https://file.aiquickdraw.com/mathbot/file/1725936269272ch720sq9.jpg

Let's solve the two problems based on the image you've provided.

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### Problem 10:
**Loan Details:**
- Loan amount: $60,000
- Interest rate: 7.7% (annual)
- Loan period: 25 years
- Monthly payment: $451.23

We can use the following formula to calculate how much of the first payment goes to interest and how much to the principal.

The interest portion of the first payment is calculated as:
\[
\text{Interest} = P \times \frac{r}{12}
\]
where:
- $ P $ is the loan amount (principal), which is $60,000
- $ r $ is the annual interest rate, 7.7% (as a decimal, 0.077)
- Dividing $ r $ by 12 converts it to a monthly interest rate.

The principal portion is the remaining part of the monthly payment:
\[
\text{Principal} = \text{Monthly Payment} - \text{Interest}
\]

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### Problem 11:
**Loan Details:**
- Loan amount: $75,000
- Interest rate: 5.3% (annual)
- Loan period: 30 years

We are asked to find:
1. The amount of the first monthly payment that goes toward repaying the principal.
2. The amount of the 121st month's payment (after 10 years) that goes toward the principal.

The formula to calculate the monthly payment for a loan is:
\[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
where:
- $ P $ is the loan amount ($75,000)
- $ r $ is the monthly interest rate (annual rate divided by 12)
- $ n $ is the total number of payments (loan term in months)

We use amortization formulas to determine the principal portion in specific months.

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Let me calculate the values for both questions.### Problem 10:
- **First month's interest**: \$385.00
- **First month's principal**: \$66.23
- **Total amount paid over 25 years**: \$135,369.00

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### Problem 11:
- **Monthly payment**: \$416.48
- **First month's principal**: \$85.23
- **Principal portion of the 121st month's payment (after 10 years)**: \$144.63

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Would you like more details or have any questions? Here are some related questions to explore:

1. How does the interest portion change over the life of a loan?
2. What are the benefits of paying extra on your mortgage each month?
3. How does the loan term impact monthly payments?
4. What is the difference between fixed and variable interest rates?
5. How can you calculate the total interest paid over the life of the loan?

**Tip**: The earlier you make extra payments on your mortgage, the more you save on interest over the loan term.

A family borrowed $60,000 to buy a house. The loan was at 7.7% and for 25 years. The monthly payments were $451.23 each. (a) How much of the first month's payment was interest, and how much was principal? (b) What was the total amount paid over the 25 years?

Learn how to calculate the interest and principal portion of the first payment for a $60,000 loan at 7.7% interest over 25 years. This step-by-step guide walks you through loan amortization concepts and key formulas to determine the total amount paid.

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