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 us start by supposing the greatest possible number speak one language.
That is, 75 speak english, 60 speak spanish and 45 speaks Swahili.
That would mean that there are a total of 45 + 60 + 75 = 180 people,
and we want to bring that number down to 100.
let's say that one guy speaks all three languages.
Then, in our previous sum that person is counted a total of three times,
that is, two extra times, which means that the new total is 180 - 2 = 178.
Now, if 40 people speak all three languages, then the total is 180 - 2*40 = 100.
So, the break down is 40 speak every language.
therefore,
45 - 40 = 5 speak only Swahili
60 - 40 = 20 speak only Spanish
75 - 40 = 35 speak only English
The number of people who speak only one language is 5 + 20 + 35 = 60.
This is the maximum.

ii)

x = the number of those who speak only English.
y = the number of those who speak only Spanish.
z = the number of those who speak only Swahili.
a = the # of those who speak only Spanish & Swahili
b = the # of those who speak only English & Swahili.
c = the # of those who speak only Spanish & English.
p = the # of those who speak all three.

Thus
100 = 75 + 60 + 45 - a - p - b - p - c - p + p
100 = 180 - p - (a + b + c + p)
(a + b + c + p) = 80 - p

Let’s also let I be the number of people who speak only one language, so
I = 100 - (a + b + c + p)
and (from above : (a + b + c + p) = 80 - p)
therefore I = 20 + p

We know that x + b + c + p = 75

Also p = 40, for b = c = 0 then x = 35

iii) From the part (ii) we have I = 20 + p

==> The greater the number of people who speak all three languages,
the greater the number of people who speak only one language

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