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) Calculate the amount to be paid at the end of the 6 months. Step 1: Identify the given values. Principal (P) = GH¢36,000 Annual interest rate (R) = 12 (1)/(2)\% = 12.5\% Time (T) = 6 months = (6)/(12) years = 0.5 years Step 2: Calculate the simple interest (I). I = (P × R × T)/(100) I = (36000 × 12.5 × 0.5)/(100) I = (225000)/(100) I = GH¢2,250 Step 3: Calculate the total amount to be paid. Amount = P + I Amount = 36000 + 2250 Amount = GH¢38,250 The amount to be paid at the end of 6 months is GH¢38,250. ii) Calculate the monthly installment. Step 1: Use the total amount and the number of months. Monthly installment = Total AmountNumber of months Monthly installment = (38250)/(6) Monthly installment = GH¢6,375 The monthly installment is GH¢6,375. b) The perimeter of a rectangular plot of land whose length is (4x + 10)m and width (2x - 20)m is 160m. Find i) Value of x Step 1: Write the formula for the perimeter of a rectangle. P = 2(L + W) Given P = 160 m, L = (4x + 10) m, W = (2x - 20) m. Step 2: Substitute the given values into the formula and solve for x. 160 = 2((4x + 10) + (2x - 20)) 160 = 2(4x + 2x + 10 - 20) 160 = 2(6x - 10) Divide both sides by 2: 80 = 6x - 10 Add 10 to both sides: 80 + 10 = 6x 90 = 6x x = (90)/(6) x = 15 The value of x is 15. ii) area of the plot. Step 1: Calculate the length and width using the value of x. L = 4x + 10 = 4(15) + 10 = 60 + 10 = 70 m W = 2x - 20 = 2(15) - 20 = 30 - 20 = 10 m Step 2: Calculate the area of the plot. Area = L × W Area = 70 m × 10 m Area = 700 m^2 The area of the plot is 700 m^2. c) A woman bought a washing machine for GH¢4,830.00 including VAT of 15%. Calculate for the i) actual cost of the machine. Step 1: Let the actual cost be C. The total cost includes VAT. Total Cost = Actual Cost + VAT Total Cost = C + 0.15C Total Cost = 1.15C Given Total Cost = GH¢4,830.00. Step 2: Solve for the actual cost C. 4830 = 1.15C C = (4830)/(1.15) C = GH¢4,200 The actual cost of the machine is GH¢4,200. ii) VAT paid on the machine. Step 1: Calculate the VAT paid. VAT Paid = Total Cost - Actual Cost VAT Paid = 4830 - 4200 VAT Paid = GH¢630 The VAT paid on the machine is GH¢630. d) Find the value of x in the diagram. Step 1: Identify the sides of the right-angled triangle. The triangle PMN is a right-angled triangle with the right angle at M. Side PM = x + 1 Side MN = 3 Hypotenuse PN = x + 2 Step 2: Apply the Pythagorean theorem. PM^2 + MN^2 = PN^2 (x + 1)^2 + 3^2 = (x + 2)^2 Step 3: Expand and solve for x. (x^2 + 2x + 1) + 9 = (x^2 + 4x + 4) x^2 + 2x + 10 = x^2 + 4x + 4 Subtract x^2 from both sides: 2x + 10 = 4x + 4 Subtract 2x from both sides: 10 = 2x + 4 Subtract 4 from both sides: 10 - 4 = 2x 6 = 2x x = (6)/(2) x = 3 The value of x is 3. Find the sum of (2sqrt(5) - 4sqrt(2)) and (3sqrt(5) + 5sqrt(2)). Step 1: Write the expression for the sum. Sum = (2sqrt(5) - 4sqrt(2)) + (3sqrt(5) + 5sqrt(2)) Step 2: Group like terms. Sum = (2sqrt(5) + 3sqrt(5)) + (-4sqrt(2) + 5sqrt(2)) Step 3: Combine the coefficients of the like terms. Sum = (2 + 3)sqrt(5) + (-4 + 5)sqrt(2) Sum = 5sqrt(5) + 1sqrt(2) Sum = 5sqrt(5) + sqrt(2) The sum is 5sqrt(5) + sqrt(2). Send me the next one 📸