If 100 people were at a party 90 spoke Spanish 80 spoke Italian and 75 spoke Mandarin At least how many spoke all three languages?

45

Check my logic. Draw the Venn diagram. Mark the seven areas S, I, M for those languages only (monoglots); a,b,c for single overlaps (bilingual); T in the middle for trilingual.

Tminimum (Words "at least") happens at abcmaximum and that happens at SIMminimum , the limit of which is 0 (zero monoglots, all partgoers bilingual).

From the diagram,

abcT = 100, that is

(1) 2abc + 2T = 200

Also, from the given

(2) abT=80

(3) bcT=75

(4) acT=90

Adding (2...4) we get

(5) 2abc + 3T = 245

Subtracting (1) we get Tmin = 45