At an international conference of 100 people, 75 speak English, 60 speak Spanish

Question

At an international conference of 100 people, 75 speak English, 60 speak Spanish and 45 speak Swahili (and everyone present speaks at least one of these languages). What is the maximum possible number of these people who can speak only one language? In this case how many people speak only English, how many speak only Spanish, how many speak only Swahili, and how many speak all three? What is the maximum number of people who speak only English? In this case what can be said about the number who speak only Spanish and the number who speak only Swahili? Prove that the greater the number of people who speak all three languages, the greater the number of people who speak only one language.

Explanation / Answer

Solution:  i) Let x = number of people who speak only English.
y = number of people who speak only Spanish.
z = number of people who speak only Swahili.
p = number of people who speak only Spanish & Swahili.
q = number of people who speak only English & Swahili..
r = number of people who speak only Spanish & English.
s = number of people who speak all three.

So,
100 = 75 + 60 + 45 - p - s - q - s - r - s + s
100 = 180 - s - (p + q + r + s)
(p + q + r + s) = 80 - s

Now,let I be the number of people who speak only one language, so
I = 100 - (p + q + r + s) and (from above : (p + q + r + s) = 80 - s)
therefore I = 20 + p

If we assume n = a + b + c (the number of people who speak exactly two languages), we
also have
(p + q + r + s) = 80 - s
(p + q + r) = 80 - 2s
2s = 80 - (p + q + r)
and n = (p + q + r)
2s = 80 - n
s = 40 - n/2

So, p and I are maximized when n = 0. In this case, s = 40 and I = 60 i.e. 40 people speak all three languages, and 60 speak exactly one language. When it happens, there are 35 people who speak English only, 20 who speak Spanish only, and 5 who speak Swahili only.

(ii) You need to maximize x. We know that x + q + r + s = 75

(iii) We already did this in the rst part, since we saw that I = 20 + p.

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