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: \$ $$ \square $$ 29,060 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: 0.075 Use the loan formula to calculate the monthly payment: $$ \mathrm{P}_{0}=\frac{d\left(1-\left(1+\frac{r}{n}\right)^{-n t}\right)}{\left(\frac{r}{n}\right)} $$ - $$ P_{0} $$ : Loan amount - $$ r $$ : Annual interest rate (decimal) - $$ n $$ : Compounding periods per year $$ (\mathrm{n}=12) $$ - $$ t $$ : Loan term in years $$ (\mathrm{t}=30) $$ Monthly Payment: \$ 1.828 .73

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UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$0.1\times 290600$$
- step1: Multiply the numbers:
  $$29060$$
Calculate or simplify the expression $$ 290600 - (0.10 * 290600) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$290600-\left(0.1\times 290600\right)$$
- step1: Multiply the numbers:
  $$290600-29060$$
- step2: Subtract the numbers:
  $$261540$$
Calculate or simplify the expression $$ (261540 * (0.075 / 12) * (1 + (0.075 / 12))^(12 * 30)) / ((1 + (0.075 / 12))^(12 * 30) - 1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(261540\left(\frac{0.075}{12}\right)\left(1+\left(\frac{0.075}{12}\right)\right)^{12\times 30}\right)}{\left(\left(1+\left(\frac{0.075}{12}\right)\right)^{12\times 30}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{261540\left(\frac{0.075}{12}\right)\left(1+\left(\frac{0.075}{12}\right)\right)^{12\times 30}}{\left(1+\left(\frac{0.075}{12}\right)\right)^{12\times 30}-1}$$
- step2: Divide the terms:
  $$\frac{261540\left(\frac{0.075}{12}\right)\left(1+\frac{1}{160}\right)^{12\times 30}}{\left(1+\left(\frac{0.075}{12}\right)\right)^{12\times 30}-1}$$
- step3: Add the numbers:
  $$\frac{261540\left(\frac{0.075}{12}\right)\left(\frac{161}{160}\right)^{12\times 30}}{\left(1+\left(\frac{0.075}{12}\right)\right)^{12\times 30}-1}$$
- step4: Divide the terms:
  $$\frac{261540\left(\frac{0.075}{12}\right)\left(\frac{161}{160}\right)^{12\times 30}}{\left(1+\frac{1}{160}\right)^{12\times 30}-1}$$
- step5: Add the numbers:
  $$\frac{261540\left(\frac{0.075}{12}\right)\left(\frac{161}{160}\right)^{12\times 30}}{\left(\frac{161}{160}\right)^{12\times 30}-1}$$
- step6: Divide the terms:
  $$\frac{261540\times \frac{1}{160}\left(\frac{161}{160}\right)^{12\times 30}}{\left(\frac{161}{160}\right)^{12\times 30}-1}$$
- step7: Multiply the numbers:
  $$\frac{261540\times \frac{1}{160}\left(\frac{161}{160}\right)^{360}}{\left(\frac{161}{160}\right)^{12\times 30}-1}$$
- step8: Multiply the numbers:
  $$\frac{261540\times \frac{1}{160}\left(\frac{161}{160}\right)^{360}}{\left(\frac{161}{160}\right)^{360}-1}$$
- step9: Multiply:
  $$\frac{\frac{13077\times 161^{360}}{8\times 160^{360}}}{\left(\frac{161}{160}\right)^{360}-1}$$
- step10: Subtract the numbers:
  $$\frac{\frac{13077\times 161^{360}}{8\times 160^{360}}}{\frac{161^{360}-160^{360}}{160^{360}}}$$
- step11: Multiply by the reciprocal:
  $$\frac{13077\times 161^{360}}{8\times 160^{360}}\times \frac{160^{360}}{161^{360}-160^{360}}$$
- step12: Rewrite the expression:
  $$\frac{13077\times 161^{360}}{8\times 160^{360}}\times \frac{32^{360}\times 5^{360}}{161^{360}-160^{360}}$$
- step13: Rewrite the expression:
  $$\frac{13077\times 161^{360}}{2^{3}\times 160^{360}}\times \frac{2^{1800}\times 5^{360}}{161^{360}-160^{360}}$$
- step14: Reduce the numbers:
  $$\frac{13077\times 161^{360}}{160^{360}}\times \frac{2^{1797}\times 5^{360}}{161^{360}-160^{360}}$$
- step15: Rewrite the expression:
  $$\frac{13077\times 161^{360}}{32^{360}\times 5^{360}}\times \frac{2^{1797}\times 5^{360}}{161^{360}-160^{360}}$$
- step16: Rewrite the expression:
  $$\frac{13077\times 161^{360}}{2^{1800}\times 5^{360}}\times \frac{2^{1797}\times 5^{360}}{161^{360}-160^{360}}$$
- step17: Reduce the numbers:
  $$\frac{13077\times 161^{360}}{2^{3}}\times \frac{1}{161^{360}-160^{360}}$$
- step18: Multiply the fractions:
  $$\frac{13077\times 161^{360}}{2^{3}\left(161^{360}-160^{360}\right)}$$
- step19: Multiply:
  $$\frac{13077\times 161^{360}}{8\times 161^{360}-8\times 160^{360}}$$
Let's break down the calculations step by step:

1. **Down Payment Calculation:**
   - The down payment is calculated as $$ 10\% $$ of the purchase price.
   $$
   \text{Down Payment} = 0.10 \times 290600 = 29060
   $$

2. **Loan Amount Calculation:**
   - The loan amount is the purchase price minus the down payment.
   $$
   \text{Loan Amount} = 290600 - 29060 = 261540
   $$

3. **Monthly Payment Calculation:**
   - We will use the loan payment formula:
   $$
   P_{0} = \frac{d\left(1-\left(1+\frac{r}{n}\right)^{-nt}\right)}{\left(\frac{r}{n}\right)}
   $$
   - Where:
     - $$ P_{0} = 261540 $$ (Loan amount)
     - $$ r = 0.075 $$ (Annual interest rate as a decimal)
     - $$ n = 12 $$ (Compounding periods per year)
     - $$ t = 30 $$ (Loan term in years)

- Plugging in the values:
   $$
   \text{Monthly Payment} = \frac{261540 \times \left(\frac{0.075}{12}\right) \times \left(1 + \frac{0.075}{12}\right)^{12 \times 30}}{\left(1 + \frac{0.075}{12}\right)^{12 \times 30} - 1}
   $$

- After calculating, we find:
   $$
   \text{Monthly Payment} \approx 1828.73
   $$

### Summary of Results:
- **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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