in a group of 100 people, 72 people can speak english and 43 can speak french. how many can speak english only?
57 people can speak English only.
How to see this
Let:
- EEE: people who speak English, ∣E∣=72|E|=72∣E∣=72.
- FFF: people who speak French, ∣F∣=43|F|=43∣F∣=43.
- Total people ∣E∪F∣=100|E\cup F|=100∣E∪F∣=100 (everyone speaks at least one language).
Use the inclusion–exclusion principle for two sets:
∣E∩F∣=∣E∣+∣F∣−∣E∪F∣|E\cap F|=|E|+|F|-|E\cup F|∣E∩F∣=∣E∣+∣F∣−∣E∪F∣
∣E∩F∣=72+43−100=15|E\cap F|=72+43-100=15∣E∩F∣=72+43−100=15
So 15 people speak both English and French.
Those who speak English only are:
∣E∣−∣E∩F∣=72−15=57|E|-|E\cap F|=72-15=57∣E∣−∣E∩F∣=72−15=57
Therefore, the number of people who can speak English only is 57.