This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — Problem 1: Language percentages Let E be the set of people who speak English and F be the set of people who speak French. Let P(X) denote the percentage of people in set X. Given information: 1. P(E F) = 12\% (speak both English and French) 2. P(neither E nor F) = 9\% (cannot speak any of the two) 3. P(only E) = P(only F) + 15\% (percentage speaking only English is 15% more than those speaking only French) From (2), the percentage of people who speak at least one language is: P(E F) = 100\% - P(neither E nor F) = 100\% - 9\% = 91\% We know that P(E F) = P(only E) + P(only F) + P(E F). Substituting the known values: 91\% = P(only E) + P(only F) + 12\% P(only E) + P(only F) = 91\% - 12\% = 79\% Now we have a system of two equations: I. P(only E) = P(only F) + 15\% II. P(only E) + P(only F) = 79\% Substitute equation I into equation II: (P(only F) + 15\%) + P(only F) = 79\% 2 · P(only F) + 15\% = 79\% 2 · P(only F) = 79\% - 15\% 2 · P(only F) = 64\% P(only F) = (64\%)/(2) = 32\% Now find P(only E) using equation I: P(only E) = 32\% + 15\% = 47\% a) Represent the information on a Venn diagram. A Venn diagram would show two overlapping circles, one for English (E) and one for French (F), within a rectangle representing the total population (100%). • The intersection of E and F would contain 12\%. • The part of circle E outside the intersection (only E) would contain 47\%. • The part of circle F outside the intersection (only F) would contain 32\%. • The area outside both circles (neither E nor F) would contain 9\%. b) Find the percentage of people who can speak: i. English P(E) = P(only E) + P(E F) = 47\% + 12\% = 59\% ii. French P(F) = P(only F) + P(E F) = 32\% + 12\% = 44\% iii. Only French P(only F) = 32\% c) Find the number of people who can speak only French if the population of the village is 4000. Number of people who speak only French = P(only F) × Total Population = 32\% × 4000 = (32)/(100) × 4000 = 32 × 40 = 1280 people Got more? Send 'em 📸