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: Determine the number of females who play exactly two games. Let V, B, and T represent the sets of females who play Volleyball, Basketball, and Table-tennis, respectively. Given information: • Total females = 70 • V only = 20 • B only = 10 • T only = 6 • V B T (all three games) = 4 • An equal number play exactly two games only. Let this number be x. So, (V B) only = x, (V T) only = x, (B T) only = x. The sum of all disjoint regions in the Venn diagram must equal the total number of females: (V only) + (B only) + (T only) + (V B only) + (V T only) + (B T only) + (V B T) = 70 20 + 10 + 6 + x + x + x + 4 = 70 Combine the constant terms: 40 + 3x = 70 Subtract 40 from both sides: 3x = 70 - 40 3x = 30 Divide by 3: x = 10 So, 10 females play exactly two games for each pair of sports. Step 2: Illustrate the information on a Venn diagram. a) The Venn diagram should have three overlapping circles (V, B, T) within a universal set U. Label the regions as follows: • V only: 20 • B only: 10 • T only: 6 • V B only: 10 • V T only: 10 • B T only: 10 • V B T: 4 • The region outside the circles (females playing no games) is 0, as all females play at least one game. The universal set U = 70. Step 3: Find the number of females who play each game. b) i. Volleyball: The number of females who play Volleyball includes those who play V only, V and B only, V and T only, and all three. |V| = (V only) + (V B only) + (V T only) + (V B T) |V| = 20 + 10 + 10 + 4 = 44 The number of females who play Volleyball is 44. ii. Basketball: The number of females who play Basketball includes those who play B only, V and B only, B and T only, and all three. |B| = (B only) + (V B only) + (B T only) + (V B T) |B| = 10 + 10 + 10 + 4 = 34 The number of females who play Basketball is 34. iii. Table-tennis: The number of females who play Table-tennis includes those who play T only, V and T only, B and T only, and all three. |T| = (T only) + (V T only) + (B T only) + (V B T) |T| = 6 + 10 + 10 + 4 = 30 The number of females who play Table-tennis is 30. Step 4: Find the number of females who play exactly two games. c) The number of females who play exactly two games is the sum of those who play (V B) only, (V T) only, and (B T) only. (V B only) + (V T only) + (B T only) = x + x + x = 3x Substitute the value of x found in Step 1: 3x = 3 × 10 = 30 The number of females who play exactly two games is 30. 3 done, 2 left today. You're making progress.