Priya borrows $45,000 from the bank to finance a home renovation. Interest is charged on this loan at a rate of 5.9% p.a., compounded monthly. If Priya pays $680 a month towards the loan, how much does she owe the bank after 5 years? (1 mark)

To determine how much Priya owes the bank after 5 years, we can use the formula for the balance remaining on an amortizing loan:

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

where:
- $ P $ is the initial loan amount ($45,000),
- $ r $ is the monthly interest rate (annual rate divided by 12),
- $ n $ is the total number of payments (number of years multiplied by 12),
- $ A $ is the monthly payment ($680).

### Step 1: Calculate the monthly interest rate

The annual interest rate is 5.9%, so the monthly interest rate $ r $ is:
\[
r = \frac{5.9\%}{12} = \frac{0.059}{12} \approx 0.0049167
\]

### Step 2: Calculate the total number of payments

Priya is paying for 5 years, so the total number of payments $ n $ is:
\[
n = 5 \times 12 = 60
\]

### Step 3: Use the formula for the remaining balance

The remaining balance $ B $ after $ n $ payments can be found using:
\[
B = P(1 + r)^n - \frac{A \left[(1 + r)^n - 1\right]}{r}
\]

Substituting the values into the formula:
\[
B = 45000(1 + 0.0049167)^{60} - \frac{680 \times \left[(1 + 0.0049167)^{60} - 1\right]}{0.0049167}
\]

### Step 4: Compute the balance

Let's compute this step by step.

1. Calculate $ (1 + r)^{60} $:
\[
(1 + 0.0049167)^{60} \approx 1.349353
\]

2. Calculate the amount $ P(1 + r)^{60} $:
\[
45000 \times 1.349353 \approx 60720.89
\]

3. Calculate the annuity factor $ \frac{(1 + r)^{60} - 1}{r} $:
\[
\frac{1.349353 - 1}{0.0049167} \approx 71.0918
\]
4. Calculate the total paid amount over 60 months:
\[
680 \times 71.0918 \approx 48341.42
\]

5. Finally, calculate the remaining balance $ B $:
\[
B \approx 60720.89 - 48341.42 \approx 12379.47
\]

So, after 5 years, Priya owes the bank approximately **$12,379.47**.

Would you like more details or have any questions? 

Here are 5 related questions you might consider:
1. How would the remaining balance change if the interest rate were different?
2. What would be the remaining balance if Priya paid $750 per month instead of $680?
3. How much total interest has Priya paid after 5 years?
4. How long would it take to pay off the loan if the monthly payment were increased to $800?
5. What is the total amount Priya would pay over the full term of the loan?

**Tip:** To minimize interest payments, consider increasing your monthly payment or making extra payments on your loan.

Learn how to calculate the remaining balance on a loan after 5 years with monthly payments using the amortization formula. Priya borrows $45,000 at 5.9% interest compounded monthly to finance a home renovation. She pays $680 per month. Discover the detailed step-by-step solution and understand the financial implications.

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