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.
ScanSolve AI Answer
Step 1: Define the sets and given information. Let A be the set of people who liked Ampesi. Let B be the set of people who liked Banku. Let F be the set of people who liked Fufu. The total number of people in the house is |U| = 20. Given: |A| = 8 |B| = 12 |F| = 12 |A F| = 5 |B F| = 6 |(A B) F| = 2 (people who liked only Ampesi and Banku) |A B F| = 3 (people who liked all three types of food) Step 2: Calculate the number of people in each distinct region of the Venn diagram. Start with the intersection of all three sets: • |A B F| = 3 (liked all three) Next, calculate the regions for exactly two types of food: • Liked only Ampesi and Banku (not Fufu): |(A B) F| = 2 (given) • Liked only Ampesi and Fufu (not Banku): |(A F) B| = |A F| - |A B F| = 5 - 3 = 2 • Liked only Banku and Fufu (not Ampesi): |(B F) A| = |B F| - |A B F| = 6 - 3 = 3 Now, calculate the regions for only one type of food: • Liked only Ampesi: |A only| = |A| - (|(A B) F| + |(A F) B| + |A B F|) |A only| = 8 - (2 + 2 + 3) = 8 - 7 = 1 • Liked only Banku: |B only| = |B| - (|(A B) F| + |(B F) A| + |A B F|) |B only| = 12 - (2 + 3 + 3) = 12 - 8 = 4 • Liked only Fufu: |F only| = |F| - (|(A F) B| + |(B F) A| + |A B F|) |F only| = 12 - (2 + 3 + 3) = 12 - 8 = 4 Finally, calculate the number of people who liked none of the foods: Sum of all calculated regions: 1 + 4 + 4 + 2 + 2 + 3 + 3 = 19 Number of people who liked none of the foods: |U| - 19 = 20 - 19 = 1 Step 3: Illustrate the information on a Venn diagram (Part a). The values for each distinct region are: • Only Ampesi: 1 • Only Banku: 4 • Only Fufu: 4 • Only Ampesi and Banku: 2 • Only Ampesi and Fufu: 2 • Only Banku and Fufu: 3 • All three (Ampesi, Banku, Fufu): 3 • None of the three foods: 1 Step 4: Find the number of people who liked only fufu (Part b-i). From Step 2, the number of people who liked only Fufu is 4. Only Fufu = 4 Step 5: Find the number of people who liked exactly two types (Part b-ii). This is the sum of people who liked only Ampesi and Banku, only Ampesi and Fufu, and only Banku and Fufu. Exactly two types = |(A B) F| + |(A F) B| + |(B F) A| Exactly two types = 2 + 2 + 3 = 7 Step 6: Find the number of people who liked only one type of food (Part b-iii). This is the sum of people who liked only Ampesi, only Banku, and only Fufu. Only one type of food = |A only| + |B only| + |F only| Only one type of food = 1 + 4 + 4 = 9 What's next?