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Calculate the future value of an investment and the profit, and determine a monthly installment for a financed purchase.

Mathematics

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Answer

4.1

R152.89\text{R}152.89

Step-by-step solution

5 steps

Step 1: Identify the given values.
Principal amount P=R10 000P = \text{R}10\,000
Interest rate i=11%=0.11i = 11\% = 0.11
Number of years n=3n = 3

Step 2: Calculate the future value using the compound interest formula.
The formula for compound interest is A=P(1+i)nA = P(1 + i)^n.

A=R10 000(1+0.11)3A = \text{R}10\,000(1 + 0.11)^3 A=R10 000(1.11)3A = \text{R}10\,000(1.11)^3 A=R10 000(1.367631)A = \text{R}10\,000(1.367631) A=R13 676.31A = \text{R}13\,676.31

Step 3: Calculate the profit.
Profit = Future Value - Principal Amount

Profit=A−P\text{Profit} = A - P Profit=R13 676.31−R10 000\text{Profit} = \text{R}13\,676.31 - \text{R}10\,000 Profit=R3 676.31\text{Profit} = \text{R}3\,676.31

The profit Roxanne will have is R3 676.31\boxed{\text{R}3\,676.31}.

Part 4.1· 5 steps

His monthly instalment over the 3-year period.

  1. Calculate the deposit amount.

    Laptop price = R3 999
    Deposit percentage = 15%

    Deposit=0.15×R3 999\text{Deposit} = 0.15 \times \text{R}3\,999 Deposit=R599.85\text{Deposit} = \text{R}599.85

  2. Calculate the amount to be financed.

    Amount financed=Laptop price−Deposit\text{Amount financed} = \text{Laptop price} - \text{Deposit} Amount financed=R3 999−R599.85\text{Amount financed} = \text{R}3\,999 - \text{R}599.85 Amount financed=R3 399.15\text{Amount financed} = \text{R}3\,399.15

  3. Determine the monthly interest rate and number of payments.

    Annual interest rate iannual=18%=0.18i_{annual} = 18\% = 0.18
    Monthly interest rate imonthly=0.1812=0.015i_{monthly} = \frac{0.18}{12} = 0.015
    Number of years nyears=3n_{years} = 3
    Number of months nmonths=3×12=36n_{months} = 3 \times 12 = 36

  4. Calculate the monthly instalment for the financed amount using the present value annuity formula.

    The formula for the present value of an annuity is P=x[1−(1+i)−n]iP = \frac{x[1 - (1 + i)^{-n}]}{i}, where P is the present value (amount financed), x is the monthly instalment, i is the monthly interest rate, and n is the number of months.
    Solving for x:

    x=P⋅i1−(1+i)−nx = \frac{P \cdot i}{1 - (1 + i)^{-n}} x=R3 399.15×0.0151−(1+0.015)−36x = \frac{\text{R}3\,399.15 \times 0.015}{1 - (1 + 0.015)^{-36}} x=50.987251−(1.015)−36x = \frac{50.98725}{1 - (1.015)^{-36}} x=50.987251−0.58509836x = \frac{50.98725}{1 - 0.58509836} x=50.987250.41490164x = \frac{50.98725}{0.41490164} x≈R122.89x \approx \text{R}122.89

    This is the monthly instalment for the loan repayment.

  5. Add the monthly insurance fee to find the total monthly instalment.

    Monthly insurance fee = R30

    Total monthly instalment=Loan instalment+Insurance fee\text{Total monthly instalment} = \text{Loan instalment} + \text{Insurance fee} Total monthly instalment=R122.89+R30\text{Total monthly instalment} = \text{R}122.89 + \text{R}30 Total monthly instalment=R152.89\text{Total monthly instalment} = \text{R}152.89

    His monthly instalment is R152.89\boxed{\text{R}152.89}.

Part 4.2

The total amount he will have paid at the end

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