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In a hotel, 1/5 of the guests do not like tea, ¼ do not like coffee and 3 people do not like tea or coffee. How many people are there in the hotel?

This is from an old school text book. I wonder if there is enough information for a unique solution because how many like both tea and coffee?

I agree you need the number who like both. If G is the number of guests, then tea lovers are .8G, coffee lovers are .75G, so

G = .8G + .75G +3 - (number who like both (N))

So you need the last term for a unique solution (G is an integer if N=14, 25, 36....)

If you can assume the same proportion who dislike tea also dislike coffee - 0.15, 0.2 and 0.05 - then there are 60 guests.

Thank you, klingore trout and drmorgans. That confirms my suspicions. The book is from the 1960s and there are no answers in the back. 60 does work with that assumption but other answers work too, I find.

Sorry, kilgore trout, not klingore!

I don't follow much of this.

It looks to me that it is a straightforward G/4-G/5=3 equation and the number who like both is irrelevant.[Happy to be proved wrong though]

Here’s a long explanation.

This is an example of a 2x2 contingency table, but I can’t show this exactly because there are formatting limitations.

As 1/5 of the guests don’t like coffee, the total number of guests must be a multiple of 5, as we can’t have fractions of guests. Similarly, as 1/4 of the guests don’t like tea, the number of guests must be a multiple of 4. As 4 and 5 don’t have any common factors, this means that the number of guests must be a multiple of 4x5 = 20. Let the number of guests be 20k, where k is an unknown integer.

Then 16k guests like coffee and 4K guests don’t like coffee. 15k guests like tea and 5k guests don’t like tea.

As 3 guests don’t like both tea and coffee, this means that 5k-3 guests don’t like coffee but like tea, and 4K-3 guests like tea but don’t like coffee.

Therefore, (5k-3) + (4k-3) + 3 guests don’t like either tea or coffee or both. Subtract this from the total number of guests 20k and you get 11k + 3, the number of guests who like both tea and coffee.

As this includes k, there is an infinite number of possible solutions. For example, if k=1, there are 20 guests, of whom 14 like both coffee and tea, 2 like coffee but not tea, 1 likes tea but not coffee, and 3 don’t like either.

That's a very comprehensive analysis, wintonian, and I think we now all agree that there is more than one answer that fits. Thanks also to tatters. Your method gives an answer of 60, which I suspect is the one the book was looking for.