To calculate the monthly payment for a loan and the total interest paid, we'll use the following steps for each interest rate and loan period:

1. Monthly Payment Formula:

The formula for calculating the monthly payment on a mortgage is the amortization formula:

$M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$ Where:

• $M$ is the monthly payment.
• $P$ is the loan principal (amount borrowed).
• $r$ is the monthly interest rate (annual interest rate divided by 12).
• $n$ is the number of monthly payments (loan term in years multiplied by 12).

2. Interest Calculation:

The total interest paid is the difference between the total amount paid over the life of the loan and the principal:

$\text{Total Interest} = M \times n - P$

Breakdown of Inputs:

• Home Price = $50,000
• 20% down payment means the loan principal $P$ is 80% of $50,000: $P = 50,000 \times 0.8 = 40,000$
• Loan Term for 25 years (300 months) and 30 years (360 months).

Step-by-Step Calculation:

Case 1: 3% Interest Rate, 25 Years

1. Monthly Interest Rate: $r = \frac{3\%}{12} = 0.0025$
2. Number of Payments: $n = 25 \times 12 = 300$

$M = 40,000 \times \frac{0.0025(1 + 0.0025)^{300}}{(1 + 0.0025)^{300} - 1}$

Case 2: 3.5% Interest Rate, 25 Years

1. Monthly Interest Rate: $r = \frac{3.5\%}{12} = 0.002917$
2. Number of Payments: $n = 300$

Case 3: 3.75% Interest Rate, 25 Years

1. Monthly Interest Rate: $r = \frac{3.75\%}{12} = 0.003125$
2. Number of Payments: $n = 300$

Case 4: 4% Interest Rate, 25 Years

1. Monthly Interest Rate: $r = \frac{4\%}{12} = 0.003333$
2. Number of Payments: $n = 300$

Case 5: 3% Interest Rate, 30 Years

1. Monthly Interest Rate: $r = \frac{3\%}{12} = 0.0025$
2. Number of Payments: $n = 360$

Case 6: 4% Interest Rate, 30 Years

1. Monthly Interest Rate: $r = \frac{4\%}{12} = 0.003333$
2. Number of Payments: $n = 360$

Solving These:

Let me compute the exact values for you.Here are the calculations based on the given conditions:

Monthly Payments for 25 Years:

1. 3% interest: $189.68
2. 3.5% interest: $200.25
3. 3.75% interest: $205.65
4. 4% interest: $211.13

Total Interest Paid Over 25 Years:

1. 3% interest: $16,905.36
2. 3.5% interest: $20,074.83
3. 3.75% interest: $21,695.74
4. 4% interest: $23,340.42

Monthly Payments for 30 Years:

1. 3% interest: $207.11
2. 4% interest: $287.48

Total Interest Paid Over 30 Years:

1. 3% interest: $20,710.98
2. 4% interest: $28,747.80

Savings in Interest Cost:

1. Between 3% and 4% (25 years): $6,435.06
2. Between 3% and 4% (30 years): $8,036.82

Key Takeaways:

• You save more on interest with a lower interest rate, especially over a longer period (30 years).
• Extending the loan from 25 years to 30 years increases the total interest, even with the same interest rate.

Would you like further details or have any questions?

Here are 5 questions you might find relevant:

1. How does down payment size affect monthly payments?
2. What if the loan term was shorter, say 20 years?
3. How would bi-weekly payments affect interest savings?
4. What happens if you make extra payments monthly?
5. What are the effects of a variable interest rate?

Tip: Making additional payments towards the principal early in the loan can significantly reduce total interest paid!