student borrowed R4800 from his father to buy a turn table and agrees to pay an amount of R5562 after fifteen months to his father in full settlement of his debt.
a) Find:
i) the effective annual interest rate, and
ii) the monthly effective interest rate
at which he borrows the money.
b) If the student would only repay his debt 18 months after borrowing the R4 800 what would he have to pay if the same rate of interest is used as in (a)(i)?

Let's solve this step-by-step:

### Part (a)

**Given Information:**
- Principal amount (loan amount): \( P = R4800 \)
- Amount to be paid after 15 months: \( A = R5562 \)
- Time period: 15 months

#### (i) Find the effective annual interest rate:

To find the effective annual interest rate, we'll first determine the monthly effective rate using the future value formula:

\[
A = P (1 + i)^n
\]

Where:
- \( A \) is the amount after time period \( n \).
- \( P \) is the principal amount.
- \( i \) is the monthly interest rate.
- \( n \) is the number of months.

Plugging in the values:

\[
5562 = 4800 (1 + i)^{15}
\]

Now, solve for \( i \):

\[
(1 + i)^{15} = \frac{5562}{4800}
\]

\[
(1 + i)^{15} = 1.15875
\]

Taking the 15th root on both sides:

\[
1 + i = (1.15875)^{\frac{1}{15}}
\]

\[
i = (1.15875)^{\frac{1}{15}} - 1
\]

Calculating the monthly effective rate:

\[
i \approx 0.009912 \text{ or } 0.9912\%
\]

To find the annual effective interest rate, we use the formula:

\[
\text{Annual rate} = (1 + i)^{12} - 1
\]

Substituting \( i \):

\[
\text{Annual rate} = (1.009912)^{12} - 1
\]

\[
\text{Annual rate} \approx 0.1257 \text{ or } 12.57\%
\]

So, the **effective annual interest rate** is approximately **12.57%**.

#### (ii) Find the monthly effective interest rate:

From the calculation above, the monthly effective interest rate \( i \) is approximately **0.9912%**.

### Part (b)

If the student repays the debt 18 months after borrowing, using the same annual rate of 12.57%, we need to find the future value:

\[
A = P (1 + i)^{n}
\]

Where:
- \( P = 4800 \)
- Monthly rate \( i = 0.9912\% \)
- Time \( n = 18 \) months

Substituting the values:

\[
A = 4800 \times (1.009912)^{18}
\]

Calculating the amount to be repaid:

\[
A \approx 4800 \times 1.18884
\]

\[
A \approx 5706.43
\]

Therefore, if the student repays the debt after 18 months, the amount he would have to pay is approximately **R5706.43**.

### Summary:

1. **Effective annual interest rate**: 12.57%
2. **Monthly effective interest rate**: 0.9912%
3. **Amount to be repaid after 18 months**: R5706.43

Do you have any questions, or would you like further details?

### Related Questions:

1. How does the interest rate affect the total repayment amount?
2. What would be the repayment amount if the loan period was extended to 24 months?
3. How would the repayment change if the interest rate was compounded quarterly instead of monthly?
4. How can the student reduce the total repayment amount?
5. What is the difference between nominal and effective interest rates?

### Tip:
Always ensure you understand the compounding frequency of an interest rate, as it significantly affects the total repayment amount.

A student borrowed R4800 from his father to buy a turntable and agrees to pay an amount of R5562 after fifteen months in full settlement. a) Find: i) the effective annual interest rate, and ii) the monthly effective interest rate at which he borrows the money. b) If the student would repay his debt 18 months after borrowing the R4800, what would he have to pay if the same rate of interest is used?

Learn how to calculate the effective annual interest rate and monthly effective interest rate for a loan of R4800. This problem also shows how to calculate the repayment amount after 18 months using a 12.57% annual interest rate. Suitable for students in Grades 10-12.

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