Amelia borrows $40000 at a rate of 4.8% p.a, compounding monthly. After 3 months, Amelia repays the loan all at once. How much money does she pay back in total?
Round your answer to the nearest cent.

To solve this problem, we'll calculate the total amount that Amelia needs to repay after 3 months, given that the interest is compounded monthly.

### Step 1: Identify the given variables
- **Principal (P):** $40,000
- **Annual interest rate (r):** 4.8% (or 0.048 as a decimal)
- **Number of compounding periods per year (n):** 12 (since it's compounded monthly)
- **Time (t):** 3 months (or $ \frac{3}{12} = 0.25 $ years)

### Step 2: Calculate the monthly interest rate
The monthly interest rate $ r_{monthly} $ is:
\[
r_{monthly} = \frac{r}{n} = \frac{0.048}{12} = 0.004
\]

### Step 3: Use the compound interest formula
The compound interest formula is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $ A $ is the amount after time $ t $
- $ P $ is the principal
- $ r $ is the annual interest rate
- $ n $ is the number of compounding periods per year
- $ t $ is the time in years

Substituting the values:
\[
A = 40000 \left(1 + 0.004\right)^{12 \times 0.25}
\]
\[
A = 40000 \left(1.004\right)^{3}
\]

### Step 4: Calculate the amount
\[
A = 40000 \times (1.012018)
\]
\[
A \approx 40000 \times 1.012018 = 40480.72
\]

### Final Answer
Amelia needs to repay approximately **$40,480.72** after 3 months.

Would you like a deeper explanation or have any questions? Here are some additional related questions:

1. What would the repayment amount be after 6 months?
2. How does the interest rate affect the repayment amount?
3. What if the loan was compounded quarterly instead of monthly?
4. How much would Amelia owe if the interest was simple instead of compound?
5. What is the effective annual rate (EAR) for this loan?

**Tip:** Always ensure you convert your time period into years when working with compound interest formulas.

Explore how to calculate the total repayment for Amelia's $40,000 loan with a 4.8% annual interest rate compounded monthly over 3 months. Get a detailed breakdown using the compound interest formula and understand the impact of compounding on loan repayment.

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