Answer :
To determine which person is the most eligible for a home loan based on the given criteria, we need to evaluate the following factors for each person: savings for the down payment and the debt-to-income ratio. Here's the detailed step-by-step solution:
### Step 1: Calculate the Down Payment Requirement
The lender requires a down payment of 20% of the home value. Let's calculate that for each person.
- Person A:
[tex]\[ \text{Down Payment} = 175,000 \times 0.20 = 35,000 \][/tex]
- Person B:
[tex]\[ \text{Down Payment} = 200,000 \times 0.20 = 40,000 \][/tex]
- Person C:
[tex]\[ \text{Down Payment} = 220,000 \times 0.20 = 44,000 \][/tex]
- Person D:
[tex]\[ \text{Down Payment} = 250,000 \times 0.20 = 50,000 \][/tex]
### Step 2: Calculate the Loan Amount Needed (Home Value minus Down Payment)
For each person:
- Person A:
[tex]\[ \text{Loan Amount} = 175,000 - 35,000 = 140,000 \][/tex]
- Person B:
[tex]\[ \text{Loan Amount} = 200,000 - 40,000 = 160,000 \][/tex]
- Person C:
[tex]\[ \text{Loan Amount} = 220,000 - 44,000 = 176,000 \][/tex]
- Person D:
[tex]\[ \text{Loan Amount} = 250,000 - 50,000 = 200,000 \][/tex]
### Step 3: Calculate the Yearly Recurring Debt
Multiply the monthly recurring debt by 12 for each person:
- Person A:
[tex]\[ \text{Yearly Recurring Debt} = 350 \times 12 = 4,200 \][/tex]
- Person B:
[tex]\[ \text{Yearly Recurring Debt} = 250 \times 12 = 3,000 \][/tex]
- Person C:
[tex]\[ \text{Yearly Recurring Debt} = 200 \times 12 = 2,400 \][/tex]
- Person D:
[tex]\[ \text{Yearly Recurring Debt} = 450 \times 12 = 5,400 \][/tex]
### Step 4: Calculate the Total Yearly Obligations Including Housing Expenses
Here, we are adding the loan amount (treated as yearly payment) to the yearly recurring debts:
- Person A:
[tex]\[ \text{Total Yearly Obligations} = 140,000 + 4,200 = 144,200 \][/tex]
- Person B:
[tex]\[ \text{Total Yearly Obligations} = 160,000 + 3,000 = 163,000 \][/tex]
- Person C:
[tex]\[ \text{Total Yearly Obligations} = 176,000 + 2,400 = 178,400 \][/tex]
- Person D:
[tex]\[ \text{Total Yearly Obligations} = 200,000 + 5,400 = 205,400 \][/tex]
### Step 5: Calculate the Debt-to-Income Ratio
Divide the total yearly obligations by the annual income for each person:
- Person A:
[tex]\[ \text{Debt-to-Income Ratio} = \frac{144,200}{51,000} \approx 2.83 \][/tex]
- Person B:
[tex]\[ \text{Debt-to-Income Ratio} = \frac{163,000}{58,000} \approx 2.81 \][/tex]
- Person C:
[tex]\[ \text{Debt-to-Income Ratio} = \frac{178,400}{63,000} \approx 2.83 \][/tex]
- Person D:
[tex]\[ \text{Debt-to-Income Ratio} = \frac{205,400}{67,000} \approx 3.07 \][/tex]
### Step 6: Verify Savings and Debt-to-Income Ratio for Eligibility
Each person must meet the down payment requirement and have a debt-to-income ratio no greater than 0.36 to be eligible.
- Person A:
- Savings: \[tex]$35,000 (meets the down payment requirement of \$[/tex]35,000)
- Debt-to-Income Ratio: 2.83 (exceeds the maximum of 0.36)
- Person B:
- Savings: \[tex]$40,000 (meets the down payment requirement of \$[/tex]40,000)
- Debt-to-Income Ratio: 2.81 (exceeds the maximum of 0.36)
- Person C:
- Savings: \[tex]$42,000 (does not meet the down payment requirement of \$[/tex]44,000)
- Debt-to-Income Ratio: 2.83 (exceeds the maximum of 0.36)
- Person D:
- Savings: \[tex]$50,000 (meets the down payment requirement of \$[/tex]50,000)
- Debt-to-Income Ratio: 3.07 (exceeds the maximum of 0.36)
### Conclusion: Highest Eligibility
Given that all four persons exceed the maximum allowable debt-to-income ratio of 0.36, none of them are eligible for the loan based on the standard criteria. However, if one were to ask which among them had enough savings for the down payment and the least severe debt-to-income overage, we would select based on who had both eligible savings and realistically coming closest to the limit.
Therefore, the scores for debt-to-income ratios are all above acceptable thresholds and treated as 'inf' (infinitely not eligible).
At this point, selecting any one option would merely be for formality. However, since Person A has the earliest conditions met (savings coverage first) despite the high ratio, we state:
a. Person A
### Step 1: Calculate the Down Payment Requirement
The lender requires a down payment of 20% of the home value. Let's calculate that for each person.
- Person A:
[tex]\[ \text{Down Payment} = 175,000 \times 0.20 = 35,000 \][/tex]
- Person B:
[tex]\[ \text{Down Payment} = 200,000 \times 0.20 = 40,000 \][/tex]
- Person C:
[tex]\[ \text{Down Payment} = 220,000 \times 0.20 = 44,000 \][/tex]
- Person D:
[tex]\[ \text{Down Payment} = 250,000 \times 0.20 = 50,000 \][/tex]
### Step 2: Calculate the Loan Amount Needed (Home Value minus Down Payment)
For each person:
- Person A:
[tex]\[ \text{Loan Amount} = 175,000 - 35,000 = 140,000 \][/tex]
- Person B:
[tex]\[ \text{Loan Amount} = 200,000 - 40,000 = 160,000 \][/tex]
- Person C:
[tex]\[ \text{Loan Amount} = 220,000 - 44,000 = 176,000 \][/tex]
- Person D:
[tex]\[ \text{Loan Amount} = 250,000 - 50,000 = 200,000 \][/tex]
### Step 3: Calculate the Yearly Recurring Debt
Multiply the monthly recurring debt by 12 for each person:
- Person A:
[tex]\[ \text{Yearly Recurring Debt} = 350 \times 12 = 4,200 \][/tex]
- Person B:
[tex]\[ \text{Yearly Recurring Debt} = 250 \times 12 = 3,000 \][/tex]
- Person C:
[tex]\[ \text{Yearly Recurring Debt} = 200 \times 12 = 2,400 \][/tex]
- Person D:
[tex]\[ \text{Yearly Recurring Debt} = 450 \times 12 = 5,400 \][/tex]
### Step 4: Calculate the Total Yearly Obligations Including Housing Expenses
Here, we are adding the loan amount (treated as yearly payment) to the yearly recurring debts:
- Person A:
[tex]\[ \text{Total Yearly Obligations} = 140,000 + 4,200 = 144,200 \][/tex]
- Person B:
[tex]\[ \text{Total Yearly Obligations} = 160,000 + 3,000 = 163,000 \][/tex]
- Person C:
[tex]\[ \text{Total Yearly Obligations} = 176,000 + 2,400 = 178,400 \][/tex]
- Person D:
[tex]\[ \text{Total Yearly Obligations} = 200,000 + 5,400 = 205,400 \][/tex]
### Step 5: Calculate the Debt-to-Income Ratio
Divide the total yearly obligations by the annual income for each person:
- Person A:
[tex]\[ \text{Debt-to-Income Ratio} = \frac{144,200}{51,000} \approx 2.83 \][/tex]
- Person B:
[tex]\[ \text{Debt-to-Income Ratio} = \frac{163,000}{58,000} \approx 2.81 \][/tex]
- Person C:
[tex]\[ \text{Debt-to-Income Ratio} = \frac{178,400}{63,000} \approx 2.83 \][/tex]
- Person D:
[tex]\[ \text{Debt-to-Income Ratio} = \frac{205,400}{67,000} \approx 3.07 \][/tex]
### Step 6: Verify Savings and Debt-to-Income Ratio for Eligibility
Each person must meet the down payment requirement and have a debt-to-income ratio no greater than 0.36 to be eligible.
- Person A:
- Savings: \[tex]$35,000 (meets the down payment requirement of \$[/tex]35,000)
- Debt-to-Income Ratio: 2.83 (exceeds the maximum of 0.36)
- Person B:
- Savings: \[tex]$40,000 (meets the down payment requirement of \$[/tex]40,000)
- Debt-to-Income Ratio: 2.81 (exceeds the maximum of 0.36)
- Person C:
- Savings: \[tex]$42,000 (does not meet the down payment requirement of \$[/tex]44,000)
- Debt-to-Income Ratio: 2.83 (exceeds the maximum of 0.36)
- Person D:
- Savings: \[tex]$50,000 (meets the down payment requirement of \$[/tex]50,000)
- Debt-to-Income Ratio: 3.07 (exceeds the maximum of 0.36)
### Conclusion: Highest Eligibility
Given that all four persons exceed the maximum allowable debt-to-income ratio of 0.36, none of them are eligible for the loan based on the standard criteria. However, if one were to ask which among them had enough savings for the down payment and the least severe debt-to-income overage, we would select based on who had both eligible savings and realistically coming closest to the limit.
Therefore, the scores for debt-to-income ratios are all above acceptable thresholds and treated as 'inf' (infinitely not eligible).
At this point, selecting any one option would merely be for formality. However, since Person A has the earliest conditions met (savings coverage first) despite the high ratio, we state:
a. Person A
