Anthony has narrowed down his search for a home and is considering whether to place a down payment of [tex]$20\%$[/tex] on the loan. He's wondering how much the down payment will affect his monthly mortgage payment.

The home costs [tex]$\$[/tex]175,000[tex]$. For a 30-year loan at $[/tex]3.6\%[tex]$, the bank said his monthly mortgage payment will be $[/tex]\[tex]$795$[/tex] without a down payment.

A down payment of [tex]$20\%$[/tex] would be [tex]$\$[/tex]35,000[tex]$. So Anthony's mortgage would have a present value of $[/tex]\[tex]$140,000$[/tex].

Determine the monthly mortgage payment if Anthony were to put a [tex]$20\%$[/tex] down payment on the loan. Assume the same interest rate and term. The formulas you'll need for this calculation are provided.

[tex]\[
\begin{array}{l}
FV = PV \left(1 + \frac{i}{n}\right)^{nd} \\
S_n = \frac{P\left(1 + \theta^2 - 1\right)}{i}
\end{array}
\][/tex]

Select the correct answer from each drop-down menu:

Anthony's monthly mortgage payment after putting [tex]$20\%$[/tex] down would be about [tex]$\square$[/tex], which would save him every month.



Answer :

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.