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This question involves calculating the monthly mortgage payment and the remaining balance after a specified period under given loan conditions. Let's break down each part of the problem.

### Given Information:
- **House Price**: $500,000
- **Down Payment**: $100,000
- **Loan Amount (Principal)**: $500,000 - $100,000 = $400,000
- **Loan Term**: 30 years (or 360 months)
- **Annual Interest Rate**: 9% compounded monthly

The monthly interest rate, $ i $, is:
\[
i = \frac{9\%}{12} = 0.75\% = 0.0075
\]

### Part (a): Monthly Payment Calculation
The formula for the monthly payment $ M $ on an amortizing loan is:
\[
M = P \frac{i (1 + i)^n}{(1 + i)^n - 1}
\]
where:
- $ P $ = $400,000 (principal),
- $ i $ = 0.0075 (monthly interest rate),
- $ n $ = 360 (total number of payments).

### Part (b): Remaining Balance after 10 Years
After 10 years (or 120 payments), the remaining balance $ B $ can be calculated using the balance formula:
\[
B = P \frac{(1 + i)^n - (1 + i)^p}{(1 + i)^n - 1}
\]
where:
- $ P $ = $400,000,
- $ i $ = 0.0075,
- $ n $ = 360,
- $ p $ = 120.

Let's calculate each part.### Solutions:

1. **Monthly Payment (Part a):** Julie's monthly payment is approximately **$3,218.49**.

2. **Remaining Balance after 10 Years (Part b):** After 120 payments (10 years), the remaining balance is approximately **$357,718.98**.

Would you like a detailed breakdown of these calculations or have any questions?

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### Related Questions:
1. How would the monthly payment change if the interest rate were 7% instead of 9%?
2. What would be the remaining balance if Julie decided to pay off the loan after 15 years instead of 10?
3. How much interest would Julie pay in total if she keeps the loan for the full 30 years?
4. What is the effect of a larger down payment on the monthly payment and total interest paid?
5. How does changing the loan term to 20 years affect the monthly payment?

### Tip:
When calculating mortgage payments, always check the compounding frequency of the interest rate, as it affects the monthly rate.

Julie a acheté une maison pour 500 000 $ et a versé une mise de fonds de 100 000 $. Elle a obtenu un prêt sur 30 ans pour le montant restant. Les paiements étaient effectués mensuellement. Le taux d'intérêt annuel nominal était de 9 %, capitalisé mensuellement. Après 10 ans (120 versements), elle a décidé de payer le solde restant du prêt. 

a. Quelle était la mensualité de son prêt ?
b. Que doit-elle payer (en plus de sa 120e mensualité habituelle) pour payer le solde restant de son prêt ?

Learn how to calculate the monthly mortgage payment and remaining balance after 10 years on a $400,000 loan with a 9% annual interest rate compounded monthly. This problem demonstrates key concepts in amortization and compound interest, ideal for Grades 11-12 students.

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