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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Here are the solutions to the problems. Problem 13: Step 1: Calculate the number of boys in the village. Total youths = 650 Number of girls = 250 Number of boys = Total youths - Number of girls Number of boys = 650 - 250 = 400 Step 2: Calculate the total number of youths who do not like music. Total youths = 650 Youths who like music = 425 Youths who do not like music = Total youths - Youths who like music Youths who do not like music = 650 - 425 = 225 Step 3: Calculate the number of boys who do not like music. Youths who do not like music = 225 Girls who do not like music = 90 Boys who do not like music = Youths who do not like music - Girls who do not like music Boys who do not like music = 225 - 90 = 135 The correct option is A. The number of boys who do not like music is A. 135. Problem 14: Step 1: Determine the number of boys who play at least one game. Total boys = 20 Boys not allowed to play games = 2 Boys who play games = Total boys - Boys not allowed to play games n(S H) = 20 - 2 = 18 Step 2: Use the Principle of Inclusion-Exclusion. Let S be the set of boys who play soccer and H be the set of boys who play hockey. We are given: n(S) = 16 n(H) = 12 n(S H) = 18 The formula is n(S H) = n(S) + n(H) - n(S H). We need to find n(S H). 18 = 16 + 12 - n(S H) 18 = 28 - n(S H) n(S H) = 28 - 18 n(S H) = 10 The correct option is C. The number of students who play soccer and hockey is C. 10. Problem 15: The set difference P - Q represents all elements that are in set P but not in set Q. This is equivalent to the intersection of set P and the complement of set Q, denoted as Q'. Therefore, P - Q = P Q'. The correct option is C. P - Q is same as C. (P Q'). Problem 16: We are given: n(A - B) = 18 n(A B) = 70 n(A B) = 25 We need to find n(B). Step 1: Use the formula for n(A - B) to find n(A). n(A - B) = n(A) - n(A B) 18 = n(A) - 25 n(A) = 18 + 25 n(A) = 43 Step 2: Use the formula for n(A B) to find n(B). n(A B) = n(A) + n(B) - n(A B) 70 = 43 + n(B) - 25 70 = 18 + n(B) n(B) = 70 - 18 n(B) = 52 The correct option is D. n(B) is D. 52. Problem 17: Let B be the set of people who buy and S be the set of people who sell. We are given: Percentage of people who buy, n(B) = 70\% Percentage of people who sell, n(S) = 90\% All people buy or sell or do both, which means n(B S) = 100\%. We need to find the percentage of people who do both, n(B S). Use the Principle of Inclusion-Exclusion for percentages: n(B S) = n(B) + n(S) - n(B S) 100\% = 70\% + 90\% - n(B S) 100\% = 160\% - n(B S) n(B S) = 160\% - 100\% n(B S) = 60\% The correct option is A. The percentage of people who do both is A. 60\%.