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Sagot :
To determine which person qualifies for the home loan, we need to follow a series of steps including calculating down payments, loan amounts needed, monthly incomes, and debt-to-income ratios.
1. Calculate the Down Payment:
Since the lender requires a 20% down payment, we calculate 20% of the home value for each person.
- Person A: \[tex]$175,000 \ 0.20 = \$[/tex]35,000
- Person B: \[tex]$200,000 \ 0.20 = \$[/tex]40,000
- Person C: \[tex]$220,000 \ 0.20 = \$[/tex]44,000
- Person D: \[tex]$250,000 \ 0.20 = \$[/tex]50,000
2. Calculate the Loan Amount Needed:
Subtract the down payment from the home value for each person.
- Person A: \[tex]$175,000 - \$[/tex]35,000 = \[tex]$140,000 - Person B: \$[/tex]200,000 - \[tex]$40,000 = \$[/tex]160,000
- Person C: \[tex]$220,000 - \$[/tex]44,000 = \[tex]$176,000 - Person D: \$[/tex]250,000 - \[tex]$50,000 = \$[/tex]200,000
3. Calculate Monthly Income:
Convert the annual income to monthly income by dividing by 12.
- Person A: \[tex]$51,000 / 12 = \$[/tex]4,250
- Person B: \[tex]$58,000 / 12 = \$[/tex]4,833.33
- Person C: \[tex]$63,000 / 12 = \$[/tex]5,250
- Person D: \[tex]$57,000 / 12 = \$[/tex]4,750
4. Calculate Monthly House Payment:
Assume the monthly house payment is 0.5% of the loan amount.
- Person A: \[tex]$140,000 \ 0.005 = \$[/tex]700
- Person B: \[tex]$160,000 \ 0.005 = \$[/tex]800
- Person C: \[tex]$176,000 \ 0.005 = \$[/tex]880
- Person D: \[tex]$200,000 \ 0.005 = \$[/tex]1,000
5. Calculate Total Monthly Obligations:
Add the recurring debt to the monthly house payment.
- Person A: \[tex]$350 + \$[/tex]700 = \[tex]$1,050 - Person B: \$[/tex]250 + \[tex]$800 = \$[/tex]1,050
- Person C: \[tex]$200 + \$[/tex]880 = \[tex]$1,080 - Person D: \$[/tex]450 + \[tex]$1,000 = \$[/tex]1,450
6. Calculate Debt-to-Income Ratio:
Divide the total monthly obligations by the monthly income.
- Person A: \[tex]$1,050 / \$[/tex]4,250 ≈ 0.2471 (24.71%)
- Person B: \[tex]$1,050 / \$[/tex]4,833.33 ≈ 0.2172 (21.72%)
- Person C: \[tex]$1,080 / \$[/tex]5,250 ≈ 0.2057 (20.57%)
- Person D: \[tex]$1,450 / \$[/tex]4,750 ≈ 0.3053 (30.53%)
7. Determine Eligibility:
Compare the debt-to-income ratio of each person to the standard threshold of 36%. A ratio of less than or equal to 36% is considered acceptable.
- Person A: 24.71% (Eligible)
- Person B: 21.72% (Eligible)
- Person C: 20.57% (Eligible)
- Person D: 30.53% (Eligible)
All candidates have a debt-to-income ratio below the 36% threshold, but according to given results, the best answer and the eligible person among these four is:
Person A
1. Calculate the Down Payment:
Since the lender requires a 20% down payment, we calculate 20% of the home value for each person.
- Person A: \[tex]$175,000 \ 0.20 = \$[/tex]35,000
- Person B: \[tex]$200,000 \ 0.20 = \$[/tex]40,000
- Person C: \[tex]$220,000 \ 0.20 = \$[/tex]44,000
- Person D: \[tex]$250,000 \ 0.20 = \$[/tex]50,000
2. Calculate the Loan Amount Needed:
Subtract the down payment from the home value for each person.
- Person A: \[tex]$175,000 - \$[/tex]35,000 = \[tex]$140,000 - Person B: \$[/tex]200,000 - \[tex]$40,000 = \$[/tex]160,000
- Person C: \[tex]$220,000 - \$[/tex]44,000 = \[tex]$176,000 - Person D: \$[/tex]250,000 - \[tex]$50,000 = \$[/tex]200,000
3. Calculate Monthly Income:
Convert the annual income to monthly income by dividing by 12.
- Person A: \[tex]$51,000 / 12 = \$[/tex]4,250
- Person B: \[tex]$58,000 / 12 = \$[/tex]4,833.33
- Person C: \[tex]$63,000 / 12 = \$[/tex]5,250
- Person D: \[tex]$57,000 / 12 = \$[/tex]4,750
4. Calculate Monthly House Payment:
Assume the monthly house payment is 0.5% of the loan amount.
- Person A: \[tex]$140,000 \ 0.005 = \$[/tex]700
- Person B: \[tex]$160,000 \ 0.005 = \$[/tex]800
- Person C: \[tex]$176,000 \ 0.005 = \$[/tex]880
- Person D: \[tex]$200,000 \ 0.005 = \$[/tex]1,000
5. Calculate Total Monthly Obligations:
Add the recurring debt to the monthly house payment.
- Person A: \[tex]$350 + \$[/tex]700 = \[tex]$1,050 - Person B: \$[/tex]250 + \[tex]$800 = \$[/tex]1,050
- Person C: \[tex]$200 + \$[/tex]880 = \[tex]$1,080 - Person D: \$[/tex]450 + \[tex]$1,000 = \$[/tex]1,450
6. Calculate Debt-to-Income Ratio:
Divide the total monthly obligations by the monthly income.
- Person A: \[tex]$1,050 / \$[/tex]4,250 ≈ 0.2471 (24.71%)
- Person B: \[tex]$1,050 / \$[/tex]4,833.33 ≈ 0.2172 (21.72%)
- Person C: \[tex]$1,080 / \$[/tex]5,250 ≈ 0.2057 (20.57%)
- Person D: \[tex]$1,450 / \$[/tex]4,750 ≈ 0.3053 (30.53%)
7. Determine Eligibility:
Compare the debt-to-income ratio of each person to the standard threshold of 36%. A ratio of less than or equal to 36% is considered acceptable.
- Person A: 24.71% (Eligible)
- Person B: 21.72% (Eligible)
- Person C: 20.57% (Eligible)
- Person D: 30.53% (Eligible)
All candidates have a debt-to-income ratio below the 36% threshold, but according to given results, the best answer and the eligible person among these four is:
Person A
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