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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a) i) Form two equations to represent the above information. Let x be the cost of a bag of maize. Let y be the cost of a bag of beans. From Musamali's purchase: Musamali bought 100 bags of maize and 80 bags of beans for a total of Ksh 694,000. 100x + 80y = 694000 (1) From Cheruiyot's purchase: Cheruiyot bought 100 + 20 = 120 bags of maize. Cheruiyot bought 50 bags of beans. Cheruiyot paid Ksh 116,500 less than Musamali. Amount paid by Cheruiyot = 694000 - 116500 = 577500 Ksh. 120x + 50y = 577500 (2) The two equations are: 100x + 80y = 694000 120x + 50y = 577500 a) ii) Find the cost of 1 bag of each item using the matrix method. Step 1: Write the system of equations in matrix form. 100 & 80 \\ 120 & 50 x \\ y = 694000 \\ 577500 Step 2: Find the determinant of the coefficient matrix A = 100 & 80 \\ 120 & 50 . det(A) = (100)(50) - (80)(120) = 5000 - 9600 = -4600 Step 3: Find the inverse of the coefficient matrix A^-1. A^-1 = (1)/(det)(A) 50 & -80 \\ -120 & 100 = (1)/(-4600) 50 & -80 \\ -120 & 100 Step 4: Multiply A^-1 by the constant matrix B = 694000 \\ 577500 to find x \\ y . x \\ y = (1)/(-4600) 50 & -80 \\ -120 & 100 694000 \\ 577500 x \\ y = (1)/(-4600) (50)(694000) + (-80)(577500) \\ (-120)(694000) + (100)(577500) x \\ y = (1)/(-4600) 34700000 - 46200000 \\ -83280000 + 57750000 x \\ y = (1)/(-4600) -11500000 \\ -25530000 Step 5: Calculate the values of x and y. x = (-11500000)/(-4600) = 2500 y = (-25530000)/(-4600) = 5550 The cost of 1 bag of maize is Ksh 2500. The cost of 1 bag of beans is Ksh 5550. Cost of maize (x) = Ksh 2500 Cost of beans (y) = Ksh 5550 b) In the subsequent month, the price of maize increased by 20% while that of beans reduced by 10%. Find the total amount he spent if he bought 40 bags of maize and 25 bags of beans. Step 1: Calculate the new price of maize. New price of maize = 2500 × (1 + 0.20) = 2500 × 1.20 = 3000 Ksh. Step 2: Calculate the new price of beans. New price of beans = 5550 × (1 - 0.10) = 5550 × 0.90 = 4995 Ksh. Step 3: Calculate the total amount spent for the new purchase. Total amount spent = (40 bags of maize × new maize price) + (25 bags of beans × new beans price) Total amount spent = (40 × 3000) + (25 × 4995) Total amount spent = 120000 + 124875 Total amount spent = 244875 Ksh. The total amount spent is Ksh 244875. 3 done, 2 left today. You're making progress.