Assume you have found a home for sale and agreed on a purchase price of $$ \$ 290,600 $$. Down Payment and Loan You will make a $$ 10 \% $$ down payment on the house. Calculate the following: Down Payment: \$ $$ \square $$ 29,080 Loan Amount: \$ $$ \square $$ 261540 Monthly Payment Calculate the monthly payment for a 30 -year loan, rounding your final answer to the nearest cent (two decimal places). Use the following details: Annual Interest Rate: 7.5\% Express the annual interest rate as a decimal: $$ \square $$ 0.075 Use the loan formula to calculate the monthly payment: $$ P_{0}=\frac{d\left(1-\left(1+\frac{r}{n}\right)^{-n t}\right)}{\left(\frac{r}{n}\right)} $$ - $$ P_{0} $$ : Loan amount - $$ \boldsymbol{r} $$ : Annual interest rate (decimal) - $$ n $$ : Compounding periods per year $$ (n=12) $$ - $$ t $$ : Loan term in years $$ (t=30) $$ Monthly Payment: \$ 1,a26.73

Question

UpStudy AI · Accepted Answer

Let's break down the problem step by step.

### Step 1: Calculate the Down Payment

The down payment is calculated as a percentage of the purchase price.

Given:
- Purchase Price = $$ \$290,600 $$
- Down Payment Percentage = $$ 10\% $$

The down payment can be calculated as follows:

$$
\text{Down Payment} = \text{Purchase Price} \times \left(\frac{10}{100}\right)
$$

### Step 2: Calculate the Loan Amount

The loan amount is the purchase price minus the down payment.

$$
\text{Loan Amount} = \text{Purchase Price} - \text{Down Payment}
$$

### Step 3: Calculate the Monthly Payment

Using the loan formula provided:

$$
P_{0} = \frac{d\left(1-\left(1+\frac{r}{n}\right)^{-n t}\right)}{\left(\frac{r}{n}\right)}
$$

Where:
- $$ P_{0} $$ = Loan amount
- $$ d $$ = Monthly payment
- $$ r $$ = Annual interest rate (decimal) = $$ 0.075 $$
- $$ n $$ = Compounding periods per year = $$ 12 $$
- $$ t $$ = Loan term in years = $$ 30 $$

We need to rearrange the formula to solve for $$ d $$:

$$
d = \frac{P_{0} \left(\frac{r}{n}\right)}{1 - \left(1+\frac{r}{n}\right)^{-n t}}
$$

Now, let's calculate the down payment, loan amount, and monthly payment using these formulas.

### Calculations

1. **Down Payment**
2. **Loan Amount**
3. **Monthly Payment**

Let's perform these calculations.

#### Down Payment Calculation

$$
\text{Down Payment} = 290600 \times 0.10
$$

#### Loan Amount Calculation

$$
\text{Loan Amount} = 290600 - \text{Down Payment}
$$

#### Monthly Payment Calculation

Using the rearranged formula for monthly payment:

$$
d = \frac{P_{0} \left(\frac{0.075}{12}\right)}{1 - \left(1+\frac{0.075}{12}\right)^{-12 \times 30}}
$$

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$290600\times 0.1$$
- step1: Multiply the numbers:
  $$29060$$
Calculate or simplify the expression $$ 290600 - (290600 * 0.10) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$290600-\left(290600\times 0.1\right)$$
- step1: Multiply the numbers:
  $$290600-29060$$
- step2: Subtract the numbers:
  $$261540$$
Calculate or simplify the expression $$ (261540 * (0.075/12)) / (1 - (1 + (0.075/12))^(-12 * 30)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(261540\left(\frac{0.075}{12}\right)\right)}{\left(1-\left(1+\left(\frac{0.075}{12}\right)\right)^{-12\times 30}\right)}$$
- step1: Remove the parentheses:
  $$\frac{261540\left(\frac{0.075}{12}\right)}{1-\left(1+\left(\frac{0.075}{12}\right)\right)^{-12\times 30}}$$
- step2: Divide the terms:
  $$\frac{261540\left(\frac{0.075}{12}\right)}{1-\left(1+\frac{1}{160}\right)^{-12\times 30}}$$
- step3: Add the numbers:
  $$\frac{261540\left(\frac{0.075}{12}\right)}{1-\left(\frac{161}{160}\right)^{-12\times 30}}$$
- step4: Divide the terms:
  $$\frac{261540\times \frac{1}{160}}{1-\left(\frac{161}{160}\right)^{-12\times 30}}$$
- step5: Multiply the numbers:
  $$\frac{261540\times \frac{1}{160}}{1-\left(\frac{161}{160}\right)^{-360}}$$
- step6: Multiply the numbers:
  $$\frac{\frac{13077}{8}}{1-\left(\frac{161}{160}\right)^{-360}}$$
- step7: Subtract the numbers:
  $$\frac{\frac{13077}{8}}{\frac{161^{360}-160^{360}}{161^{360}}}$$
- step8: Multiply by the reciprocal:
  $$\frac{13077}{8}\times \frac{161^{360}}{161^{360}-160^{360}}$$
- step9: Multiply the fractions:
  $$\frac{13077\times 161^{360}}{8\left(161^{360}-160^{360}\right)}$$
- step10: Multiply:
  $$\frac{13077\times 161^{360}}{8\times 161^{360}-8\times 160^{360}}$$
### Step 1: Down Payment Calculation

The down payment is calculated as follows:

$$
\text{Down Payment} = 290600 \times 0.10 = 29060
$$

### Step 2: Loan Amount Calculation

The loan amount is:

$$
\text{Loan Amount} = 290600 - 29060 = 261540
$$

### Step 3: Monthly Payment Calculation

Using the rearranged formula for the monthly payment:

$$
d = \frac{261540 \times \left(\frac{0.075}{12}\right)}{1 - \left(1 + \frac{0.075}{12}\right)^{-12 \times 30}}
$$

Calculating this gives us:

$$
d \approx 1828.73
$$

### Summary of Results

- **Down Payment:** $$ \$29,060 $$
- **Loan Amount:** $$ \$261,540 $$
- **Monthly Payment:** $$ \$1,828.73 $$

Thus, the final answers are:
- Down Payment: $$ \$29,060 $$
- Loan Amount: $$ \$261,540 $$
- Monthly Payment: $$ \$1,828.73 $$
Down Payment: \$29,060 Loan Amount: \$261,540 Monthly Payment: \$1,828.73

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