SOLUTION: Aya and Harumi would like to buy a house and their dream house costs $500,000. They have $50,000 saved up for a down payment but would still need to take out a mortgage loan for t

Question 1173889: Aya and Harumi would like to buy a house and their dream house costs $500,000. They have $50,000 saved up for a down payment but would still need to take out a mortgage loan for the remaining $450,000 and they’re not sure whether they could afford the monthly loan payments. The bank has offered them an interest rate of 4.25%, compounded monthly.
Q1 How much would they have to be able to afford to pay each month in order to pay off their mortgage in 25 years?
Q2 What is the total amount that would be paid to the lender after 25 years of payments?
Q3 What if Aya and Harumi could only afford a monthly payment of $2,000? What would be the maximum mortgage amount they could afford to borrow from the bank, if all the other conditions were the same?
Q4 What is the total amount that would be paid to the lender over 25 years?
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's break down this mortgage calculation problem step-by-step.
**Q1: Monthly Payment for a $450,000 Loan (25 years, 4.25%)**
* **Loan Amount (P):** $450,000
* **Annual Interest Rate (r):** 4.25% or 0.0425
* **Monthly Interest Rate (i):** 0.0425 / 12 ≈ 0.00354167
* **Number of Payments (n):** 25 years * 12 months/year = 300 months
We'll use the mortgage payment formula:
M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1]
Where:
* M = Monthly payment
* P = Principal loan amount
* i = Monthly interest rate
* n = Total number of payments
Let's calculate:
M = 450000 [ 0.00354167(1 + 0.00354167)^300 ] / [ (1 + 0.00354167)^300 - 1]
M = 450000 [ 0.00354167(2.871036) ] / [ 2.871036 - 1 ]
M = 450000 [ 0.010196 ] / [ 1.871036 ]
M = 450000 * 0.00545
M ≈ $2,452.50
Therefore, Aya and Harumi would have to afford to pay approximately **$2,452.50** each month.
**Q2: Total Amount Paid to the Lender**
* Total Paid = Monthly Payment * Number of Payments
* Total Paid = $2,452.50 * 300
* Total Paid = $735,750
The total amount paid to the lender would be **$735,750**.
**Q3: Maximum Mortgage Amount with $2,000 Monthly Payment**
* **Monthly Payment (M):** $2,000
* **Monthly Interest Rate (i):** 0.00354167
* **Number of Payments (n):** 300
We'll rearrange the mortgage payment formula to solve for P:
P = M [ (1 + i)^n - 1 ] / [ i(1 + i)^n ]
P = 2000 [ (1 + 0.00354167)^300 - 1 ] / [ 0.00354167(1 + 0.00354167)^300 ]
P = 2000 [ 1.871036 ] / [ 0.010196 ]
P = 2000 * 183.506
P ≈ $367,012
The maximum mortgage amount they could afford to borrow would be approximately **$367,012**.
**Q4: Total Amount Paid to the Lender (with $2,000 Payment)**
* Total Paid = Monthly Payment * Number of Payments
* Total Paid = $2,000 * 300
* Total Paid = $600,000
The total amount paid to the lender would be **$600,000**.

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