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: Question 6: A woman gave (1)/(3) of her farm to her daughter and sold (1)/(4) of the remainder. a) Calculate the fraction sold. Step 1: Determine the fraction of the farm remaining after giving to her daughter. Fraction remaining = 1 - (1)/(3) = (3)/(3) - (1)/(3) = (2)/(3) Step 2: Calculate the fraction of the farm sold, which is (1)/(4) of the remainder. Fraction sold = (1)/(4) × (2)/(3) = (2)/(12) = (1)/(6) The fraction sold is (1)/(6). b) Calculate the fraction of the farm remaining. Step 1: The fraction remaining after giving to her daughter was (2)/(3). She sold (1)/(4) of this remainder, meaning (3)/(4) of this remainder is still left. Fraction remaining = (3)/(4) × (2)/(3) Step 2: Multiply the fractions. Fraction remaining = (6)/(12) = (1)/(2) The fraction of the farm remaining is (1)/(2). Question 7: Solve for x in the equation 4^x+1 = 32^x-1. Step 1: Express both bases as powers of the same prime number, which is 2. 4 = 2^2 32 = 2^5 Step 2: Substitute these into the equation. (2^2)^x+1 = (2^5)^x-1 Step 3: Apply the exponent rule (a^m)^n = a^mn. 2^2(x+1) = 2^5(x-1) 2^2x+2 = 2^5x-5 Step 4: Since the bases are equal, equate the exponents. 2x+2 = 5x-5 Step 5: Solve for x. 2+5 = 5x-2x 7 = 3x x = (7)/(3) The value of x is (7)/(3). Question 8: A trader took a loan of 3,000,000 FCFA from a bank at the rate of 2% simple interest per annum. a) Calculate how much interest he pays per year. Step 1: Use the simple interest formula for one year: I = P × R. I = 3,000,000 FCFA × 0.02 I = 60,000 FCFA The interest paid per year is 60,000 FCFA. b) Calculate the total interest paid in three years. Step 1: Use the simple interest formula: I = P × R × T. I = 3,000,000 FCFA × 0.02 × 3 I = 60,000 FCFA × 3 I = 180,000 FCFA The total interest paid in three years is 180,000 FCFA. c) Calculate the total amount paid at the end of three years. Step 1: The total amount paid is the principal plus the total interest. Total Amount = Principal + Total Interest Total Amount = 3,000,000 FCFA + 180,000 FCFA Total Amount = 3,180,000 FCFA The total amount paid at the end of three years is 3,180,000 FCFA. Question 9: The diagram below represents roads linking the offices in an establishment. a) State the number of roads (ARCS) on the diagram by identifying their endpoints (nodes). Step 1: Identify the nodes (offices): Principal, V Principal, BC Workshop, Bursary. Step 2: Count the lines connecting these nodes, which represent the roads (arcs). • Principal to V Principal • Principal to Bursary • Principal to BC Workshop • V Principal to Bursary • BC Workshop to Bursary There are 5 roads (arcs) on the diagram. b) State the number of regions in the network. Step 1: Identify the number of vertices (nodes, V) and edges (arcs, E) in the planar graph. V = 4 (Principal, V Principal, BC Workshop, Bursary) E = 5 (The 5 roads identified in part a) Step 2: Apply Euler's formula for planar graphs, which states V - E + F = 2, where F is the number of faces (regions). 4 - 5 + F = 2 -1 + F = 2 F = 2 + 1 F = 3 The number of regions in the network is 3. What's next?