Aaron borrows a lean of $4000000 from Aspire Bank at an interest rate of 4% per annum, compounded monthly. For each successive month after the day when the loan is taken, loan interest is calculated and then a monthly instalment of $20000 is immediately paid to the bank until the loan is fully repaid. Find the number of months for Aaron to fully relay the loan

To solve this problem, we'll need to set up a model that accounts for the monthly compounding interest and the monthly payments Aaron makes.

### Step 1: Determine the monthly interest rate.
Since the interest rate is compounded monthly, the monthly interest rate is:
\[
\text{Monthly interest rate} = \frac{4\%}{12} = \frac{0.04}{12} \approx 0.0033333
\]

### Step 2: Establish the recurrence relation.
Each month, the interest is added to the remaining loan balance, and then the payment is subtracted. Let $ B_n $ represent the loan balance at the end of the $ n $-th month after the payment is made. The balance at the start of the $ n $-th month before the payment is made is $ B_{n-1} $. The recurrence relation is:

\[
B_n = B_{n-1} \times (1 + 0.0033333) - 20000
\]

### Step 3: Compute the number of months required.
We need to find the smallest $ n $ such that $ B_n \leq 0 $.

Let's solve for $ n $ using a formula for the balance after $ n $ months:

\[
B_n = P \times (1 + r)^n - \frac{M \times ((1 + r)^n - 1)}{r}
\]
Where:
- $ P = 4000000 $ (initial principal)
- $ r = 0.0033333 $ (monthly interest rate)
- $ M = 20000 $ (monthly payment)
- $ B_n = 0 $ (when the loan is fully paid off)

Set $ B_n = 0 $ and solve for $ n $:

\[
0 = 4000000 \times (1 + 0.0033333)^n - \frac{20000 \times ((1 + 0.0033333)^n - 1)}{0.0033333}
\]

Now, let's calculate this numerically to find the number of months.It seems there was an issue with solving the equation directly. Instead, we can calculate the loan balance month by month iteratively until it reaches zero. Let's do that to find the number of months required for Aaron to fully repay the loan.Aaron will take **331 months** to fully repay the loan.

Would you like more details on the calculations or have any other questions?

Here are 8 related questions you might find interesting:
1. How much total interest does Aaron pay over the 331 months?
2. What would happen if the interest rate increased to 5% per annum?
3. How many months would it take if Aaron increased his monthly payment to $30,000?
4. What is the remaining balance after 100 months?
5. How much does Aaron pay in interest during the first year?
6. How does the repayment period change with bi-weekly payments instead of monthly?
7. Can the loan be repaid in less than 20 years if Aaron increases his monthly payment slightly?
8. What would be the total payment if the interest were compounded annually instead of monthly?

**Tip:** If you’re repaying a loan with regular payments, increasing your payment amount even slightly can significantly reduce the total interest paid over the life of the loan.

Learn how to calculate loan repayment with monthly compounding interest. Solve the problem of Aaron repaying a $4,000,000 loan at 4% annual interest rate compounded monthly with $20,000 monthly instalments. Discover how long it takes to fully repay the loan and explore related questions.

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