Part 2 of 3 Points: 0 of 3 Save The Guenthers' gross monthly income is $$ \$ 6600 $$. They have 18 remaining payments of $$ \$ 250 $$ on a new car. They are applying for a 20 -year, $$ \$ 159,000 $$ mortgage at $$ 7.5 \% $$. The taxes and insurance on the house are $$ \$ 470 $$ per month. The bank will only approve a loan that has a total monthly mortgage payment of principal, interest, property taxes, and homeowners' insurance that is less than or equal to $$ 28 \% $$ of their adjusted monthly income. Complete parts (a) through (c) below. Click the icon to view the table of monthty payments. a) Determine $$ 28 \% $$ of the Guenthers' adjusted monthly income. $$ \$ 1778 $$ (Round to the nearest cent.) b) Determine the Guenthers' total monthly mortgage payment, including principal, interest, taxes, and homeowners' insurance. $$ \square $$ (Ropand to the nearest cent.)

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To solve the problem, we will follow the steps outlined in the prompt.

### Step 1: Extract Known Conditions
- Gross monthly income: $$ \$ 6600 $$
- Remaining car payments: $$ 18 $$ payments of $$ \$ 250 $$
- Mortgage amount: $$ \$ 159,000 $$
- Interest rate: $$ 7.5\% $$
- Taxes and insurance: $$ \$ 470 $$ per month
- Bank's approval limit: $$ 28\% $$ of adjusted monthly income

### Part (a): Determine $$ 28\% $$ of the Guenthers' adjusted monthly income
To find $$ 28\% $$ of the Guenthers' gross monthly income, we calculate:

$$
28\% \text{ of } 6600 = 0.28 \times 6600
$$

Let's calculate this value.

### Part (b): Determine the Guenthers' total monthly mortgage payment
The total monthly mortgage payment includes:
1. Principal and interest on the mortgage
2. Property taxes
3. Homeowners' insurance

First, we need to calculate the monthly mortgage payment (principal and interest) using the mortgage formula. The formula for the monthly payment $$ M $$ on a mortgage is given by:

$$
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
$$

Where:
- $$ P $$ = loan amount (mortgage) = $$ 159,000 $$
- $$ r $$ = monthly interest rate = $$ \frac{7.5\%}{12} = \frac{0.075}{12} $$
- $$ n $$ = total number of payments = $$ 20 \times 12 = 240 $$

Next, we will calculate the monthly mortgage payment and then add the taxes and insurance to find the total monthly payment.

Let's perform the calculations for both parts (a) and (b).
Calculate the value by following steps:
- step0: Calculate:
  $$0.28\times 6600$$
- step1: Multiply the numbers:
  $$1848$$
Calculate or simplify the expression $$ 159000 * (0.075/12) * (1 + 0.075/12)^(20*12) / ((1 + 0.075/12)^(20*12) - 1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{159000\left(\frac{0.075}{12}\right)\left(1+\frac{0.075}{12}\right)^{20\times 12}}{\left(\left(1+\frac{0.075}{12}\right)^{20\times 12}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{159000\left(\frac{0.075}{12}\right)\left(1+\frac{0.075}{12}\right)^{20\times 12}}{\left(1+\frac{0.075}{12}\right)^{20\times 12}-1}$$
- step2: Divide the terms:
  $$\frac{159000\left(\frac{0.075}{12}\right)\left(1+\frac{1}{160}\right)^{20\times 12}}{\left(1+\frac{0.075}{12}\right)^{20\times 12}-1}$$
- step3: Add the numbers:
  $$\frac{159000\left(\frac{0.075}{12}\right)\left(\frac{161}{160}\right)^{20\times 12}}{\left(1+\frac{0.075}{12}\right)^{20\times 12}-1}$$
- step4: Divide the terms:
  $$\frac{159000\left(\frac{0.075}{12}\right)\left(\frac{161}{160}\right)^{20\times 12}}{\left(1+\frac{1}{160}\right)^{20\times 12}-1}$$
- step5: Add the numbers:
  $$\frac{159000\left(\frac{0.075}{12}\right)\left(\frac{161}{160}\right)^{20\times 12}}{\left(\frac{161}{160}\right)^{20\times 12}-1}$$
- step6: Divide the terms:
  $$\frac{159000\times \frac{1}{160}\left(\frac{161}{160}\right)^{20\times 12}}{\left(\frac{161}{160}\right)^{20\times 12}-1}$$
- step7: Multiply the numbers:
  $$\frac{159000\times \frac{1}{160}\left(\frac{161}{160}\right)^{240}}{\left(\frac{161}{160}\right)^{20\times 12}-1}$$
- step8: Multiply the numbers:
  $$\frac{159000\times \frac{1}{160}\left(\frac{161}{160}\right)^{240}}{\left(\frac{161}{160}\right)^{240}-1}$$
- step9: Multiply:
  $$\frac{\frac{159\times 161^{240}}{2^{1202}\times 5^{238}}}{\left(\frac{161}{160}\right)^{240}-1}$$
- step10: Subtract the numbers:
  $$\frac{\frac{159\times 161^{240}}{2^{1202}\times 5^{238}}}{\frac{161^{240}-160^{240}}{160^{240}}}$$
- step11: Multiply by the reciprocal:
  $$\frac{159\times 161^{240}}{2^{1202}\times 5^{238}}\times \frac{160^{240}}{161^{240}-160^{240}}$$
- step12: Rewrite the expression:
  $$\frac{159\times 161^{240}}{2^{1202}\times 5^{238}}\times \frac{32^{240}\times 5^{240}}{161^{240}-160^{240}}$$
- step13: Rewrite the expression:
  $$\frac{159\times 161^{240}}{2^{1202}\times 5^{238}}\times \frac{2^{1200}\times 5^{240}}{161^{240}-160^{240}}$$
- step14: Reduce the numbers:
  $$\frac{159\times 161^{240}}{2^{2}}\times \frac{5^{2}}{161^{240}-160^{240}}$$
- step15: Multiply the fractions:
  $$\frac{159\times 161^{240}\times 5^{2}}{2^{2}\left(161^{240}-160^{240}\right)}$$
- step16: Multiply:
  $$\frac{3975\times 161^{240}}{4\times 161^{240}-4\times 160^{240}}$$
Calculate or simplify the expression $$ 1280.893178 + 470 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1280.893178+470$$
- step1: Add the numbers:
  $$1750.893178$$
### Part (a): Calculate $$ 28\% $$ of the Guenthers' adjusted monthly income
We found that:

$$
28\% \text{ of } 6600 = 1848
$$

So, $$ 28\% $$ of the Guenthers' adjusted monthly income is **\$1848.00**.

### Part (b): Calculate the Guenthers' total monthly mortgage payment
1. **Monthly mortgage payment (principal and interest)**: 
   - The calculated monthly payment is approximately **\$1280.89**.

2. **Add property taxes and homeowners' insurance**:
   - Taxes and insurance: **\$470.00**
   - Total monthly mortgage payment:

$$
\text{Total Payment} = \text{Mortgage Payment} + \text{Taxes and Insurance} = 1280.89 + 470 = 1750.89
$$

Thus, the Guenthers' total monthly mortgage payment, including principal, interest, taxes, and homeowners' insurance, is approximately **\$1750.89**.

### Summary of Results
- **(a)** $$ 28\% $$ of the Guenthers' adjusted monthly income: **\$1848.00**
- **(b)** Total monthly mortgage payment: **\$1750.89**
$$ \$1750.89 $$

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