To solve this problem, we need to determine the number of monthly payments Jonathan needs to make to fully repay the car loan. This involves calculating the total number of payments required for a loan with fixed monthly payments and an interest rate compounded monthly.

Step 1: Identify the given variables

• Loan amount (Principal, $P$): R445,222
• Monthly payment ($R$): R12,117
• Annual interest rate ($r$): 7.947% per year
• Monthly interest rate ($i$): $\frac{7.947\%}{12}$ per month

Step 2: Convert the annual interest rate to a monthly interest rate

$i = \frac{7.947\%}{12} = \frac{0.07947}{12} \approx 0.0066225$

Step 3: Use the loan amortization formula

The loan balance after $n$ payments is given by:

$P_n = P \times (1 + i)^n - \frac{R \times ((1 + i)^n - 1)}{i}$

Since we need to find the point where the remaining balance $P_n$ becomes zero, we rearrange the formula to solve for $n$:

$P = \frac{R \times ((1 + i)^n - 1)}{i \times (1 + i)^n}$

Simplifying further, we solve for $n$:

$(1 + i)^n = \frac{R}{R - P \times i}$

$n = \frac{\log \left(\frac{R}{R - P \times i}\right)}{\log(1 + i)}$

Step 4: Substitute the values and solve for $n$

Substituting the given values:

$n = \frac{\log \left(\frac{12,117}{12,117 - 445,222 \times 0.0066225}\right)}{\log(1.0066225)}$

Let's calculate this value.Jonathan needs to make approximately 42.24 payments to repay the car loan. Since the number of payments must be a whole number, he will make 42 regular payments of R12,117, and the final (43rd) payment will be slightly smaller.

Would you like further details or have any questions?

Here are 5 related questions you might consider:

1. How much will Jonathan's final payment be?
2. What would happen if the interest rate were different?
3. How much interest will Jonathan pay over the course of the loan?
4. How would a down payment affect the number of payments?
5. What would be the impact if Jonathan increased his monthly payments?

Tip: Understanding how interest compounds on a loan can help you make better financial decisions, especially when it comes to long-term payments.