Zinzi is buying a car today, 01/03/2020. She can afford to make monthly payments of R3100 at the end of every month, but she can only make the first payment on 01/07/2020. If she must pay interest at a rate of 11.39% per year, compounded monthly, and she wants to make exactly 13 payments, what is the cash-price of the car that she can buy?

This problem involves calculating the present value of an annuity due, where the payments are made at the end of each period, and the interest is compounded monthly.

We are given the following information:

- Monthly payment: \( R3100 \)
- Number of payments: \( 13 \)
- Annual interest rate: \( 11.39\% \)
- First payment on: \( 01/07/2020 \)
- Payments are made at the end of each month.

Since the interest is compounded monthly, we first need to convert the annual interest rate into a monthly rate:
\[
i = \frac{11.39\%}{12} = \frac{0.1139}{12} \approx 0.00949167 \, \text{(monthly rate)}
\]

Next, we use the formula for the present value \( PV \) of an ordinary annuity, which is:
\[
PV = P \times \frac{1 - (1 + i)^{-n}}{i}
\]
where:
- \( P \) is the monthly payment,
- \( i \) is the monthly interest rate,
- \( n \) is the number of payments.

### Substituting the values:

- \( P = 3100 \)
- \( i = 0.00949167 \)
- \( n = 13 \)

Let's calculate the cash price of the car that Zinzi can afford.The cash price of the car that Zinzi can afford is approximately **R37,744.80**.

Would you like more details or have any further questions?

Here are 5 related questions that might help expand on this:

1. How does the number of payments affect the cash price she can afford?
2. What would happen if the interest rate increased to 12% per year?
3. How would the cash price change if she could start making payments immediately instead of waiting?
4. What if Zinzi could make a down payment? How would that affect the loan amount and payments?
5. How do we calculate the total interest paid over the loan period?

**Tip:** Always double-check whether payments are made at the beginning or end of the period, as it can significantly affect the calculation.

Learn how to calculate the present value of an ordinary annuity to determine the cash price of a car Zinzi can afford. With 13 monthly payments of R3100, an interest rate of 11.39% compounded monthly, and the first payment on 01/07/2020, find the maximum car price she can afford. Ideal for students in Grades 10-12.

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