To determine how much money Archie owes the bank 6 years after taking the loan, we need to account for the compound interest on the loan and subtract the payments he made. Let's go through the steps.

Step 1: Calculate the amount owed after 1 year before the first payment

The loan amount is R1,200,000 with a 10% annual interest rate compounded monthly. The monthly interest rate is:

$\text{Monthly interest rate} = \frac{10\%}{12} = \frac{0.10}{12} = 0.0083333\ \text{per month}$

After 1 year (12 months), the amount owed is:

$A_1 = 1,200,000 \times \left(1 + 0.0083333\right)^{12}$

Step 2: Subtract the first payment after 1 year

After 1 year, Archie pays back R400,000, so the remaining amount owed is:

$A_1' = A_1 - 400,000$

Step 3: Calculate the amount owed after 5 years (4 years after first payment)

We need to calculate the amount owed after 5 years total (4 years after the first payment), with the interest continuing to compound monthly:

$A_5 = A_1' \times \left(1 + 0.0083333\right)^{48}$

Step 4: Subtract the second payment after 5 years

Archie pays R750,000 after 5 years, so the remaining amount owed is:

$A_5' = A_5 - 750,000$

Step 5: Calculate the amount owed after 6 years

Finally, calculate the amount owed 1 year later (at the end of year 6), with interest continuing to compound monthly:

$A_6 = A_5' \times \left(1 + 0.0083333\right)^{12}$

Let's calculate these amounts step by step.After 6 years, Archie owes the bank approximately R694,454.76.

Would you like more details on how this was calculated or have any other questions?

Here are five related questions you might find interesting:

1. How would the remaining balance change if the interest rate were 12% instead of 10%?
2. What would happen to the total amount owed if Archie made an additional payment after 2 years?
3. How much interest did Archie effectively pay after 6 years?
4. What would the remaining balance be if the interest was compounded annually instead of monthly?
5. How would the remaining balance change if the second payment were R800,000 instead of R750,000?

Tip: When dealing with loans, paying more upfront or making additional payments early on can significantly reduce the total amount of interest paid over the life of the loan.