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's the solution to the problem: Let U be the set of all households surveyed. Let H be the set of households doing homestay business. Let A be the set of households doing agriculture business. Given information: Total households, n(U) = 120. Households doing homestay business, n(H) = 70. Households doing agriculture business, n(A) = 50. Households doing other work (neither homestay nor agriculture), n(H A)' = 30. Step 1: Calculate the number of households doing at least one business. n(H A) = n(U) - n(H A)' n(H A) = 120 - 30 n(H A) = 90 Step 2: Calculate the number of households doing both homestay and agriculture business. n(H A) = n(H) + n(A) - n(H A) 90 = 70 + 50 - n(H A) 90 = 120 - n(H A) n(H A) = 120 - 90 n(H A) = 30 Step 3: Calculate the number of households doing only homestay business and only agriculture business. Households doing only homestay business: n(H only) = n(H) - n(H A) = 70 - 30 = 40 Households doing only agriculture business: n(A only) = n(A) - n(H A) = 50 - 30 = 20 a) Write the cardinality notation of the number of households who do not do any of the homestay and agriculture business. The cardinality notation for the number of households who do not do any of the homestay and agriculture business is n(H A)'. n(H A)' b) Present the above information in a Venn-diagram. The Venn diagram would show: • A universal set U containing 120 households. • Two overlapping circles, H (Homestay) and A (Agriculture). • The intersection H A contains 30 households. • The part of H only contains n(H only) = 40 households. • The part of A only contains n(A only) = 20 households. • The region outside both circles, n(H A)', contains 30 households. ` +---------------------------------+ | U = 120 | | +-----------------------+ | | | H | | | | (40) (30) | | | | +-------+ | | | | | | | | | | | A | | | | | | (20) | | | | +-----------+-------+---+ | | (30) | +---------------------------------+ ` c) Find the number of households doing only one business. The number of households doing only one business is the sum of households doing only homestay business and households doing only agriculture business. n(only one business) = n(H only) + n(A only) n(only one business) = 40 + 20 n(only one business) = 60 The number of households doing only one business is 60. d) If 10 households who do other work start homestay business, find the ratio of households who do homestay business to agriculture business. Original number of households doing homestay business, n(H)_old = 70. Original number of households doing agriculture business, n(A)_old = 50. Original number of households doing other work, n(H A)'_old = 30. If 10 households from those doing other work start homestay business: The new number of households doing homestay business, n(H)_new = n(H)_old + 10 = 70 + 10 = 80. The number of households doing agriculture business remains unchanged, n(A)_new = 50. The ratio of households doing homestay business to agriculture business is: Ratio = n(H)_new : n(A)_new Ratio = 80 : 50 Ratio = 8 : 5 The ratio of households doing homestay business to agriculture business is 8:5.