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.