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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19. A trader bought 2 cows and 9 goats for a total of ksh. 98,200. If she had bought 3 cows and 4 goats, she would have spent ksh. 2,200 less. a) Form two equations to represent the above information. Step 1: Define variables. Let c be the cost of one cow in Ksh. Let g be the cost of one goat in Ksh. Step 2: Form the first equation. "A trader bought 2 cows and 9 goats for a total of ksh. 98,200." 2c + 9g = 98200 Step 3: Form the second equation. "If she had bought 3 cows and 4 goats, she would have spent ksh. 2,200 less." Ksh. 2,200 less than Ksh. 98,200 is 98200 - 2200 = 96000. 3c + 4g = 96000 The two equations are: 2c + 9g &= 98200 \\ 3c + 4g &= 96000 b) Use the matrix method to determine the cost of a cow and that of a goat. Step 1: Write the system of equations in matrix form. 2 & 9 \\ 3 & 4 c \\ g = 98200 \\ 96000 Step 2: Find the determinant of the coefficient matrix. Let A = 2 & 9 \\ 3 & 4 . det(A) = (2)(4) - (9)(3) = 8 - 27 = -19 Step 3: Find the inverse of the coefficient matrix. A^-1 = (1)/(det)(A) 4 & -9 \\ -3 & 2 = (1)/(-19) 4 & -9 \\ -3 & 2 Step 4: Multiply the inverse matrix by the constant matrix to find c and g. c \\ g = (1)/(-19) 4 & -9 \\ -3 & 2 98200 \\ 96000 c \\ g = -(1)/(19) (4)(98200) + (-9)(96000) \\ (-3)(98200) + (2)(96000) c \\ g = -(1)/(19) 392800 - 864000 \\ -294600 + 192000 c \\ g = -(1)/(19) -471200 \\ -102600 c \\ g = (-471200)/(-19) \\ (-102600)/(-19) c \\ g = 24800 \\ 5400 The cost of a cow is Ksh 24,800 and the cost of a goat is Ksh 5,400. c) The trader later sold the animals she had bought making a profit of 30% per cow and 40% per goat. (i) Calculate the total amount of money she received. Step 1: Calculate the selling price of one cow. Profit on cow = 30% Selling Price per cow (SP_c) = 24800 × (1 + 0.30) = 24800 × 1.30 = 32240 Ksh. Step 2: Calculate the selling price of one goat. Profit on goat = 40% Selling Price per goat (SP_g) = 5400 × (1 + 0.40) = 5400 × 1.40 = 7560 Ksh. Step 3: Calculate the total money received from selling 2 cows and 9 goats. Total received = (2 × SP_c) + (9 × SP_g) Total received = (2 × 32240) + (9 × 7560) Total received = 64480 + 68040 Total received = 132520 Ksh. The total amount of money she received is Ksh 132,520. (ii) Determine correct to 4 significant figures the percentage profit the trader made from the sale of animals. Step 1: Calculate the total cost of the animals. Total cost = 2c + 9g = 2(24800) + 9(5400) = 49600 + 48600 = 98200 Ksh. (This matches the initial total cost given in the problem statement). Step 2: Calculate the total profit. Total profit = Total received - Total cost Total profit = 132520 - 98200 = 34320 Ksh. Step 3: Calculate the percentage profit. Percentage Profit = Total ProfitTotal Cost × 100\% Percentage Profit = (34320)/(98200) × 100\% Percentage Profit ≈ 0.349490835 × 100\% Percentage Profit ≈ 34.9490835\% Step 4: Round the percentage profit to 4 significant figures. The first four significant figures are 3, 4, 9, 4. The fifth digit is 9, so we round up the fourth digit. Percentage Profit ≈ 34.95\% The percentage profit is 34.95\%. What's next?