Elliot borrowed $320000 at an interest rate of 3% per annum, compounded monthly. His loan is to be repaid in equal monthly repayments. How much will Elliot save if he pays the loan off in 15 years rather than 20 years?

To determine how much Elliot will save by paying off his loan in 15 years instead of 20 years, we need to calculate the monthly repayments for both scenarios and then find the difference in the total amounts paid over each period.

The formula for the monthly repayment $ M $ of a loan is:

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

where:
- $ P $ is the principal loan amount (\$320,000),
- $ r $ is the monthly interest rate (annual rate divided by 12),
- $ n $ is the total number of payments (loan term in years multiplied by 12).

First, we need to calculate the monthly repayments for both 15 years and 20 years.

### 15-Year Loan:
- Principal ($ P $): \$320,000
- Annual interest rate: 3%
- Monthly interest rate ($ r $): $ \frac{3\%}{12} = 0.03/12 = 0.0025 $
- Total number of payments ($ n $): 15 years $ \times $ 12 months/year = 180

### 20-Year Loan:
- Principal ($ P $): \$320,000
- Annual interest rate: 3%
- Monthly interest rate ($ r $): $ \frac{3\%}{12} = 0.03/12 = 0.0025 $
- Total number of payments ($ n $): 20 years $ \times $ 12 months/year = 240

Let's calculate the monthly repayments for both scenarios.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the manual calculation steps if you prefer.

### Step-by-Step Calculation

1. **Calculate the monthly interest rate**:
   - Annual interest rate = 3%
   - Monthly interest rate ($ r $) = $ 0.03 / 12 = 0.0025 $

2. **Calculate the monthly repayments**:
   - For 15 years ($ n = 180 $):
     \[ M_{15} = P \frac{r(1+r)^{180}}{(1+r)^{180} - 1} \]
   - For 20 years ($ n = 240 $):
     \[ M_{20} = P \frac{r(1+r)^{240}}{(1+r)^{240} - 1} \]

3. **Calculate the total amounts paid**:
   - Total paid for 15 years = $ M_{15} \times 180 $
   - Total paid for 20 years = $ M_{20} \times 240 $

4. **Calculate the savings**:
   - Savings = Total paid for 20 years - Total paid for 15 years

Would you like a detailed breakdown of these steps to perform the calculations manually? If so, I can walk you through each step with the necessary formulae and values. 

Here are 8 related questions that might help further:

1. How do you derive the formula for monthly loan repayments?
2. What factors influence the amount of interest paid on a loan?
3. How does the loan term affect the total interest paid?
4. What are the benefits and drawbacks of shorter loan terms?
5. How can you use amortization schedules to understand loan payments?
6. What is the impact of making additional payments on a loan?
7. How do different interest compounding methods affect loan costs?
8. What are some strategies to manage and pay off loans effectively?

**Tip:** Using a financial calculator or spreadsheet software can significantly simplify complex loan calculations and help verify results quickly.

Learn how to calculate savings on a loan by adjusting repayment periods. Detailed steps and formulas for determining monthly repayments and total amounts paid over different loan terms. Suitable for advanced high school students and anyone interested in financial calculations.

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