Inclusion - exclusion principle

threeder

1. The problem statement, all variables and given/known data

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 the three languages).What is the maximum number of people who speak only English?

3. The attempt at a solution

The first thing to do is to draw Venn diagram and mark the sectors, say:

x = speak only English.
y = speak only Spanish.
z = speak only Swahili.
a = speak only Spanish & Swahili
b = speak only English & Swahili
c = speak only Spanish & English
p = speak all three

So we want to maximize x. x=75 - b - p -c = 75 - (b+p+c) so the task is to find an expresion which would allow us to minimize b+p+c but when I try to find it I just get into vicious circle... Any help?

Simon Bridge

You are over-thinking the problem...
Play with the numbers:

If 50 people only speak English - would that fit?

Ray Vickson

You CAN formulate and solve this question as a linear programming problem (but that is probably overkill). Just to show you how, I'll give the formulation. To make it easier to read, I will use the following variable names:
E = English alone number, S = Spanish alone number, W = sWahili alone number
ES, EW and SW mean English and Spanish only number, etc.
ESW = number speaking all three.

The problem is to maximize E, subject to the constraints:
E+ES+EW+ESW = 75, S+ES+SW+ESW = 60, W+EW+SW+ESW = 45, E+ES+EW+S+SW+W+ESW = 100, all variables >= 0. This can be solved using a linear programming package (such as the EXCEL Solver, etc.)

RGV

Villyer

I don't know if you want a mathematical method for doing a problem like this, but I would approach it by saying that if a person speaks only English, than they don't speak Spanish or Swahili.
So there are 100 people, and at least 60 of them don't fit in the set of people who only speak English because they speak Spanish.
45 people don't fit into the set of people who only speak English either, because they speak Swahili, but if you were to assume that all people who speak Swahili also speak Spanish you can ignore this number (as 45 < 60).

So using these assumptions you can find the maximum number of people who can speak just English without the complicated equations that were mentioned, but this may not work in all cases so I can understand wanting to know a more mathematical solution to the problem.

Simon Bridge

Hey threeder, how are you getting on?

Hi Villyer - that's what I was thinking and since the idea is to maximize only-English speakers it will work every time.

However, lets let OP play with the figures a bit and get used to how they combine and then get back to us before providing any more detailed hint: it's perilously close to doing the homework for OP as it is ;)

threeder

Well I wanted somewhat more formal approach but oh well. Since it is not homework, but rather individual study, I will stick to more efficient way then.

So taking this approach, the answer to me is obvious - 40 people can speak only english at most, because there has to be at least 60 people not speaking english. And this is a viable option, since 45 out of those 60 could speak Swahili as well, right?

Simon Bridge

And there you go.
Note: there can be no more than 25 people who don't speak English. You mean there have to be at least 60 people who speak a language other than English.

The ven diagrams were the right approach, you would have ended up with the Swahili one inside the Spanish one. Which is basically what you did.