The price of a home is $$ \$ 230,000 $$. The bank requires a $$ 15 \% $$ down payment. The buyer is offered two mortgage options: 15 -year fixed at $$ 6.5 \% $$ or 30 -year fixed at $$ 6.5 \% $$. Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 15 -year option? Use the following formula to determine the regular payment amount. PMT $$ =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ Find the monthly payment for the 15-year option. $$ \$ \square $$ (Round to the nearest dollar as needed.)

Question

UpStudy AI · Accepted Answer

To solve this problem, we will follow these steps:

1. **Calculate the down payment and loan amount.**
2. **Use the provided formula to calculate the monthly payment for both mortgage options.**
3. **Calculate the total payment and interest paid for each option.**
4. **Determine the savings in interest with the 15-year option.**

### Step 1: Calculate the Down Payment and Loan Amount

The price of the home is $$ P = 230,000 $$ and the down payment is $$ 15\% $$.

$$
\text{Down Payment} = 0.15 \times P
$$

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

### Step 2: Calculate Monthly Payment for the 15-Year Option

The formula for the monthly payment $$ PMT $$ is given by:

$$
PMT = \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]}
$$

Where:
- $$ P $$ is the loan amount.
- $$ r $$ is the annual interest rate (as a decimal).
- $$ n $$ is the number of payments per year (12 for monthly).
- $$ t $$ is the number of years.

For the 15-year option:
- $$ r = 0.065 $$
- $$ n = 12 $$
- $$ t = 15 $$

### Step 3: Calculate Total Payment and Interest Paid

1. Calculate the monthly payment for the 15-year option.
2. Calculate the total payment over 15 years.
3. Calculate the total interest paid.

### Step 4: Calculate Monthly Payment for the 30-Year Option

Repeat the same calculations for the 30-year option:
- $$ t = 30 $$

### Step 5: Calculate Savings in Interest

Finally, subtract the total interest paid for the 15-year option from the total interest paid for the 30-year option.

Let's perform the calculations.

#### Step 1: Calculate Down Payment and Loan Amount

$$
\text{Down Payment} = 0.15 \times 230,000
$$
$$
\text{Loan Amount} = 230,000 - \text{Down Payment}
$$

#### Step 2: Calculate Monthly Payment for the 15-Year Option

Now, we will calculate the down payment and loan amount, and then the monthly payment for the 15-year option.

Let's calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$0.15\times 230000$$
- step1: Multiply the numbers:
  $$34500$$
Calculate or simplify the expression $$ 230000 - (0.15 * 230000) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$230000-\left(0.15\times 230000\right)$$
- step1: Multiply the numbers:
  $$230000-34500$$
- step2: Subtract the numbers:
  $$195500$$
Calculate or simplify the expression $$ 195500 * (0.065 / 12) / (1 - (1 + (0.065 / 12))^(-12 * 15)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{195500\left(\frac{0.065}{12}\right)}{\left(1-\left(1+\left(\frac{0.065}{12}\right)\right)^{-12\times 15}\right)}$$
- step1: Remove the parentheses:
  $$\frac{195500\left(\frac{0.065}{12}\right)}{1-\left(1+\left(\frac{0.065}{12}\right)\right)^{-12\times 15}}$$
- step2: Divide the terms:
  $$\frac{195500\left(\frac{0.065}{12}\right)}{1-\left(1+\frac{13}{2400}\right)^{-12\times 15}}$$
- step3: Add the numbers:
  $$\frac{195500\left(\frac{0.065}{12}\right)}{1-\left(\frac{2413}{2400}\right)^{-12\times 15}}$$
- step4: Divide the terms:
  $$\frac{195500\times \frac{13}{2400}}{1-\left(\frac{2413}{2400}\right)^{-12\times 15}}$$
- step5: Multiply the numbers:
  $$\frac{195500\times \frac{13}{2400}}{1-\left(\frac{2413}{2400}\right)^{-180}}$$
- step6: Multiply the numbers:
  $$\frac{\frac{25415}{24}}{1-\left(\frac{2413}{2400}\right)^{-180}}$$
- step7: Subtract the numbers:
  $$\frac{\frac{25415}{24}}{\frac{2413^{180}-2400^{180}}{2413^{180}}}$$
- step8: Multiply by the reciprocal:
  $$\frac{25415}{24}\times \frac{2413^{180}}{2413^{180}-2400^{180}}$$
- step9: Multiply the fractions:
  $$\frac{25415\times 2413^{180}}{24\left(2413^{180}-2400^{180}\right)}$$
- step10: Multiply:
  $$\frac{25415\times 2413^{180}}{24\times 2413^{180}-24\times 2400^{180}}$$
Calculate or simplify the expression $$ 195500 * (0.065 / 12) / (1 - (1 + (0.065 / 12))^(-12 * 30)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{195500\left(\frac{0.065}{12}\right)}{\left(1-\left(1+\left(\frac{0.065}{12}\right)\right)^{-12\times 30}\right)}$$
- step1: Remove the parentheses:
  $$\frac{195500\left(\frac{0.065}{12}\right)}{1-\left(1+\left(\frac{0.065}{12}\right)\right)^{-12\times 30}}$$
- step2: Divide the terms:
  $$\frac{195500\left(\frac{0.065}{12}\right)}{1-\left(1+\frac{13}{2400}\right)^{-12\times 30}}$$
- step3: Add the numbers:
  $$\frac{195500\left(\frac{0.065}{12}\right)}{1-\left(\frac{2413}{2400}\right)^{-12\times 30}}$$
- step4: Divide the terms:
  $$\frac{195500\times \frac{13}{2400}}{1-\left(\frac{2413}{2400}\right)^{-12\times 30}}$$
- step5: Multiply the numbers:
  $$\frac{195500\times \frac{13}{2400}}{1-\left(\frac{2413}{2400}\right)^{-360}}$$
- step6: Multiply the numbers:
  $$\frac{\frac{25415}{24}}{1-\left(\frac{2413}{2400}\right)^{-360}}$$
- step7: Subtract the numbers:
  $$\frac{\frac{25415}{24}}{\frac{2413^{360}-2400^{360}}{2413^{360}}}$$
- step8: Multiply by the reciprocal:
  $$\frac{25415}{24}\times \frac{2413^{360}}{2413^{360}-2400^{360}}$$
- step9: Multiply the fractions:
  $$\frac{25415\times 2413^{360}}{24\left(2413^{360}-2400^{360}\right)}$$
- step10: Multiply:
  $$\frac{25415\times 2413^{360}}{24\times 2413^{360}-24\times 2400^{360}}$$
Calculate or simplify the expression $$ 1235.692986 * 12 * 30 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1235.692986\times 12\times 30$$
- step1: Multiply the terms:
  $$14828.315832\times 30$$
- step2: Multiply the numbers:
  $$444849.47496$$
Calculate or simplify the expression $$ 1703.014899 * 12 * 15 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1703.014899\times 12\times 15$$
- step1: Multiply the terms:
  $$20436.178788\times 15$$
- step2: Multiply the numbers:
  $$306542.68182$$
Calculate or simplify the expression $$ 444849.47496 - (195500) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$444849.47496-195500$$
- step1: Subtract the numbers:
  $$249349.47496$$
Calculate or simplify the expression $$ 306542.68182 - (195500) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$306542.68182-195500$$
- step1: Subtract the numbers:
  $$111042.68182$$
Let's summarize the calculations step by step:

### Step 1: Calculate Down Payment and Loan Amount

1. **Down Payment**:
   $$
   \text{Down Payment} = 0.15 \times 230,000 = 34,500
   $$

2. **Loan Amount**:
   $$
   \text{Loan Amount} = 230,000 - 34,500 = 195,500
   $$

### Step 2: Calculate Monthly Payment

1. **Monthly Payment for the 15-Year Option**:
   $$
   PMT_{15} = \frac{195500 \left(\frac{0.065}{12}\right)}{1 - \left(1 + \frac{0.065}{12}\right)^{-12 \times 15}} \approx 1,703
   $$

2. **Monthly Payment for the 30-Year Option**:
   $$
   PMT_{30} = \frac{195500 \left(\frac{0.065}{12}\right)}{1 - \left(1 + \frac{0.065}{12}\right)^{-12 \times 30}} \approx 1,236
   $$

### Step 3: Calculate Total Payment and Interest Paid

1. **Total Payment for the 15-Year Option**:
   $$
   \text{Total Payment}_{15} = 1,703 \times 12 \times 15 \approx 306,543
   $$

2. **Total Payment for the 30-Year Option**:
   $$
   \text{Total Payment}_{30} = 1,236 \times 12 \times 30 \approx 444,849
   $$

3. **Total Interest Paid**:
   - For the 15-Year Option:
     $$
     \text{Interest}_{15} = 306,543 - 195,500 \approx 111,043
     $$
   - For the 30-Year Option:
     $$
     \text{Interest}_{30} = 444,849 - 195,500 \approx 249,349
     $$

### Step 4: Calculate Savings in Interest

$$
\text{Savings} = \text{Interest}_{30} - \text{Interest}_{15} \approx 249,349 - 111,043 \approx 138,306
$$

### Final Results

- **Monthly Payment for the 15-Year Option**: $$ \$ 1,703 $$
- **Total Interest Paid for the 15-Year Option**: $$ \$ 111,043 $$
- **Total Interest Paid for the 30-Year Option**: $$ \$ 249,349 $$
- **Savings in Interest with the 15-Year Option**: $$ \$ 138,306 $$

If you have any further questions or need additional calculations, feel free to ask!
The monthly payment for the 15-year option is \$1,703.

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