A 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 break down the problem and solve it step by step:

### Given Information:
- Principal (Initial loan amount): \( P = R4800 \)
- Final repayment amount after 15 months: \( A = R5562 \)
- Time period for repayment: \( t = 15 \) months

We will assume that the interest compounds monthly for simplicity unless otherwise stated.

### (a) Finding the interest rates:

#### (i) Effective annual interest rate

The formula for compound interest is given by:

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

Where:
- \( A \) is the final amount,
- \( P \) is the principal amount,
- \( i \) is the monthly effective interest rate,
- \( t \) is the time period in months.

For the given situation:
\[
5562 = 4800 \times (1 + i)^{15}
\]

First, we need to solve for \( i \), the monthly effective interest rate. Rearranging the equation:

\[
(1 + i)^{15} = \frac{5562}{4800}
\]
\[
(1 + i)^{15} = 1.15875
\]

Now, take the 15th root of both sides:

\[
1 + i = (1.15875)^{\frac{1}{15}}
\]
\[
1 + i = 1.009912
\]
\[
i = 0.009912
\]

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

#### (ii) Effective annual interest rate

To find the **effective annual interest rate**, we use the relationship between the monthly and annual interest rates. The effective annual interest rate \( r_{\text{annual}} \) is related to the monthly interest rate \( i \) by the formula:

\[
r_{\text{annual}} = (1 + i)^{12} - 1
\]

Substitute \( i = 0.009912 \):

\[
r_{\text{annual}} = (1 + 0.009912)^{12} - 1
\]
\[
r_{\text{annual}} = 1.12574 - 1
\]
\[
r_{\text{annual}} = 0.12574
\]

Thus, the **effective annual interest rate** is approximately \( 12.57\% \).

### (b) Amount to repay after 18 months

If the student repays the loan after 18 months instead of 15 months, using the same monthly interest rate \( i = 0.009912 \), the formula for the final amount becomes:

\[
A = P \times (1 + i)^{18}
\]

Substitute \( P = 4800 \) and \( i = 0.009912 \):

\[
A = 4800 \times (1 + 0.009912)^{18}
\]
\[
A = 4800 \times (1.009912)^{18}
\]
\[
A = 4800 \times 1.184454
\]
\[
A = 5684.38
\]

Thus, the student would have to repay approximately **R5684.38** after 18 months if the same interest rate is applied.

### Summary:
- (a)(i) The **effective annual interest rate** is approximately **12.57%**.
- (a)(ii) The **monthly effective interest rate** is approximately **0.9912%**.
- (b) The student would have to repay **R5684.38** after 18 months.

---

Would you like more details or have any questions on this solution? Here are 5 follow-up questions that can deepen your understanding:

1. What formula can you use to calculate compound interest for any time period?
2. How does increasing the repayment time affect the total amount repaid in compound interest scenarios?
3. What is the difference between the nominal interest rate and the effective interest rate?
4. Can you calculate the amount if the student repays the loan after 12 months at the same interest rate?
5. How would the repayment amount change if the interest compounded quarterly instead of monthly?

**Tip:** Always clarify whether interest is compounded annually, monthly, or at some other frequency when dealing with financial calculations.

A 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 R4800, what would he have to pay if the same rate of interest is used as in (a)(i)?

Solve a problem involving compound interest where a student borrows R4800 and repays R5562 after 15 months. Learn how to calculate the effective annual interest rate, the monthly interest rate, and the repayment amount for 18 months using compound interest formulas. Suitable for Grades 10-12.

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