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Sure, let's walk through the problem step-by-step to determine Anthony's monthly mortgage payment if he makes a 20% down payment on the loan, which will save him money each month.
### Step-by-Step Solution
1. Determine the Home Cost and Down Payment:
- The home costs [tex]\( \$175,000 \)[/tex].
- A 20% down payment on the home is:
[tex]\[ \text{Down Payment} = 0.20 \times 175{,}000 = \$35{,}000 \][/tex]
2. Calculate the Mortgage Amount After Down Payment:
- The mortgage amount (or loan principal) after the 20% down payment is:
[tex]\[ \text{Loan Principal (PV)} = 175{,}000 - 35{,}000 = \$140{,}000 \][/tex]
3. Interest Rate and Loan Term:
- Annual interest rate ([tex]\(r\)[/tex]) = 3.6% or 0.036.
- Loan term = 30 years.
- The number of monthly payments ([tex]\(n\)[/tex]):
[tex]\[ n = 30 \times 12 = 360 \text{ payments} \][/tex]
4. Convert Annual Interest Rate to Monthly Interest Rate:
- Monthly interest rate ([tex]\(r_m\)[/tex]) is:
[tex]\[ r_m = \frac{0.036}{12} = 0.003 \][/tex]
5. Calculate the Monthly Mortgage Payment:
- Using the formula for monthly mortgage payment:
[tex]\[ \text{PMT} = PV \times \frac{r_m (1 + r_m)^n}{(1 + r_m)^n - 1} \][/tex]
- Plugging in the values:
[tex]\[ \text{PMT} = 140{,}000 \times \frac{0.003 \times (1 + 0.003)^{360}}{(1 + 0.003)^{360} - 1} \][/tex]
- After evaluating the expression, the monthly mortgage payment (PMT) turns out to be approximately:
[tex]\[ \text{PMT} = \$636.50 \text{ (rounded to two decimal places)} \][/tex]
6. Compare Monthly Payments and Calculate Savings:
- Monthly mortgage payment without down payment is [tex]\( \$795 \)[/tex].
- Monthly mortgage payment with down payment is [tex]\( \$636.50 \)[/tex].
The savings per month by making the down payment:
[tex]\[ \text{Monthly Savings} = 795 - 636.50 = \$158.50 \][/tex]
### Conclusion
Anthony's monthly mortgage payment after putting a 20% down payment would be about [tex]\( \$636.50 \)[/tex]. This would save him approximately [tex]\( \$158.50 \)[/tex] every month compared to not making the down payment.
### Step-by-Step Solution
1. Determine the Home Cost and Down Payment:
- The home costs [tex]\( \$175,000 \)[/tex].
- A 20% down payment on the home is:
[tex]\[ \text{Down Payment} = 0.20 \times 175{,}000 = \$35{,}000 \][/tex]
2. Calculate the Mortgage Amount After Down Payment:
- The mortgage amount (or loan principal) after the 20% down payment is:
[tex]\[ \text{Loan Principal (PV)} = 175{,}000 - 35{,}000 = \$140{,}000 \][/tex]
3. Interest Rate and Loan Term:
- Annual interest rate ([tex]\(r\)[/tex]) = 3.6% or 0.036.
- Loan term = 30 years.
- The number of monthly payments ([tex]\(n\)[/tex]):
[tex]\[ n = 30 \times 12 = 360 \text{ payments} \][/tex]
4. Convert Annual Interest Rate to Monthly Interest Rate:
- Monthly interest rate ([tex]\(r_m\)[/tex]) is:
[tex]\[ r_m = \frac{0.036}{12} = 0.003 \][/tex]
5. Calculate the Monthly Mortgage Payment:
- Using the formula for monthly mortgage payment:
[tex]\[ \text{PMT} = PV \times \frac{r_m (1 + r_m)^n}{(1 + r_m)^n - 1} \][/tex]
- Plugging in the values:
[tex]\[ \text{PMT} = 140{,}000 \times \frac{0.003 \times (1 + 0.003)^{360}}{(1 + 0.003)^{360} - 1} \][/tex]
- After evaluating the expression, the monthly mortgage payment (PMT) turns out to be approximately:
[tex]\[ \text{PMT} = \$636.50 \text{ (rounded to two decimal places)} \][/tex]
6. Compare Monthly Payments and Calculate Savings:
- Monthly mortgage payment without down payment is [tex]\( \$795 \)[/tex].
- Monthly mortgage payment with down payment is [tex]\( \$636.50 \)[/tex].
The savings per month by making the down payment:
[tex]\[ \text{Monthly Savings} = 795 - 636.50 = \$158.50 \][/tex]
### Conclusion
Anthony's monthly mortgage payment after putting a 20% down payment would be about [tex]\( \$636.50 \)[/tex]. This would save him approximately [tex]\( \$158.50 \)[/tex] every month compared to not making the down payment.
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