A family has a $88,570​, 30​-year mortgage at 5.7% compounded monthly. Find the monthly payment. Also find the unpaid balance after the following periods of time.
​(A) 10 years    ​ (B) 20 years    ​ (C) 25 years


To solve this problem, we need to apply the formula for the monthly mortgage payment and the formula for the remaining balance after a certain period of time. 

### 1. Monthly Payment Calculation
The formula for calculating the monthly payment $ M $ on a mortgage is:

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

where:
- $ P $ is the principal (loan amount),
- $ r $ is the monthly interest rate,
- $ n $ is the total number of payments (months).

**Given:**
- Principal $ P = 88,570 $ dollars,
- Annual interest rate $ = 5.7\% $, so monthly interest rate $ r = \frac{5.7\%}{12} = \frac{0.057}{12} $,
- Loan term $ = 30 $ years, so total payments $ n = 30 \times 12 = 360 $ months.

### 2. Unpaid Balance Formula
The unpaid balance after $ t $ years is calculated using:

\[
B_t = P \times \frac{(1 + r)^n - (1 + r)^{t \times 12}}{(1 + r)^n - 1}
\]

where:
- $ B_t $ is the unpaid balance after $ t $ years,
- $ t \times 12 $ is the number of payments already made after $ t $ years,
- All other terms remain the same.

We will compute the monthly payment first, then calculate the unpaid balance after 10, 20, and 25 years. Let me calculate these values.### Results:

1. **Monthly Payment:**
   The monthly mortgage payment is approximately **$514.06**.

2. **Unpaid Balance:**
   - After **10 years** (120 payments), the unpaid balance is approximately **$73,517.89**.
   - After **20 years** (240 payments), the unpaid balance is approximately **$46,937.62**.
   - After **25 years** (300 payments), the unpaid balance is approximately **$26,782.90**.

Would you like further details or explanations?

Here are some related questions:
1. How is compound interest calculated over time?
2. What happens if the interest rate changes during the mortgage period?
3. How would early payments affect the balance?
4. How does the term length affect the monthly payments and total interest paid?
5. What options exist to pay off a mortgage faster?

**Tip:** Making extra payments towards your mortgage principal can reduce your overall interest and shorten the loan term.

A family has a $88,570​, 30​-year mortgage at 5.7% compounded monthly. Find the monthly payment. Also find the unpaid balance after the following periods of time. (A) 10 years (B) 20 years (C) 25 years.

Learn how to calculate the monthly payment for a $88,570, 30-year mortgage at 5.7% interest compounded monthly. This example provides a step-by-step solution, including the unpaid balance after 10, 20, and 25 years. Suitable for high school and college-level students studying finance or algebra.

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