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Population Percentages
By Anthony Curtis (P684) on Wednesday, March 3, 1999 - 10:35 am:

Here's a problem that might interest you.

If 70% of the population have poor eyesight, 75% have poor hearing, 85% suffer from allergies and 80% are unfit then what is the minimum percentage of the population that fit into all of these categories?

Thanks,
Anthony

By Alex Barnard (Agb21) on Thursday, March 11, 1999 - 04:29 pm:

Do you know how to do this problem if there are only 2 categories? If you can't do this then you will find the higher order cases too hard. The answer is that if a% of people have property A and b% of people have property B then at least (a+b-100)% of people must be in A and B and this is the best you can do. I'm not going to show you why this is true - it really isn't that hard to work out for yourself!

Now try and do 3 categories... so a,b,c% of people are in A,B,C what's the minimum in all of them. We can use what we did before to say that we must have (a+b-100)% of people in A and B. And using it again with C and (A and B) we see that at least:

(c+a+b-100-100)% of people are in all 3.

But now you have to check this is the best you can do - and I'm leaving that for you to do.

I'm sure there is enough information here for you to work out how to do 4,5,6,... categories.

Let me know how you get on.

AlexB.