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# Comparing Total Cost of Borrowing R5,425 for 2 Years under Simple and Compound Interest

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Question: Tebogo's parents decided that since they cannot afford to cover the total cost of R 14 000 needed for the trip to Cape Town. They decided to borrow the remaining balance of R5 425 at Mbibane Bank. The bank offered them the following options. Option 1:. R 5 425 at 9.7 % per , simple interest for 2 years . Option 2: R5 425 at 9.5% per year compounded yearly for 2 years. Tebogo's father said that option 1 will be the better option. Verify with calculations if his statement is valid. (8)

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To determine which loan option is better, we need to compare the total amount of money that will be owed under each option after 2 years. We'll start with calculating the total amount for each option. ### Option 1: Simple Interest The formula for simple interest is: $A = P(1 + rt)$ where: - $$A$$ is the total amount owed - $$P$$ is the principal amount (R5 425) - $$r$$ is the annual interest rate (9.7% or 0.097) - $$t$$ is the time in years (2 years) $A = 5,425(1 + 0.097 \times 2)$ $A = 5,425(1 + 0.194)$ $A = 5,425 \times 1.194$ $A = 6,479.45$ So, the total amount to be repaid under Option 1 is R6,479.45. ### Option 2: Compound Interest The formula for compound interest is: $A = P \left(1 + \frac{r}{n}\right)^{nt}$ where: - $$A$$ is the total amount owed - $$P$$ is the principal amount (R5 425) - $$r$$ is the annual interest rate (9.5% or 0.095) - $$n$$ is the number of times the interest is compounded per year (1 in this case since it's compounded yearly) - $$t$$ is the time in years (2 years) Since the interest is compounded yearly, $$n = 1$$: $A = 5,425 \left(1 + \frac{0.095}{1}\right)^{1 \times 2}$ $A = 5,425 \left(1 + 0.095\right)^{2}$ $A = 5,425 \left(1.095\right)^{2}$ $A = 5,425 \left(1.199025\right)$ $A = 6,504.91$ (approximately) So, the total amount to be repaid under Option 2 is R6,504.91. ### Comparing the Options - Option 1 total repayment: R6,479.45 - Option 2 total repayment: R6,504.91 ### Conclusion Since R6,479.45 < R6,504.91, Option 1 with simple interest is indeed the cheaper option. Tebogo’s father is correct in stating that Option 1 is the better option.

Aug. 12, 2024, 6:49 a.m.

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