1. The problem statement, all variables and given/known data At an international conference of 100 people, 75 speak English, 60 speak Spanish, and 45 speak Swahili (and everyone present speaks at least one of the three languages).What is the maximum number of people who speak only English? 3. The attempt at a solution The first thing to do is to draw Venn diagram and mark the sectors, say: x = speak only English. y = speak only Spanish. z = speak only Swahili. a = speak only Spanish & Swahili b = speak only English & Swahili c = speak only Spanish & English p = speak all three So we want to maximize x. x=75 - b - p -c = 75 - (b+p+c) so the task is to find an expresion which would allow us to minimize b+p+c but when I try to find it I just get into vicious circle... Any help?