To solve this problem, we need to determine how much Cindy will still owe the bank immediately before making her 4th payment. Given that payments are made at the beginning of each month, we will use the formula for the present value of an annuity due.

Step 1: Define the given values

• Monthly payment (P): R1260
• Number of payments (n): 5
• Annual interest rate: 13.357%
• Monthly interest rate (r): $\frac{13.357\%}{12}$

Step 2: Calculate the monthly interest rate

$r = \frac{13.357\%}{12} = \frac{0.13357}{12} \approx 0.01113$

Step 3: Calculate the present value of the remaining payments

Since we want to know how much Cindy owes before making her 4th payment, we will consider the present value of the remaining two payments (4th and 5th).

The formula for the present value of an annuity due (where payments are made at the beginning of each period) is:

$PV_{\text{due}} = P \times \left[\frac{1 - (1 + r)^{-m}}{r} \right] \times (1 + r)$

Where:

• $P = 1260$ (the payment amount)
• $r \approx 0.01113$ (monthly interest rate)
• $m = 2$ (number of remaining payments after the 3rd payment)

Plugging in the values:

$PV_{\text{due}} = 1260 \times \left[\frac{1 - (1 + 0.01113)^{-2}}{0.01113}\right] \times (1 + 0.01113)$

Step 4: Perform the calculations

First, calculate the part inside the brackets:

$\frac{1 - (1 + 0.01113)^{-2}}{0.01113} = \frac{1 - (1.01113)^{-2}}{0.01113} \approx \frac{1 - 0.97807}{0.01113} \approx \frac{0.02193}{0.01113} \approx 1.970$

Next, multiply by $P$ and adjust for the annuity due:

$PV_{\text{due}} = 1260 \times 1.970 \times 1.01113 \approx 1260 \times 1.991 \approx 2509.32$

Conclusion

Cindy will owe the bank R2509.32 immediately before making her 4th payment.

Would you like further details on any step, or do you have any questions?

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Here are 5 related questions you might find helpful:

1. How would the amount owed change if Cindy made payments at the end of each month instead?
2. What is the total amount of interest Cindy will pay over the course of the loan?
3. How would a different interest rate affect the amount owed before the 4th payment?
4. What would be the present value of the entire loan if there were more payments?
5. How does the compounding frequency affect the loan repayment?

Tip: When calculating the present value of an annuity due, remember that payments are made at the beginning of each period, which slightly increases the present value compared to an ordinary annuity.