish or French; 38 speak French or German; 48 speak German or Spanish; 8 speak Spanish, French, and German; and 7 speak English only. (Round your answer to one decimal place.)

(a) What percent of the employees speak at least one language other than English?
  %

(b) What percent of the employees speak at least two languages other than English?
  %

Explanation / Answer

Let A denote speaking German, B denote speaking French, C denote speaking Spanish.

n(A) = 23 n(B) = 25 n(C) = 33

n(B U C) = 45 n(A U B) = 38 n(A U C) = 48

n(A B C) = 8

n(A U B U C)' = 7

n(B C) = n(B) + n(C) - N(B U C)

= 25 + 33 - 45

= 13

n(B A) = n(B) + n(A) - N(B U A)

= 25 + 23 - 38

= 10

n(A C) = n(A) + n(C) - N(A U C)

= 23 + 33 - 48

= 8

(a) n(A U B U C) = n(A) + n(B) + n(C) - n(A B) - n(B C) - n(A C) - n(A B C)

= 23 + 25 + 33 - 10 - 13 - 8 - 8

= 42.

Total number of employees = n(A U B U C) + n(A U B U C)' = 42 + 7 = 49.

=> Percentage of employees speaking atleast one language other than English = 42/49 = 0.8571.

(b) n(A B) + n(B C) + n(C A) - 2 n(A B C)

= 10 + 13 + 8 - 2 * 8

= 15.

=> Percentage of employees who speak atleast two languages other than English

= 15 / 49

= 0.3061.

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