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Sagot :
To determine the eligibility for a home loan for each person, we need to follow these steps:
1. Calculate the required down payment for each person:
[tex]\[ \text{down payment} = \text{home value} \times 0.15 \][/tex]
2. Calculate the loan amount for each person:
[tex]\[ \text{loan amount} = \text{home value} - \text{down payment} \][/tex]
3. Calculate the annual debt for each person:
[tex]\[ \text{annual debt} = \text{recurring debt} \times 12 \][/tex]
4. Calculate the maximum loan amount based on income considering the debt-to-income ratio threshold (43%):
[tex]\[ \text{max loan based on income} = \frac{\text{income} \times 0.43 - \text{annual debt}}{0.43} \][/tex]
5. Check if each person has sufficient savings for the down payment. If savings are less than the required down payment, their eligibility score will be zero.
6. Determine the eligibility score by finding the minimum between the loan amount and max loan based on income.
7. Find the person with the highest eligibility score.
Let's break down each step for each person:
### Person A:
1. Down payment:
[tex]\[ 95,000 \times 0.15 = 14,250 \][/tex]
2. Loan amount:
[tex]\[ 95,000 - 14,250 = 80,750 \][/tex]
3. Annual debt:
[tex]\[ 310 \times 12 = 3,720 \][/tex]
4. Max loan based on income:
[tex]\[ \frac{46,000 \times 0.43 - 3,720}{0.43} = \frac{19,780 - 3,720}{0.43} = \frac{16,060}{0.43} \approx 37,349 \][/tex]
5. Savings:
[tex]\[ 20,000 > 14,250 \quad (\text{Sufficient for down payment}) \][/tex]
6. Eligibility score:
[tex]\[ \min(80,750, 37,349) = 37,349 \][/tex]
### Person B:
1. Down payment:
[tex]\[ 107,000 \times 0.15 = 16,050 \][/tex]
2. Loan amount:
[tex]\[ 107,000 - 16,050 = 90,950 \][/tex]
3. Annual debt:
[tex]\[ 198 \times 12 = 2,376 \][/tex]
4. Max loan based on income:
[tex]\[ \frac{53,000 \times 0.43 - 2,376}{0.43} = \frac{22,790 - 2,376}{0.43} = \frac{20,414}{0.43} \approx 47,473 \][/tex]
5. Savings:
[tex]\[ 13,910 < 16,050 \quad (\text{Not sufficient for down payment}) \][/tex]
6. Eligibility score:
[tex]\[ 0 \quad (\text{Due to insufficient savings}) \][/tex]
### Person C:
1. Down payment:
[tex]\[ 120,000 \times 0.15 = 18,000 \][/tex]
2. Loan amount:
[tex]\[ 120,000 - 18,000 = 102,000 \][/tex]
3. Annual debt:
[tex]\[ 265 \times 12 = 3,180 \][/tex]
4. Max loan based on income:
[tex]\[ \frac{58,000 \times 0.43 - 3,180}{0.43} = \frac{24,940 - 3,180}{0.43} = \frac{21,760}{0.43} \approx 50,605 \][/tex]
5. Savings:
[tex]\[ 18,000 = 18,000 \quad (\text{Exact amount sufficient for down payment}) \][/tex]
6. Eligibility score:
[tex]\[ \min(102,000, 50,605) = 50,605 \][/tex]
### Person D:
1. Down payment:
[tex]\[ 128,000 \times 0.15 = 19,200 \][/tex]
2. Loan amount:
[tex]\[ 128,000 - 19,200 = 108,800 \][/tex]
3. Annual debt:
[tex]\[ 400 \times 12 = 4,800 \][/tex]
4. Max loan based on income:
[tex]\[ \frac{60,000 \times 0.43 - 4,800}{0.43} = \frac{25,800 - 4,800}{0.43} = \frac{21,000}{0.43} \approx 48,837 \][/tex]
5. Savings:
[tex]\[ 19,200 = 19,200 \quad (\text{Exact amount sufficient for down payment}) \][/tex]
6. Eligibility score:
[tex]\[ \min(108,800, 48,837) = 48,837 \][/tex]
Comparing the eligibility scores:
- Person A: 37,349
- Person B: 0
- Person C: 50,605
- Person D: 48,837
The highest eligibility score is 50,605 for Person C.
Therefore, the correct answer is:
c. Person C
1. Calculate the required down payment for each person:
[tex]\[ \text{down payment} = \text{home value} \times 0.15 \][/tex]
2. Calculate the loan amount for each person:
[tex]\[ \text{loan amount} = \text{home value} - \text{down payment} \][/tex]
3. Calculate the annual debt for each person:
[tex]\[ \text{annual debt} = \text{recurring debt} \times 12 \][/tex]
4. Calculate the maximum loan amount based on income considering the debt-to-income ratio threshold (43%):
[tex]\[ \text{max loan based on income} = \frac{\text{income} \times 0.43 - \text{annual debt}}{0.43} \][/tex]
5. Check if each person has sufficient savings for the down payment. If savings are less than the required down payment, their eligibility score will be zero.
6. Determine the eligibility score by finding the minimum between the loan amount and max loan based on income.
7. Find the person with the highest eligibility score.
Let's break down each step for each person:
### Person A:
1. Down payment:
[tex]\[ 95,000 \times 0.15 = 14,250 \][/tex]
2. Loan amount:
[tex]\[ 95,000 - 14,250 = 80,750 \][/tex]
3. Annual debt:
[tex]\[ 310 \times 12 = 3,720 \][/tex]
4. Max loan based on income:
[tex]\[ \frac{46,000 \times 0.43 - 3,720}{0.43} = \frac{19,780 - 3,720}{0.43} = \frac{16,060}{0.43} \approx 37,349 \][/tex]
5. Savings:
[tex]\[ 20,000 > 14,250 \quad (\text{Sufficient for down payment}) \][/tex]
6. Eligibility score:
[tex]\[ \min(80,750, 37,349) = 37,349 \][/tex]
### Person B:
1. Down payment:
[tex]\[ 107,000 \times 0.15 = 16,050 \][/tex]
2. Loan amount:
[tex]\[ 107,000 - 16,050 = 90,950 \][/tex]
3. Annual debt:
[tex]\[ 198 \times 12 = 2,376 \][/tex]
4. Max loan based on income:
[tex]\[ \frac{53,000 \times 0.43 - 2,376}{0.43} = \frac{22,790 - 2,376}{0.43} = \frac{20,414}{0.43} \approx 47,473 \][/tex]
5. Savings:
[tex]\[ 13,910 < 16,050 \quad (\text{Not sufficient for down payment}) \][/tex]
6. Eligibility score:
[tex]\[ 0 \quad (\text{Due to insufficient savings}) \][/tex]
### Person C:
1. Down payment:
[tex]\[ 120,000 \times 0.15 = 18,000 \][/tex]
2. Loan amount:
[tex]\[ 120,000 - 18,000 = 102,000 \][/tex]
3. Annual debt:
[tex]\[ 265 \times 12 = 3,180 \][/tex]
4. Max loan based on income:
[tex]\[ \frac{58,000 \times 0.43 - 3,180}{0.43} = \frac{24,940 - 3,180}{0.43} = \frac{21,760}{0.43} \approx 50,605 \][/tex]
5. Savings:
[tex]\[ 18,000 = 18,000 \quad (\text{Exact amount sufficient for down payment}) \][/tex]
6. Eligibility score:
[tex]\[ \min(102,000, 50,605) = 50,605 \][/tex]
### Person D:
1. Down payment:
[tex]\[ 128,000 \times 0.15 = 19,200 \][/tex]
2. Loan amount:
[tex]\[ 128,000 - 19,200 = 108,800 \][/tex]
3. Annual debt:
[tex]\[ 400 \times 12 = 4,800 \][/tex]
4. Max loan based on income:
[tex]\[ \frac{60,000 \times 0.43 - 4,800}{0.43} = \frac{25,800 - 4,800}{0.43} = \frac{21,000}{0.43} \approx 48,837 \][/tex]
5. Savings:
[tex]\[ 19,200 = 19,200 \quad (\text{Exact amount sufficient for down payment}) \][/tex]
6. Eligibility score:
[tex]\[ \min(108,800, 48,837) = 48,837 \][/tex]
Comparing the eligibility scores:
- Person A: 37,349
- Person B: 0
- Person C: 50,605
- Person D: 48,837
The highest eligibility score is 50,605 for Person C.
Therefore, the correct answer is:
c. Person C
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