Problem 17: Venn Diagram

Here are the solutions to the problems: Problem 17: Venn Diagram a) 18 guests were served with only one meal, find the value of a. Step 1: Identify the expressions for guests served with only one meal. From the Venn diagram: Guests served only Matooke (M) = 7a - 4 Guests served only Rice (R) = 5a - 2 Step 2: Set up the equation based on the given information. The total number of guests served with only one meal is 18. (7a - 4) + (5a - 2) = 18 Step 3: Solve the equation for a. Combine like terms: 12a - 6 = 18 Add 6 to both sides: 12a = 18 + 6 12a = 24 Divide by 12: a = (24)/(12) a = 2 The value of a is 2. b) Find the total number of guests who attended the party. Step 1: Identify all regions in the Venn diagram. Guests served only Matooke (M) = 7a - 4 Guests served both Matooke and Rice (M and R) = 4a Guests served only Rice (R) = 5a - 2 Guests served neither meal = 3a Step 2: Substitute the value of a=2 into each expression. Guests served only M = 7(2) - 4 = 14 - 4 = 10 Guests served both M and R = 4(2) = 8 Guests served only R = 5(2) - 2 = 10 - 2 = 8 Guests served neither meal = 3(2) = 6 Step 3: Sum all the values to find the total number of guests. Total guests = (only M) + (both M and R) + (only R) + (neither) Total guests = 10 + 8 + 8 + 6 Total guests = 32 The total number of guests who attended the party is 32. --- Problem 18: Consecutive Even Numbers a) The sum of three consecutive even numbers is 12. Find the numbers. Step 1: Define the three consecutive even numbers. Let the first even number be x. The next consecutive even number is x+2. The third consecutive even number is x+4. Step 2: Set up the equation based on their sum. x + (x+2) + (x+4) = 12 Step 3: Solve the equation for x. Combine like terms: 3x + 6 = 12 Subtract 6 from both sides: 3x = 12 - 6 3x = 6 Divide by 3: x = (6)/(3) x = 2 Step 4: Find the three numbers. First number: x = 2 Second number: x+2 = 2+2 = 4 Third number: x+4 = 2+4 = 6 The three consecutive even numbers are 2, 4, 6. (The phrase "if largest numbers is (m-4)" appears to be an unrelated statement or a typo, as it does not fit the context of finding the numbers whose sum is 12.) b) Work out their range. Step 1: Identify the largest and smallest numbers. The numbers are 2, 4, and 6. Largest number = 6 Smallest number = 2 Step 2: Calculate the range. Range = Largest number - Smallest number Range = 6 - 2 Range = 4 The range of the numbers is 4.