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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Question 3.1 Show that the effective interest rate is 12.86% p.a. compounded monthly. Step 1: Convert the nominal interest rate compounded quarterly to an effective annual interest rate. The given nominal interest rate is 13% p.a. compounded quarterly. Nominal rate (i_nom,q) = 0.13. Number of compounding periods per year (m_q) = 4. The formula for the effective annual interest rate (i_eff) is: 1 + i_eff = (1 + i_nom,qm_q)^m_q 1 + i_eff = (1 + (0.13)/(4))^4 1 + i_eff = (1 + 0.0325)^4 1 + i_eff = (1.0325)^4 1 + i_eff = 1.13647990625 Step 2: Convert the effective annual interest rate to an equivalent nominal interest rate compounded monthly. Let the nominal interest rate compounded monthly be i_nom,m. Number of compounding periods per year (m_m) = 12. The formula relating effective annual rate to nominal monthly rate is: 1 + i_eff = (1 + i_nom,mm_m)^m_m 1.13647990625 = (1 + i_nom,m12)^12 Take the 12th root of both sides: (1.13647990625)^1/12 = 1 + i_nom,m12 1.010720000 = 1 + i_nom,m12 i_nom,m12 = 1.010720000 - 1 i_nom,m12 = 0.010720000 i_nom,m = 0.010720000 × 12 i_nom,m = 0.12864 Convert to a percentage and round to two decimal places: i_nom,m = 0.12864 × 100\% = 12.864\% ≈ 12.86\% Thus, the effective interest rate is 12.86\% p.a. compounded monthly. Question 3.2 How many full monthly repayments would Jake pay? Step 1: Identify the loan details and the monthly interest rate. Loan amount (P) = R600 000. Monthly payment (X) = R9 000. The nominal interest rate compounded monthly is 12.864\% (from 3.1, using the unrounded value for accuracy). Monthly interest rate (i) = (0.12864)/(12) = 0.01072. Step 2: Use the present value annuity formula to solve for the number of payments (n). The formula is: P = X 1 - (1 + i)^-ni Substitute the known values: 600\,000 = 9\,000 1 - (1 + 0.01072)^-n0.01072 Divide by 9 000 and multiply by 0.01072: (600\,000 × 0.01072)/(9\,000) = 1 - (1.01072)^-n 0.714666667 = 1 - (1.01072)^-n Rearrange to solve for (1.01072)^-n: (1.01072)^-n = 1 - 0.714666667 (1.01072)^-n = 0.285333333 Take the natural logarithm of both sides: -n (1.01072) = (0.285333333) -n (0.01066310) = -1.25441800 n = (-1.25441800)/(-0.01066310) n ≈ 117.639 The number of full monthly repayments Jake would pay is 117. Question 3.3 What will the final payment be? Step 1: Calculate the outstanding balance after 117 full payments. Number of full payments (k) = 117. Loan amount (P) = R600 000. Monthly payment (X) = R9 000. Monthly interest rate (i) = 0.01072. The formula for the outstanding balance (B_k) after k payments is: B_k = P(1+i)^k - X ((1+i)^k - 1)/(i) B_117 = 600\,000(1.01072)^117 - 9\,000 (1.01072)^117 - 10.01072 Calculate (1.01072)^117 ≈ 3.46879800. B_117 = 600\,000(3.46879800) - 9\,000 (3.46879800 - 1)/(0.01072) B_117 = 2\,081\,278.80 - 9\,000 (2.46879800)/(0.01072) B_117 = 2\,081\,278.80 - 9\,000(230.2983208955) B_117 = 2\,081\,278.80 - 2\,072\,684.89 B_117 = R8\,593.91 Step 2: Calculate the final payment, which includes interest on the outstanding balance for the last period. Final payment = B_117(1+i) Final payment = R8\,593.91 × (1 + 0.01072) Final payment = R8\,593.91 × 1.01072 Final payment = R8\,685.99 The value of Jake's final payment is R8 685.99. Question 3.4 What did the car cost Jake in total by the time it is paid off? Step 1: Calculate the total amount paid. Total payments = (Number of full payments × Monthly payment) + Final payment Total payments = (117 × R9\,000) + R8\,685.99 Total payments = R1\,053\,000 + R8\,685.99 Total payments = R1\,061\,685.99 The car cost Jake a total of R1 061 685.99. What's next? 📸