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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Step 1: Identify the given values. The principal amount borrowed is P = N8000. The annual interest rate is r = 7\% = 0.07. The loan term is T = 2 years. The payments are made monthly. Step 2: Calculate the monthly interest rate and the total number of payments. The monthly interest rate i is the annual rate divided by 12: i = (0.07)/(12) The total number of payments n is the loan term in years multiplied by 12: n = 2 years × 12 months/year = 24 months Step 3: Use the loan payment formula for an amortizing loan. The formula for the monthly payment M is: M = P (i(1+i)^n)/((1+i)^n - 1) Substitute the values: M = 8000 (0.07)/(12)(1+(0.07)/(12))^24(1+(0.07)/(12))^24 - 1 Calculate the value of 1+(0.07)/(12): 1+(0.07)/(12) ≈ 1 + 0.005833333 = 1.005833333 Calculate (1+(0.07)/(12))^24: (1.005833333)^24 ≈ 1.149716 Now substitute these values back into the formula for M: M = 8000 (0.005833333 × 1.149716)/(1.149716 - 1) M = 8000 (0.006706676)/(0.149716) M = 8000 × 0.044795 M ≈ 358.36 The monthly payment is approximately N358.36. The monthly payment is N358.36. a) Evaluate (1)/(4)A - 2B. Step 1: Calculate (1)/(4)A. (1)/(4)A = (1)/(4) 4 & -1 \\ 0 & 3 = (1)/(4) × 4 & (1)/(4) × (-1) \\ (1)/(4) × 0 & (1)/(4) × 3 = 1 & -(1)/(4) \\ 0 & (3)/(4) Step 2: Calculate 2B. 2B = 2 -2 & 1 \\ 1 & 0 = 2 × (-2) & 2 × 1 \\ 2 × 1 & 2 × 0 = -4 & 2 \\ 2 & 0 Step 3: Subtract 2B from (1)/(4)A. (1)/(4)A - 2B = 1 & -(1)/(4) \\ 0 & (3)/(4) - -4 & 2 \\ 2 & 0 = 1 - (-4) & -(1)/(4) - 2 \\ 0 - 2 & (3)/(4) - 0 = 1 + 4 & -(1)/(4) - (8)/(4) \\ -2 & (3)/(4) = 5 & -(9)/(4) \\ -2 & (3)/(4) The result is 5 & -(9)/(4) \\ -2 & (3)/(4) . b) Evaluate B + 3A. Step 1: Calculate 3A. 3A = 3 4 & -1 \\ 0 & 3 = 3 × 4 & 3 × (-1) \\ 3 × 0 & 3 × 3 = 12 & -3 \\ 0 & 9 Step 2: Add B to 3A. B + 3A = -2 & 1 \\ 1 & 0 + 12 & -3 \\ 0 & 9 = -2 + 12 & 1 + (-3) \\ 1 + 0 & 0 + 9 = 10 & -2 \\ 1 & 9 The result is 10 & -2 \\ 1 & 9 . Step 1: Apply the logarithm property _b M - _b N = _b ((M)/(N)). The given equation is: _3(x^2) - _3 x = 2 Applying the property: _3 ((x^2)/(x)) = 2 Step 2: Simplify the expression inside the logarithm. _3 x = 2 Step 3: Convert the logarithmic equation to an exponential equation. If _b Y = X, then b^X = Y. Here, b=3, Y=x, and X=2. x = 3^2 Step 4: Calculate the value of x. x = 9 For the logarithms to be defined, x^2 > 0 and x > 0. Our solution x=9 satisfies these conditions. The value of x is 9. Send me the next one 📸