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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Here are the solutions to your questions: Question 2 2.1 Step 1: Convert the nominal interest rate of 13% p.a. compounded quarterly to an equivalent nominal interest rate compounded monthly. We use the formula for equivalent interest rates: (1 + i_nom1m_1)^m_1 = (1 + i_nom2m_2)^m_2 Given i_nom1 = 0.13 (13%), m_1 = 4 (quarterly). We want to find i_nom2 (monthly rate), with m_2 = 12. (1 + (0.13)/(4))^4 = (1 + i_monthly12)^12 Step 2: Calculate the left side of the equation. (1 + 0.0325)^4 = (1.0325)^4 = 1.13647990625 Step 3: Solve for i_monthly. 1.13647990625 = (1 + i_monthly12)^12 Take the 12th root of both sides: (1.13647990625)^1/12 = 1 + i_monthly12 1.010720333 = 1 + i_monthly12 0.010720333 = i_monthly12 i_monthly = 0.010720333 × 12 i_monthly = 0.128643996 Expressed as a percentage, this is 12.8643996\% ≈ 12.86\%. Thus, the effective interest rate is 12.86\% p.a. compounded monthly. 2.2 Step 1: Identify the known values for the present value annuity formula. Loan amount (P) = R600 000 Monthly instalment (x) = R9 000 Monthly interest rate (i) = (0.128643996)/(12) = 0.010720333 We need to find the number of payments (n). The present value annuity formula is: P = x[1 - (1 + i)^-n]i Step 2: Substitute the values into the formula. 600000 = 9000[1 - (1 + 0.010720333)^-n]0.010720333 Step 3: Solve for n. 600000 × 0.010720333 = 9000[1 - (1.010720333)^-n] 6432.1998 = 9000[1 - (1.010720333)^-n] (6432.1998)/(9000) = 1 - (1.010720333)^-n 0.71468886 = 1 - (1.010720333)^-n (1.010720333)^-n = 1 - 0.71468886 (1.010720333)^-n = 0.28531114 Take the natural logarithm of both sides: -n (1.010720333) = (0.28531114) -n × 0.01066310 = -1.254790 n = (-1.254790)/(-0.01066310) n = 117.674 Since n is not an integer, there will be 117 full instalments of R9 000. The number of instalments of R9 000 that must be paid is 117. 2.3 Step 1: Calculate the outstanding balance after 117 payments. The balance outstanding after k payments (B_k) is given by: B_k = P(1+i)^k - x ((1+i)^k - 1)/(i) Using P = 600000, x = 9000, i = 0.010720333, and k = 117: B_117 = 600000(1.010720333)^117 - 9000 (1.010720333)^117 - 10.010720333 First, calculate (1.010720333)^117 ≈ 3.437990. B_117 = 600000(3.437990) - 9000 (3.437990 - 1)/(0.010720333) B_117 = 2062794 - 9000 (2.437990)/(0.010720333) B_117 = 2062794 - 9000 × 227.4170 B_117 = 2062794 - 2046753 B_117 = 16041 Step 2: Calculate the final payment. The final payment will be this outstanding balance plus one month's interest. This payment occurs at the end of the 118th month. Final payment = B_117(1+i) Final payment = 16041(1 + 0.010720333) Final payment = 16041(1.010720333) Final payment = 16213.38 The final payment will be R16 213.38. 2.4 Step 1: Calculate the total amount paid. Total cost = (Number of full payments × Instalment amount) + Final payment Total cost = (117 × R9000) + R16213.38 Total cost = R1053000 + R16213.38 Total cost = R1069213.38 The car cost Jake in total R1 069 213.38. Question 3 3.1 Step 1: Identify the nominal interest rate and compounding frequency. Nominal rate (i_nom) = 0.14 (14%) Number of compounding periods per year (m) = 12 (monthly) Step 2: Use the formula for effective annual interest rate. (1 + i_eff) = (1 + i_nomm)^m (1 + i_eff) = (1 + (0.14)/(12))^12 (1 + i_eff) = (1 + 0.01166666666)^12 (1 + i_eff) = (1.01166666666)^12 (1 + i_eff) = 1.149342026 Step 3: Calculate the effective annual rate. i_eff = 1.149342026 - 1 i_eff = 0.149342026 Expressed as a percentage, this is 14.9342026\% ≈ 14.93\%. The effective interest rate per annum is 14.93\%. 3.2 Step 1: Identify the known values for the future value annuity formula. Future Value (FV) = R7003007.22 Monthly payment (x) = R800 Nominal interest rate = 16% p.a. compounded monthly. Monthly interest rate (i) = (0.16)/(12) = 0.01333333333 We need to find the number of payments (n). The future value annuity formula is: FV = (x[(1 + i)^n - 1])/(i) Step 2: Substitute the values into the formula. 7003007.22 = (800[(1 + 0.01333333333)^n - 1])/(0.01333333333) Step 3: Solve for n. 7003007.22 × 0.01333333333 = 800[(1.01333333333)^n - 1] 93373.4296 = 800[(1.01333333333)^n - 1] (93373.4296)/(800) = (1.01333333333)^n - 1 116.716787 = (1.01333333333)^n - 1 117.716787 = (1.01333333333)^n Take the natural logarithm of both sides: (117.716787) = n (1.01333333333) 4.768390 = n × 0.0132440 n = (4.768390)/(0.0132440) n = 360 Skomota made 360 payments. 3 done, 2 left today. You're making progress.