In a certain school there are \(127\) boys in one particular year. They may study one or more (or none) of the languages Latin, French, and German. No boys study both Latin and German; \(18\) study both French and German, half as many French and Latin; \(21\) study Latin only; \(70\) study a language but not German; \(82\) study one or more of the two modern languages. By using Venn diagrams, or otherwise, find how many boys study each language and how many study none.

Which of the areas in the Venn diagram can you fill in straight away from the information above?

How can we manage the remaining areas?