A new university graduate determines that she can afford a car payment of
R3 500 per month. If the car manufacturer is offering a special of 2,1% annual financing rate, compounded monthly for 4 years, how much can she borrow and still have a R3 500 monthly payment?
Round your answer to the nearest ten cents.

To determine how much the university graduate can borrow, we need to use the formula for the present value of an annuity, which is commonly used to calculate the principal amount that can be borrowed given a fixed monthly payment, interest rate, and loan term.

The formula for the present value \( P \) of an annuity is:

\[
P = \frac{R \left(1 - (1 + i)^{-n}\right)}{i}
\]

Where:
- \( P \) is the principal amount (the amount to be borrowed),
- \( R \) is the monthly payment (R3,500),
- \( i \) is the monthly interest rate (annual rate divided by 12),
- \( n \) is the total number of payments (number of months).

### Given:
- **Monthly payment \( R \)**: R3,500
- **Annual interest rate**: 2.1% (compounded monthly)
- **Loan term**: 4 years

### Calculations:

1. **Convert the annual interest rate to a monthly interest rate**:
   \[
   i = \frac{2.1\%}{12} = \frac{0.021}{12} = 0.00175
   \]

2. **Calculate the total number of payments \( n \)**:
   \[
   n = 4 \text{ years} \times 12 \text{ months/year} = 48 \text{ months}
   \]

3. **Substitute the values into the formula**:
   \[
   P = \frac{3500 \left(1 - (1 + 0.00175)^{-48}\right)}{0.00175}
   \]

Let's compute this.

The graduate can borrow **R161,002.48** and still have a monthly payment of R3,500 over 4 years with an annual financing rate of 2.1%, compounded monthly.

Would you like more details or have any questions?

### Related Questions
1. How would the maximum loan amount change if the interest rate increased to 3%?
2. What if the graduate could only afford R3,000 per month? How much could they borrow then?
3. How does the length of the loan term affect the amount she can borrow?
4. What would the monthly payment be if she borrowed R200,000 under the same terms?
5. How much total interest will be paid over the 4 years?
6. What is the impact of changing from monthly to bi-weekly payments?
7. How does the present value of an annuity formula apply to mortgages?
8. What is the formula to calculate the future value of an annuity?

**Tip:** When calculating loan payments, it's essential to understand how interest compounding affects the total cost of borrowing and the amount you can afford to borrow.

Learn how to calculate how much you can borrow with a fixed monthly payment and annual financing rate using the present value of annuity formula. Example: A university graduate can borrow R161,002.48 with a monthly payment of R3,500 over 4 years at 2.1% annual interest rate, compounded monthly.

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