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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Question 22: a) Represent the information on the Venn diagram. The Venn diagram provided in the question already correctly represents the given information: • The rectangle represents the universal set of all guests, N() = 50. • The left circle represents guests who took soda, n(S) = 30. The portion taking only soda is 30 - Y. • The right circle represents guests who took beer, n(B) = 25. The portion taking only beer is 25 - Y. • The overlapping region (intersection) represents guests who took both soda and beer, denoted by Y. • The region outside both circles but inside the rectangle represents guests who took none of the drinks, which is 3. b) How many guests took both drinks? Step 1: Identify the given information from the problem and the Venn diagram. Total guests N() = 50. Guests who took soda n(S) = 30. Guests who took beer n(B) = 25. Guests who took both drinks = Y. Guests who took only soda = 30 - Y. Guests who took only beer = 25 - Y. Guests who took none of the drinks = 3. Step 2: Set up an equation using the total number of guests. The total number of guests is the sum of those who took only soda, only beer, both drinks, and none of the drinks. N() = (only soda) + (only beer) + (both) + (none) 50 = (30 - Y) + (25 - Y) + Y + 3 Step 3: Simplify the equation. 50 = 30 - Y + 25 - Y + Y + 3 50 = (30 + 25 + 3) + (-Y - Y + Y) 50 = 58 - Y Step 4: Solve for Y. Y = 58 - 50 Y = 8 The number of guests who took both drinks is 8. That's 2 down. 3 left today — send the next one.