To solve this problem, we need to determine how many months it will take the teacher to repay the loan, given the repayment amount, interest rate, and loan amount.

Formula for Loan Repayment

The formula for the monthly repayment $P$ of a loan is derived from the formula for the present value of an annuity:

$P = \frac{r \cdot PV}{1 - (1 + r)^{-n}}$

Where:

• $P$ = monthly repayment (R700)
• $r$ = monthly interest rate (18% per annum / 12 months = 0.015 or 1.5% per month)
• $PV$ = loan amount (R12,800)
• $n$ = number of months (what we are solving for)

Rearranging the formula to solve for $n$:

$n = \frac{\ln\left(\frac{P}{P - r \cdot PV}\right)}{\ln(1 + r)}$

Substituting Values

• $P = 700$
• $PV = 12,800$
• $r = 0.015$

[ n = \frac{\ln\left(\frac{700}{700 - 0.015 \cdot 12800}\right)}{\ln(1.015)}

Given values

P = 700 # Monthly repayment PV = 12800 # Loan amount r = 0.015 # Monthly interest rate

Calculate number of months (n)

numerator = math.log(P / (P - r * PV)) denominator = math.log(1 + r) The calculation shows that it will take approximately 21.53 months to repay the loan. Since the teacher can only pay in full months, it would take 22 months to completely repay the loan.

Let me know if you'd like further explanations or calculations!

Expanding Questions:

1. How does the compounding frequency affect the total interest paid?
2. What would happen if the teacher increased the monthly payment to R800?
3. How would the repayment period change if the interest rate decreased to 15% p.a.?
4. Can you explain the difference between nominal and effective interest rates in this context?
5. What is the total amount paid over the 22 months?

Tip:

Always verify the terms of a loan to understand how changes in interest