Three Languages, Four Clues

Difficulty: Hard · Topics: Discrete Mathematics, Sets, Word Problems

In a group of 60 people — some of whom know English, French or German, while others know none of them — everyone who knows German knows English or French, and everyone who knows English and French knows German.

The number of people who know all three languages is equal to each of the following.

I. The number of people who know none of them.

II. Half of the number of people who know German and French.

III. One third of the number of people who know German and English.

IV. One fifth of the number of people who know exactly one language.

How many people know 3 languages?