Help regarding amounts!

Quote:

Originally Posted by gossen

Total amount of students: 102.
47 study french, where 23 study only french.
46 study german, where 20 study only german.
43 study spanish, where 16 study only spanish.
15 study both french and german.

a) How many study all 3 languages?
b) How many study neither of the languages?
c) How many study both french and spanish?

I have tried really hard using a venn-diagram to get the answers but failed, if someone could explain how you reach the answer and not just the answers it would be very very appreciated since I need to comprehend the procedure for future calculation. Once again, thanks in advance.

I haven't actually solved the system, but this is the setup. Using the notation:
(F) = studies French only
(FG) = studies French and German
(FGS) = studies French, German, and Spanish
and their obvious extensions to the other sets in the Venn diagram.

The conditions immediately give us:
(F) = 23
(G) = 20
(S) = 16
(FG) + (FGS) = 15 (from the last condition)

The other conditions give the equations:
(F) + (FG) + (FS) + (FGS) = 47
(G) + (FG) + (GS) + (FGS) = 46
(S) + (FS) + (GS) + (FGS) = 43
(F) + (G) + (S) + (FG) + (FS) + (GS) + (FGS) = 102 (from the initial statement)

-Dan