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.
ScanSolve AI Answer
Step 1: Identify the given values and determine the number of periods and periodic interest rate. The future value (FV) Sarah wishes to accumulate is R10 000 000. The nominal annual interest rate (i_nom) is 15% or 0.15. The interest is compounded monthly, so the compounding frequency (m) is 12. The monthly interest rate (i) is calculated as: i = i_nomm = (0.15)/(12) = 0.0125 Sarah starts making payments on her 18th birthday and the last payment is one month before her 50th birthday. This means payments are made for 50 - 18 = 32 years. The total number of monthly payments (n) is: n = 32 years × 12 months/year = 384 months Step 2: Apply the future value of an ordinary annuity formula. The formula for the future value of an ordinary annuity is: FV = P [ ((1+i)^n - 1)/(i) ] Where P is the monthly instalment. Substitute the known values into the formula: 10,000,000 = P [ (1+0.0125)^384 - 10.0125 ] Step 3: Calculate the term inside the brackets. First, calculate (1+0.0125)^384: (1.0125)^384 ≈ 117.3248006 Now, substitute this back into the bracketed term: (117.3248006 - 1)/(0.0125) = (116.3248006)/(0.0125) ≈ 9305.984048 Step 4: Solve for P, the monthly instalment. 10,000,000 = P × 9305.984048 P = (10,000,000)/(9305.984048) P ≈ 1074.57 The monthly instalment Sarah will make is: R1074.57 Drop the next question!