In a survey of 90 students about their favorite meals, 63 liked Banku, 55 liked Rice, and 50 liked Fufu. 13 liked only Banku, 8 liked only Rice, and 5 liked only Fufu. If 22 students liked all three m

Step 1: Define the sets and given information. Let B be the set of students who liked Banku, R be the set of students who liked Rice, and F be the set of students who liked Fufu. Total number of students = 90. Given: |B| = 63 |R| = 55 |F| = 50 |B only| = 13 |R only| = 8 |F only| = 5 |B R F| = 22 (students who liked all three meals) Let x be the number of students who liked Banku and Rice only (|B R only|). Let y be the number of students who liked Banku and Fufu only (|B F only|). Let z be the number of students who liked Rice and Fufu only (|R F only|). Step 2: Set up equations to find x, y, z. The total number of students who liked Banku is the sum of those who liked Banku only, Banku and Rice only, Banku and Fufu only, and all three: |B| = |B only| + x + y + |B R F| 63 = 13 + x + y + 22 63 = 35 + x + y x + y = 28 (1) Similarly for Rice: |R| = |R only| + x + z + |B R F| 55 = 8 + x + z + 22 55 = 30 + x + z x + z = 25 (2) And for Fufu: |F| = |F only| + y + z + |B R F| 50 = 5 + y + z + 22 50 = 27 + y + z y + z = 23 (3) Step 3: Solve the system of equations for x, y, z. From equation (1), y = 28 - x. Substitute this into equation (3): (28 - x) + z = 23 z - x = 23 - 28 z - x = -5 x - z = 5 (4) Now we have a system with equations (2) and (4): x + z = 25 (2) x - z = 5 (4) Add equation (2) and equation (4): (x + z) + (x - z) = 25 + 5 2x = 30 x = 15 Substitute x = 15 into equation (2): 15 + z = 25 z = 25 - 15 z = 10 Substitute x = 15 into equation (1): 15 + y = 28 y = 28 - 15 y = 13 So, the number of students who liked: Banku and Rice only (x) = 15 Banku and Fufu only (y) = 13 Rice and Fufu only (z) = 10 All three meals = 22 Banku only = 13 Rice only = 8 Fufu only = 5 a) Draw a Venn diagram to illustrate the information. A Venn diagram would consist of three overlapping circles labeled B, R, and F. The numbers in each region would be: • Region B only: 13 • Region R only: 8 • Region F only: 5 • Region B R only (intersection of B and R, outside F): 15 • Region B F only (intersection of B and F, outside R): 13 • Region R F only (intersection of R and F, outside B): 10 • Region B R F (intersection of all three): 22 b) Find the number of students who liked: i) Banku and Rice: This includes students who liked Banku and Rice only, and students who liked all three. Number of students who liked Banku and Rice = x + |B R F| = 15 + 22 = 37. ii) only two of the meals: This is the sum of students who liked (Banku and Rice only), (Banku and Fufu only), and (Rice and Fufu only). Number of students who liked only two meals = x + y + z = 15 + 13 + 10 = 38. iii) none of the three meals: First, find the total number of students who liked at least one meal: |B R F| = |B only| + |R only| + |F only| + x + y + z + |B R F| |B R F| = 13 + 8 + 5 + 15 + 13 + 10 + 22 = 86. Number of students who liked none of the meals = Total students - |B R F| Number of students who liked none of the meals = 90 - 86 = 4. 3 done, 2 left today. You'