A motor car costing R200 000 depreciated at a rate of 8% per annum on the reducing balance method. Calculate how long it took for the car to depreciate to a value of R90 000 under these conditions.

Question

Asked on July 23, 2026Mathematics

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.

Accepted Answer · 2026-07-23T06:14:18.376Z

3.1 To calculate how long it took for the car to depreciate: The formula for depreciation on a reducing balance is A = P(1 - i)^n. Given: P = R200\,000 (initial value) A = R90\,000 (depreciated value) i = 8\% = 0.08 (annual depreciation rate) Step 1: Substitute the given values into the formula. 90\,000 = 200\,000(1 - 0.08)^n 90\,000 = 200\,000(0.92)^n Step 2: Isolate the term with n. (90\,000)/(200\,000) = (0.92)^n 0.45 = (0.92)^n Step 3: Use logarithms to solve for n. (0.45) = n (0.92) n = ((0.45))/((0.92)) Step 4: Calculate the value of n. n ≈ (-0.34678)/(-0.03621) n ≈ 9.576 It took approximately 9.58 years for the car to depreciate to R90 000. 3.2.1 To calculate the monthly interest rate: The annual interest rate is 6.25% compounded semi-annually. Nominal annual rate i_nom = 0.0625. Compounding periods per year m = 2. Step 1: Calculate the effective annual interest rate (i_eff). (1 + i_eff) = (1 + i_nomm)^m (1 + i_eff) = (1 + (0.0625)/(2))^2 (1 + i_eff) = (1 + 0.03125)^2 (1 + i_eff) = (1.03125)^2 = 1.0634765625 Step 2: Convert the effective annual rate to a monthly rate (i_monthly). (1 + i_monthly)^12 = (1 + i_eff) (1 + i_monthly)^12 = 1.0634765625 1 + i_monthly = (1.0634765625)^1/12 1 + i_monthly ≈ 1.005139908 Step 3: Calculate the monthly interest rate. i_monthly ≈ 1.005139908 - 1 i_monthly ≈ 0.005139908 The monthly interest rate is approximately 0.00514 or 0.514\%. 3.2.2 To calculate the future value of his investment at the end of the five-year period: The total period is 5 years × 12 months/year = 60 months. Monthly interest rate i = 0.005139908. Phetolo makes an initial deposit of R2000 at the beginning of the first month, and then R2000 at the end of the first month, and continues with R2000 at the end of each month for 60 months. Step 1: Calculate the future value of the initial R2000 deposit. This deposit earns interest for the full 60 months. FV_initial = P(1+i)^n FV_initial = 2000(1 + 0.005139908)^60 FV_initial = 2000(1.005139908)^60 FV_initial ≈ 2000 × 1.35620901 ≈ R2712.42 Step 2: Calculate the future value of the 60 monthly payments of R2000 (ordinary annuity). FV_annuity = X ((1+i)^n - 1)/(i) FV_annuity = 2000 (1.005139908)^60 - 10.005139908 FV_annuity = 2000 (1.35620901 - 1)/(0.005139908) FV_annuity = 2000 (0.35620901)/(0.005139908) ≈ 2000 × 69.39000 ≈ R138780.00 Step 3: Calculate the total future value. FV_total = FV_initial + FV_annuity FV_total = 2712.42 + 138780.00 ≈ R141492.42 The future value of his investment is R141\,492.42. 3.2.3 To calculate the value of his investment if he misses the last two payments: This means only 60 - 2 = 58 monthly payments were made for the annuity part. The initial R2000 still accumulates for 60 months. Step 1: The future value of the initial R2000 deposit remains the same. FV_initial ≈ R2712.42 Step 2: Calculate the future value of the 58 monthly payments of R2000 (ordinary annuity). FV_annuity = X ((1+i)^n - 1)/(i) FV_annuity = 2000 (1.005139908)^58 - 10.005139908 (1.005139908)^58 ≈ 1.34107001 FV_annuity = 2000 (1.34107001 - 1)/(0.005139908) FV_annuity = 2000 (0.34107001)/(0.005139908) ≈ 2000 × 66.35400 ≈ R132708.00 Step 3: Calculate the total future value. FV_total = FV_initial + FV_annuity FV_total = 2712.42 + 132708.00 ≈ R135420.42 The value of his investment is R135\,420.42. 3.3.1 To determine the number of payments required to settle the loan: Loan amount P = R300\,000. Monthly payment x = R5\,000. Annual interest rate = 18% compounded monthly. Monthly interest rate i = (0.18)/(12) = 0.015. Step 1: Use the present value annuity formula. P = x 1 - (1+i)^-ni 300\,000 = 5\,000 1 - (1+0.015)^-n0.015 Step 2: Isolate the term with n. (300\,000 × 0.015)/(5\,000) = 1 - (1.015)^-n (4500)/(5000) = 1 - (1.015)^-n 0.9 = 1 - (1.015)^-n (1.015)^-n = 1 - 0.9 (1.015)^-n = 0.1 Step 3: Use logarithms to solve for n. -n (1.015) = (0.1) -n = ((0.1))/((1.015)) Step 4: Calculate the value of n. -n ≈ (-1)/(0.00646604) -n ≈ -154.644 n ≈ 154.644 Since a payment cannot be a fraction, Jill will make 154 full payments of R5000 and one final smaller payment. Therefore, the total number of payments required is 155. 3.3.2 To calculate the balance outstanding after Jill has paid the last R5 000: This refers to the balance outstanding after 154 full payments of R5000. We use the formula for the balance outstanding on a loan: A_n = P(1+i)^n - x ((1+i)^n - 1)/(i). Here, n = 154 (number of full payments). P = R300\,000. x = R5\,000. i = 0.015. Step 1: Calculate the future value of the original loan after 154 months. FV_loan = 300\,000(1+0.015)^154 FV_loan = 300\,000(1.015)^154 (1.015)^154 ≈ 9.8732000 FV_loan ≈ 300\,000 × 9.8732000 ≈ R2\,961\,960.00 Step 2: Calculate the future value of the 154 payments made. FV_payments = 5000 (1.015)^154 - 10.015 FV_payments = 5000 (9.8732000 - 1)/(0.015) FV_payments = 5000 (8.8732000)/(0.015) ≈ 5000 × 591.5466667 ≈ R2\,957\,733.33 Step 3: Calculate the balance outstanding. Balance = FV_loan - FV_payments Balance = 2\,961\,960.00 - 2\,957\,733.33 ≈ R4226.67 The balance outstanding after Jill has paid the last R5 000 is R4226.67. 3 done, 2 left today. You're making progress.

Accepted Answer · 2026-07-23T06:14:18.376Z

3.1 To calculate how long it took for the car to depreciate: The formula for depreciation on a reducing balance is A = P(1 - i)^n. Given: P = R200\,000 (initial value) A = R90\,000 (depreciated value) i = 8\% = 0.08 (annual depreciation rate) Step 1: Substitute the given values into the formula. 90\,000 = 200\,000(1 - 0.08)^n 90\,000 = 200\,000(0.92)^n Step 2: Isolate the term with n. (90\,000)/(200\,000) = (0.92)^n 0.45 = (0.92)^n Step 3: Use logarithms to solve for n. (0.45) = n (0.92) n = ((0.45))/((0.92)) Step 4: Calculate the value of n. n ≈ (-0.34678)/(-0.03621) n ≈ 9.576 It took approximately 9.58 years for the car to depreciate to R90 000. 3.2.1 To calculate the monthly interest rate: The annual interest rate is 6.25% compounded semi-annually. Nominal annual rate i_nom = 0.0625. Compounding periods per year m = 2. Step 1: Calculate the effective annual interest rate (i_eff). (1 + i_eff) = (1 + i_nomm)^m (1 + i_eff) = (1 + (0.0625)/(2))^2 (1 + i_eff) = (1 + 0.03125)^2 (1 + i_eff) = (1.03125)^2 = 1.0634765625 Step 2: Convert the effective annual rate to a monthly rate (i_monthly). (1 + i_monthly)^12 = (1 + i_eff) (1 + i_monthly)^12 = 1.0634765625 1 + i_monthly = (1.0634765625)^1/12 1 + i_monthly ≈ 1.005139908 Step 3: Calculate the monthly interest rate. i_monthly ≈ 1.005139908 - 1 i_monthly ≈ 0.005139908 The monthly interest rate is approximately 0.00514 or 0.514\%. 3.2.2 To calculate the future value of his investment at the end of the five-year period: The total period is 5 years × 12 months/year = 60 months. Monthly interest rate i = 0.005139908. Phetolo makes an initial deposit of R2000 at the beginning of the first month, and then R2000 at the end of the first month, and continues with R2000 at the end of each month for 60 months. Step 1: Calculate the future value of the initial R2000 deposit. This deposit earns interest for the full 60 months. FV_initial = P(1+i)^n FV_initial = 2000(1 + 0.005139908)^60 FV_initial = 2000(1.005139908)^60 FV_initial ≈ 2000 × 1.35620901 ≈ R2712.42 Step 2: Calculate the future value of the 60 monthly payments of R2000 (ordinary annuity). FV_annuity = X ((1+i)^n - 1)/(i) FV_annuity = 2000 (1.005139908)^60 - 10.005139908 FV_annuity = 2000 (1.35620901 - 1)/(0.005139908) FV_annuity = 2000 (0.35620901)/(0.005139908) ≈ 2000 × 69.39000 ≈ R138780.00 Step 3: Calculate the total future value. FV_total = FV_initial + FV_annuity FV_total = 2712.42 + 138780.00 ≈ R141492.42 The future value of his investment is R141\,492.42. 3.2.3 To calculate the value of his investment if he misses the last two payments: This means only 60 - 2 = 58 monthly payments were made for the annuity part. The initial R2000 still accumulates for 60 months. Step 1: The future value of the initial R2000 deposit remains the same. FV_initial ≈ R2712.42 Step 2: Calculate the future value of the 58 monthly payments of R2000 (ordinary annuity). FV_annuity = X ((1+i)^n - 1)/(i) FV_annuity = 2000 (1.005139908)^58 - 10.005139908 (1.005139908)^58 ≈ 1.34107001 FV_annuity = 2000 (1.34107001 - 1)/(0.005139908) FV_annuity = 2000 (0.34107001)/(0.005139908) ≈ 2000 × 66.35400 ≈ R132708.00 Step 3: Calculate the total future value. FV_total = FV_initial + FV_annuity FV_total = 2712.42 + 132708.00 ≈ R135420.42 The value of his investment is R135\,420.42. 3.3.1 To determine the number of payments required to settle the loan: Loan amount P = R300\,000. Monthly payment x = R5\,000. Annual interest rate = 18% compounded monthly. Monthly interest rate i = (0.18)/(12) = 0.015. Step 1: Use the present value annuity formula. P = x 1 - (1+i)^-ni 300\,000 = 5\,000 1 - (1+0.015)^-n0.015 Step 2: Isolate the term with n. (300\,000 × 0.015)/(5\,000) = 1 - (1.015)^-n (4500)/(5000) = 1 - (1.015)^-n 0.9 = 1 - (1.015)^-n (1.015)^-n = 1 - 0.9 (1.015)^-n = 0.1 Step 3: Use logarithms to solve for n. -n (1.015) = (0.1) -n = ((0.1))/((1.015)) Step 4: Calculate the value of n. -n ≈ (-1)/(0.00646604) -n ≈ -154.644 n ≈ 154.644 Since a payment cannot be a fraction, Jill will make 154 full payments of R5000 and one final smaller payment. Therefore, the total number of payments required is 155. 3.3.2 To calculate the balance outstanding after Jill has paid the last R5 000: This refers to the balance outstanding after 154 full payments of R5000. We use the formula for the balance outstanding on a loan: A_n = P(1+i)^n - x ((1+i)^n - 1)/(i). Here, n = 154 (number of full payments). P = R300\,000. x = R5\,000. i = 0.015. Step 1: Calculate the future value of the original loan after 154 months. FV_loan = 300\,000(1+0.015)^154 FV_loan = 300\,000(1.015)^154 (1.015)^154 ≈ 9.8732000 FV_loan ≈ 300\,000 × 9.8732000 ≈ R2\,961\,960.00 Step 2: Calculate the future value of the 154 payments made. FV_payments = 5000 (1.015)^154 - 10.015 FV_payments = 5000 (9.8732000 - 1)/(0.015) FV_payments = 5000 (8.8732000)/(0.015) ≈ 5000 × 591.5466667 ≈ R2\,957\,733.33 Step 3: Calculate the balance outstanding. Balance = FV_loan - FV_payments Balance = 2\,961\,960.00 - 2\,957\,733.33 ≈ R4226.67 The balance outstanding after Jill has paid the last R5 000 is R4226.67. 3 done, 2 left today. You're making progress.

A motor car costing R200 000 depreciated at a rate of 8% per annum on the reducing balance method. Calculate how long it took for the car to depreciate to a value of R90 000 under these conditions.

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A motor car costing R200 000 depreciated at a rate of 8% per annum on the reducing balance method. Calculate how long it took for the car to depreciate to a value of R90 000 under these conditions.

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