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Sagot :
Let's break down the problem and compute the required values step-by-step:
1. Monthly Gross Income: [tex]$3,475 2. Other Debt Payments: $[/tex]235
3. Loan Interest Rate: 6% (which is 0.06 in decimal form)
4. Down Payment Percentage: 15% (which is 0.15 in decimal form)
5. Monthly Estimate for Property Taxes and Insurance: [tex]$210 ### Step 1: Calculate the Affordable Monthly Mortgage Payment To begin, we need to determine how much of the monthly gross income can be allocated towards mortgage payments, considering a cap of 28% and deducting the costs for other debt payments and estimated property taxes and insurance. ``` Affordable monthly mortgage payment = 0.28 * Monthly gross income - Other debt payments - Monthly estimate for property taxes and insurance ``` Substituting the given values: \[ Affordable monthly mortgage payment = 0.28 \times 3475 - 235 - 210 = 973 - 235 - 210 = 528 \] So, the affordable monthly mortgage payment is $[/tex]528.
### Step 2: Calculate the Affordable Mortgage Amount
To find out how much can be borrowed over a 30-year term, we need to use the mortgage payment formula:
[tex]\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \][/tex]
Where:
- [tex]\( M \)[/tex] is the monthly mortgage payment, which is [tex]$528 - \( r \) is the monthly interest rate, calculated as the annual interest rate divided by 12. Here, it would be \( 0.06 / 12 = 0.005 \) - \( n \) is the total number of monthly payments, which is \( 30 \times 12 = 360 \) for a 30-year loan Rearranging to solve for \( P \) (the mortgage amount): \[ P = \frac{M \cdot \left(\frac{(1 + r)^n - 1}{r \cdot (1 + r)^n}\right)} = \frac{528 \times \left(\frac{(1 + 0.005)^{360} - 1}{0.005 \times (1 + 0.005)^{360}}\right)} \] This simplifies further to yield the mortgage amount: \[ P = 88066 \] So, the affordable mortgage amount is $[/tex]88,066.
### Step 3: Calculate the Affordable Home Purchase Price
The home purchase price includes both the amount borrowed (mortgage amount) and the down payment. Using the given down payment percentage:
[tex]\[ Affordable home purchase price = \frac{Mortgage amount}{1 - Down payment percentage} = \frac{88066}{1 - 0.15} = \frac{88066}{0.85} = 103607 \][/tex]
So, the affordable home purchase price is [tex]$103,607. ### Summary: To summarize the results: - Affordable monthly mortgage payment: $[/tex]528
- Affordable mortgage amount: [tex]$88,066 - Affordable home purchase price: $[/tex]103,607
These calculations will guide in determining what kind of house can be afforded based on the given financial constraints.
1. Monthly Gross Income: [tex]$3,475 2. Other Debt Payments: $[/tex]235
3. Loan Interest Rate: 6% (which is 0.06 in decimal form)
4. Down Payment Percentage: 15% (which is 0.15 in decimal form)
5. Monthly Estimate for Property Taxes and Insurance: [tex]$210 ### Step 1: Calculate the Affordable Monthly Mortgage Payment To begin, we need to determine how much of the monthly gross income can be allocated towards mortgage payments, considering a cap of 28% and deducting the costs for other debt payments and estimated property taxes and insurance. ``` Affordable monthly mortgage payment = 0.28 * Monthly gross income - Other debt payments - Monthly estimate for property taxes and insurance ``` Substituting the given values: \[ Affordable monthly mortgage payment = 0.28 \times 3475 - 235 - 210 = 973 - 235 - 210 = 528 \] So, the affordable monthly mortgage payment is $[/tex]528.
### Step 2: Calculate the Affordable Mortgage Amount
To find out how much can be borrowed over a 30-year term, we need to use the mortgage payment formula:
[tex]\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \][/tex]
Where:
- [tex]\( M \)[/tex] is the monthly mortgage payment, which is [tex]$528 - \( r \) is the monthly interest rate, calculated as the annual interest rate divided by 12. Here, it would be \( 0.06 / 12 = 0.005 \) - \( n \) is the total number of monthly payments, which is \( 30 \times 12 = 360 \) for a 30-year loan Rearranging to solve for \( P \) (the mortgage amount): \[ P = \frac{M \cdot \left(\frac{(1 + r)^n - 1}{r \cdot (1 + r)^n}\right)} = \frac{528 \times \left(\frac{(1 + 0.005)^{360} - 1}{0.005 \times (1 + 0.005)^{360}}\right)} \] This simplifies further to yield the mortgage amount: \[ P = 88066 \] So, the affordable mortgage amount is $[/tex]88,066.
### Step 3: Calculate the Affordable Home Purchase Price
The home purchase price includes both the amount borrowed (mortgage amount) and the down payment. Using the given down payment percentage:
[tex]\[ Affordable home purchase price = \frac{Mortgage amount}{1 - Down payment percentage} = \frac{88066}{1 - 0.15} = \frac{88066}{0.85} = 103607 \][/tex]
So, the affordable home purchase price is [tex]$103,607. ### Summary: To summarize the results: - Affordable monthly mortgage payment: $[/tex]528
- Affordable mortgage amount: [tex]$88,066 - Affordable home purchase price: $[/tex]103,607
These calculations will guide in determining what kind of house can be afforded based on the given financial constraints.
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