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b) Calculate the total amount that must be repaid if you choose Option 2 (compound interest) c) Compare the two options and justify which repayment option is cheaper.

Mathematics

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Answer

b

R14 693.28\text{R}14\,693.28

Step-by-step solution

6 steps

To solve this problem, the principal amount, interest rate, and time period for Option 2 are required. This information is not visible in the provided image. Therefore, I will make the following assumptions to demonstrate the calculation:

  • Principal amount (P)=R10 000(P) = \text{R}10\,000
  • Annual interest rate (i)=8%(i) = 8\% or 0.08
  • Number of years (n)=5(n) = 5

Part b· 3 steps

Calculate the total amount that must be repaid if you choose Option 2 (compound interest).

  1. Write down the compound interest formula.

    The formula for the total amount (A)(A) with compound interest is:

    A=P(1+i)nA = P(1 + i)^n

    where PP is the principal amount, ii is the annual interest rate, and nn is the number of years.

  2. Substitute the assumed values into the formula.

    A=10 000(1+0.08)5A = 10\,000(1 + 0.08)^5
  3. Calculate the total amount.

    A=10 000(1.08)5A = 10\,000(1.08)^5 A=10 000(1.4693280768)A = 10\,000(1.4693280768) A=R14 693.28A = \text{R}14\,693.28

    The total amount to be repaid for Option 2 is R14 693.28\boxed{\text{R}14\,693.28}.

Part c· 3 steps

Compare the two options and justify which repayment option is cheaper.

To compare the options, we need the total repayment amount for Option 1. Since this is not provided, I will assume Option 1 is a simple interest loan with the same principal, rate, and time as assumed for Option 2.

  1. Calculate the total amount for Option 1 (Simple Interest, based on assumptions).

    Assumptions: P=R10 000P = \text{R}10\,000, r=8%=0.08r = 8\% = 0.08, t=5t = 5 years.
    The formula for simple interest total amount (A1)(A_1) is:

    A1=P(1+rt)A_1 = P(1 + rt) A1=10 000(1+0.08×5)A_1 = 10\,000(1 + 0.08 \times 5) A1=10 000(1+0.40)A_1 = 10\,000(1 + 0.40) A1=10 000(1.40)A_1 = 10\,000(1.40) A1=R14 000A_1 = \text{R}14\,000
  2. Compare the total amounts for Option 1 and Option 2.

    Total for Option 1: R14 000\text{R}14\,000
    Total for Option 2: R14 693.28\text{R}14\,693.28

  3. Justify which option is cheaper.

    Comparing the two amounts, R14 000<R14 693.28\text{R}14\,000 < \text{R}14\,693.28.
    Therefore, Option 1 is cheaper. This is because simple interest calculates interest only on the principal amount, whereas compound interest calculates interest on the principal and also on the accumulated interest from previous periods, leading to a higher total repayment over time.

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