Jonathan is buying a car today. The car costs R445222 and he can afford to make monthly payments of R12117 at the end of every month. He must pay interest at a rate of 7.947% per year compounded monthly. If the last payment Jonathan makes can be larger than R12117, how many payments must Jonathan make to repay the car loan?

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.

Learn how to calculate the number of payments needed to repay a car loan using compound interest. Jonathan plans to buy a car costing R445,222 with monthly payments of R12,117 at an interest rate of 7.947% per year compounded monthly. Find out how many payments he needs to make, including details on final payment variations.

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