Jenelle bought a home for $$ \$ 210,000 $$, paying $$ 18 \% $$ as a down payment, and financing the rest at $$ 5 \% $$ interest for 30 years. Round your answers to the nearest cent. How much money did Jenelle pay as a down payment? $$ \$ $$ What was the original amount financed? \$ What is her monthly payment? \$ If Jenelle makes these payments every month for thirty years, determine the total amount of money she will spend on this home. Include the down payment in your answer. \$

Question

UpStudy AI · Accepted Answer

Let's break down the problem step by step.

### Step 1: Calculate the Down Payment
Jenelle paid $$ 18\% $$ of the home price as a down payment. The home price is $$ \$ 210,000 $$.

The down payment can be calculated as:
$$
\text{Down Payment} = \text{Home Price} \times \text{Down Payment Percentage}
$$
$$
\text{Down Payment} = 210,000 \times 0.18
$$

### Step 2: Calculate the Original Amount Financed
The amount financed is the home price minus the down payment:
$$
\text{Amount Financed} = \text{Home Price} - \text{Down Payment}
$$

### Step 3: Calculate the Monthly Payment
To calculate the monthly payment for a loan, we can use the formula for a fixed-rate mortgage:
$$
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
$$
Where:
- $$ M $$ = monthly payment
- $$ P $$ = principal (amount financed)
- $$ r $$ = monthly interest rate (annual rate divided by 12)
- $$ n $$ = number of payments (loan term in months)

In this case:
- The annual interest rate is $$ 5\% $$, so the monthly interest rate $$ r $$ is:
$$
r = \frac{0.05}{12}
$$
- The loan term is $$ 30 $$ years, which is $$ 30 \times 12 = 360 $$ months.

### Step 4: Calculate the Total Amount Spent
The total amount spent on the home includes the down payment and the total of all monthly payments:
$$
\text{Total Amount Spent} = \text{Down Payment} + (M \times n)
$$

Now, let's perform the calculations.

1. Calculate the down payment.
2. Calculate the amount financed.
3. Calculate the monthly payment.
4. Calculate the total amount spent.

Let's execute these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$210000\times 0.18$$
- step1: Multiply the numbers:
  $$37800$$
Calculate or simplify the expression $$ 210000 - (210000 * 0.18) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$210000-\left(210000\times 0.18\right)$$
- step1: Multiply the numbers:
  $$210000-37800$$
- step2: Subtract the numbers:
  $$172200$$
Calculate or simplify the expression $$ (210000 - (210000 * 0.18)) * (0.05/12) * (1 + (0.05/12))^(30*12) / ((1 + (0.05/12))^(30*12) - 1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}}{\left(\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}}{\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}-1}$$
- step2: Divide the terms:
  $$\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(1+\frac{1}{240}\right)^{30\times 12}}{\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}-1}$$
- step3: Add the numbers:
  $$\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{30\times 12}}{\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}-1}$$
- step4: Divide the terms:
  $$\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{30\times 12}}{\left(1+\frac{1}{240}\right)^{30\times 12}-1}$$
- step5: Add the numbers:
  $$\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{30\times 12}}{\left(\frac{241}{240}\right)^{30\times 12}-1}$$
- step6: Multiply the numbers:
  $$\frac{\left(210000-37800\right)\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{30\times 12}}{\left(\frac{241}{240}\right)^{30\times 12}-1}$$
- step7: Subtract the numbers:
  $$\frac{172200\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{30\times 12}}{\left(\frac{241}{240}\right)^{30\times 12}-1}$$
- step8: Divide the terms:
  $$\frac{172200\times \frac{1}{240}\left(\frac{241}{240}\right)^{30\times 12}}{\left(\frac{241}{240}\right)^{30\times 12}-1}$$
- step9: Multiply the numbers:
  $$\frac{172200\times \frac{1}{240}\left(\frac{241}{240}\right)^{360}}{\left(\frac{241}{240}\right)^{30\times 12}-1}$$
- step10: Multiply the numbers:
  $$\frac{172200\times \frac{1}{240}\left(\frac{241}{240}\right)^{360}}{\left(\frac{241}{240}\right)^{360}-1}$$
- step11: Multiply:
  $$\frac{\frac{287\times 241^{360}}{2\times 5^{359}\times 48^{360}}}{\left(\frac{241}{240}\right)^{360}-1}$$
- step12: Subtract the numbers:
  $$\frac{\frac{287\times 241^{360}}{2\times 5^{359}\times 48^{360}}}{\frac{241^{360}-240^{360}}{240^{360}}}$$
- step13: Multiply by the reciprocal:
  $$\frac{287\times 241^{360}}{2\times 5^{359}\times 48^{360}}\times \frac{240^{360}}{241^{360}-240^{360}}$$
- step14: Rewrite the expression:
  $$\frac{287\times 241^{360}}{2\times 5^{359}\times 48^{360}}\times \frac{16^{360}\times 15^{360}}{241^{360}-240^{360}}$$
- step15: Rewrite the expression:
  $$\frac{287\times 241^{360}}{2\times 5^{359}\times 48^{360}}\times \frac{2^{1440}\times 15^{360}}{241^{360}-240^{360}}$$
- step16: Reduce the numbers:
  $$\frac{287\times 241^{360}}{5^{359}\times 48^{360}}\times \frac{2^{1439}\times 15^{360}}{241^{360}-240^{360}}$$
- step17: Rewrite the expression:
  $$\frac{287\times 241^{360}}{5^{359}\times 16^{360}\times 3^{360}}\times \frac{2^{1439}\times 15^{360}}{241^{360}-240^{360}}$$
- step18: Rewrite the expression:
  $$\frac{287\times 241^{360}}{5^{359}\times 2^{1440}\times 3^{360}}\times \frac{2^{1439}\times 15^{360}}{241^{360}-240^{360}}$$
- step19: Reduce the numbers:
  $$\frac{287\times 241^{360}}{5^{359}\times 2\times 3^{360}}\times \frac{15^{360}}{241^{360}-240^{360}}$$
- step20: Rewrite the expression:
  $$\frac{287\times 241^{360}}{5^{359}\times 2\times 3^{360}}\times \frac{5^{360}\times 3^{360}}{241^{360}-240^{360}}$$
- step21: Reduce the numbers:
  $$\frac{287\times 241^{360}}{2}\times \frac{5}{241^{360}-240^{360}}$$
- step22: Multiply the fractions:
  $$\frac{287\times 241^{360}\times 5}{2\left(241^{360}-240^{360}\right)}$$
- step23: Multiply:
  $$\frac{1435\times 241^{360}}{2\times 241^{360}-2\times 240^{360}}$$
Calculate or simplify the expression $$ (210000 * 0.18) + (((210000 - (210000 * 0.18)) * (0.05/12) * (1 + (0.05/12))^(30*12) / ((1 + (0.05/12))^(30*12) - 1)) * (30*12)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(210000\times 0.18\right)+\left(\left(\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}}{\left(\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}-1\right)}\right)\left(30\times 12\right)\right)$$
- step1: Remove the parentheses:
  $$\left(210000\times 0.18\right)+\left(\left(\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}}{\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}-1}\right)\times 30\times 12\right)$$
- step2: Multiply the numbers:
  $$37800+\left(\left(\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}}{\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}-1}\right)\times 30\times 12\right)$$
- step3: Divide the terms:
  $$37800+\left(\left(\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(1+\frac{1}{240}\right)^{30\times 12}}{\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}-1}\right)\times 30\times 12\right)$$
- step4: Add the numbers:
  $$37800+\left(\left(\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{30\times 12}}{\left(1+\left(\frac{0.05}{12}\right)\right)^{30\times 12}-1}\right)\times 30\times 12\right)$$
- step5: Divide the terms:
  $$37800+\left(\left(\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{30\times 12}}{\left(1+\frac{1}{240}\right)^{30\times 12}-1}\right)\times 30\times 12\right)$$
- step6: Add the numbers:
  $$37800+\left(\left(\frac{\left(210000-\left(210000\times 0.18\right)\right)\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{30\times 12}}{\left(\frac{241}{240}\right)^{30\times 12}-1}\right)\times 30\times 12\right)$$
- step7: Multiply the numbers:
  $$37800+\left(\left(\frac{\left(210000-37800\right)\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{30\times 12}}{\left(\frac{241}{240}\right)^{30\times 12}-1}\right)\times 30\times 12\right)$$
- step8: Subtract the numbers:
  $$37800+\left(\left(\frac{172200\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{30\times 12}}{\left(\frac{241}{240}\right)^{30\times 12}-1}\right)\times 30\times 12\right)$$
- step9: Divide the terms:
  $$37800+\left(\left(\frac{172200\times \frac{1}{240}\left(\frac{241}{240}\right)^{30\times 12}}{\left(\frac{241}{240}\right)^{30\times 12}-1}\right)\times 30\times 12\right)$$
- step10: Multiply the numbers:
  $$37800+\left(\left(\frac{172200\times \frac{1}{240}\left(\frac{241}{240}\right)^{360}}{\left(\frac{241}{240}\right)^{30\times 12}-1}\right)\times 30\times 12\right)$$
- step11: Multiply the numbers:
  $$37800+\left(\left(\frac{172200\times \frac{1}{240}\left(\frac{241}{240}\right)^{360}}{\left(\frac{241}{240}\right)^{360}-1}\right)\times 30\times 12\right)$$
- step12: Multiply:
  $$37800+\left(\left(\frac{\frac{287\times 241^{360}}{2\times 5^{359}\times 48^{360}}}{\left(\frac{241}{240}\right)^{360}-1}\right)\times 30\times 12\right)$$
- step13: Subtract the numbers:
  $$37800+\left(\left(\frac{\frac{287\times 241^{360}}{2\times 5^{359}\times 48^{360}}}{\frac{241^{360}-240^{360}}{240^{360}}}\right)\times 30\times 12\right)$$
- step14: Divide the terms:
  $$37800+\left(\frac{1435\times 241^{360}}{2\times 241^{360}-2\times 240^{360}}\times 30\times 12\right)$$
- step15: Multiply the terms:
  $$37800+\frac{258300\times 241^{360}}{241^{360}-240^{360}}$$
- step16: Reduce fractions to a common denominator:
  $$\frac{37800\left(241^{360}-240^{360}\right)}{241^{360}-240^{360}}+\frac{258300\times 241^{360}}{241^{360}-240^{360}}$$
- step17: Transform the expression:
  $$\frac{37800\left(241^{360}-240^{360}\right)+258300\times 241^{360}}{241^{360}-240^{360}}$$
- step18: Apply the distributive property:
  $$\frac{37800\times 241^{360}-37800\times 240^{360}+258300\times 241^{360}}{241^{360}-240^{360}}$$
- step19: Add the numbers:
  $$\frac{296100\times 241^{360}-37800\times 240^{360}}{241^{360}-240^{360}}$$
### Step 1: Calculate the Down Payment
The down payment is:
$$
\text{Down Payment} = 210,000 \times 0.18 = 37,800
$$
So, Jenelle paid **\$37,800** as a down payment.

### Step 2: Calculate the Original Amount Financed
The amount financed is:
$$
\text{Amount Financed} = 210,000 - 37,800 = 172,200
$$
Thus, the original amount financed is **\$172,200**.

### Step 3: Calculate the Monthly Payment
Using the mortgage payment formula, we find:
$$
M \approx 924.41
$$
So, Jenelle's monthly payment is approximately **\$924.41**.

### Step 4: Calculate the Total Amount Spent
The total amount spent on the home is:
$$
\text{Total Amount Spent} = 37,800 + (924.41 \times 360) \approx 370,586.46
$$
Therefore, the total amount of money Jenelle will spend on this home is approximately **\$370,586.46**.

### Summary of Results
- Down Payment: **\$37,800**
- Original Amount Financed: **\$172,200**
- Monthly Payment: **\$924.41**
- Total Amount Spent: **\$370,586.46**
- Down Payment: \$37,800 - Original Amount Financed: \$172,200 - Monthly Payment: \$924.41 - Total Amount Spent: \$370,586.46

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