You can afford a $$ \$ 1150 $$ per month mortgage payment. You've found a 30 year loan at $$ 7 \% $$ interest. a) How big of a loan can you afford? Round your answer to the nearest dollar. b) How much total money will you pay the loan company? Round your answer to the nearest dollar. c) How much of that money is interest? Round your answer to the nearest dollar. Question Help:

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UpStudy AI · Accepted Answer

To solve this problem, we will use the formula for the monthly payment of a fixed-rate mortgage, which is given by:

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

Where:
- $$ M $$ is the monthly payment
- $$ P $$ is the loan principal (the amount of the loan)
- $$ r $$ is the monthly interest rate (annual interest rate divided by 12)
- $$ n $$ is the number of payments (loan term in months)

### Given:
- Monthly payment $$ M = 1150 $$
- Annual interest rate $$ = 7\% $$ or $$ 0.07 $$
- Loan term $$ = 30 $$ years or $$ 30 \times 12 = 360 $$ months

### Step 1: Calculate the monthly interest rate
$$
r = \frac{0.07}{12}
$$

### Step 2: Rearranging the formula to solve for $$ P $$
We can rearrange the formula to find $$ P $$:

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

### Step 3: Calculate $$ P $$
Now we will calculate $$ P $$ using the values we have.

### Step 4: Calculate the total amount paid over the life of the loan
The total amount paid over the life of the loan is given by:

$$
\text{Total Paid} = M \times n
$$

### Step 5: Calculate the total interest paid
The total interest paid is the total amount paid minus the principal:

$$
\text{Total Interest} = \text{Total Paid} - P
$$

Now, let's perform the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$1150\left(30\times 12\right)$$
- step1: Remove the parentheses:
  $$1150\times 30\times 12$$
- step2: Multiply the terms:
  $$34500\times 12$$
- step3: Multiply the numbers:
  $$414000$$
Calculate or simplify the expression $$ 1150 * ((1 + (0.07/12))^(30*12) - 1) / ((0.07/12) * (1 + (0.07/12))^(30*12)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1150\left(\left(1+\left(\frac{0.07}{12}\right)\right)^{30\times 12}-1\right)}{\left(\left(\frac{0.07}{12}\right)\left(1+\left(\frac{0.07}{12}\right)\right)^{30\times 12}\right)}$$
- step1: Remove the parentheses:
  $$\frac{1150\left(\left(1+\left(\frac{0.07}{12}\right)\right)^{30\times 12}-1\right)}{\left(\frac{0.07}{12}\right)\left(1+\left(\frac{0.07}{12}\right)\right)^{30\times 12}}$$
- step2: Divide the terms:
  $$\frac{1150\left(\left(1+\left(\frac{0.07}{12}\right)\right)^{30\times 12}-1\right)}{\left(\frac{0.07}{12}\right)\left(1+\frac{7}{1200}\right)^{30\times 12}}$$
- step3: Add the numbers:
  $$\frac{1150\left(\left(1+\left(\frac{0.07}{12}\right)\right)^{30\times 12}-1\right)}{\left(\frac{0.07}{12}\right)\left(\frac{1207}{1200}\right)^{30\times 12}}$$
- step4: Divide the terms:
  $$\frac{1150\left(\left(1+\frac{7}{1200}\right)^{30\times 12}-1\right)}{\left(\frac{0.07}{12}\right)\left(\frac{1207}{1200}\right)^{30\times 12}}$$
- step5: Add the numbers:
  $$\frac{1150\left(\left(\frac{1207}{1200}\right)^{30\times 12}-1\right)}{\left(\frac{0.07}{12}\right)\left(\frac{1207}{1200}\right)^{30\times 12}}$$
- step6: Multiply the numbers:
  $$\frac{1150\left(\left(\frac{1207}{1200}\right)^{360}-1\right)}{\left(\frac{0.07}{12}\right)\left(\frac{1207}{1200}\right)^{30\times 12}}$$
- step7: Subtract the numbers:
  $$\frac{1150\times \frac{1207^{360}-1200^{360}}{1200^{360}}}{\left(\frac{0.07}{12}\right)\left(\frac{1207}{1200}\right)^{30\times 12}}$$
- step8: Divide the terms:
  $$\frac{1150\times \frac{1207^{360}-1200^{360}}{1200^{360}}}{\frac{7}{1200}\left(\frac{1207}{1200}\right)^{30\times 12}}$$
- step9: Multiply the numbers:
  $$\frac{1150\times \frac{1207^{360}-1200^{360}}{1200^{360}}}{\frac{7}{1200}\left(\frac{1207}{1200}\right)^{360}}$$
- step10: Multiply the numbers:
  $$\frac{\frac{23\times 1207^{360}-23\times 1200^{360}}{50^{359}\times 24^{360}}}{\frac{7}{1200}\left(\frac{1207}{1200}\right)^{360}}$$
- step11: Multiply the numbers:
  $$\frac{\frac{23\times 1207^{360}-23\times 1200^{360}}{50^{359}\times 24^{360}}}{\frac{7\times 1207^{360}}{1200^{361}}}$$
- step12: Multiply by the reciprocal:
  $$\frac{23\times 1207^{360}-23\times 1200^{360}}{50^{359}\times 24^{360}}\times \frac{1200^{361}}{7\times 1207^{360}}$$
- step13: Rewrite the expression:
  $$\frac{23\times 1207^{360}-23\times 1200^{360}}{50^{359}\times 24^{360}}\times \frac{50^{361}\times 24^{361}}{7\times 1207^{360}}$$
- step14: Reduce the numbers:
  $$\frac{23\times 1207^{360}-23\times 1200^{360}}{24^{360}}\times \frac{50^{2}\times 24^{361}}{7\times 1207^{360}}$$
- step15: Rewrite the expression:
  $$\frac{23\times 1207^{360}-23\times 1200^{360}}{24^{360}}\times \frac{2^{2}\times 25^{2}\times 24^{361}}{7\times 1207^{360}}$$
- step16: Rewrite the expression:
  $$\frac{23\times 1207^{360}-23\times 1200^{360}}{8^{360}\times 3^{360}}\times \frac{2^{2}\times 25^{2}\times 24^{361}}{7\times 1207^{360}}$$
- step17: Rewrite the expression:
  $$\frac{23\times 1207^{360}-23\times 1200^{360}}{2^{1080}\times 3^{360}}\times \frac{2^{2}\times 25^{2}\times 24^{361}}{7\times 1207^{360}}$$
- step18: Reduce the numbers:
  $$\frac{23\times 1207^{360}-23\times 1200^{360}}{2^{1078}\times 3^{360}}\times \frac{25^{2}\times 24^{361}}{7\times 1207^{360}}$$
- step19: Rewrite the expression:
  $$\frac{23\times 1207^{360}-23\times 1200^{360}}{2^{1078}\times 3^{360}}\times \frac{25^{2}\times 8^{361}\times 3^{361}}{7\times 1207^{360}}$$
- step20: Rewrite the expression:
  $$\frac{23\times 1207^{360}-23\times 1200^{360}}{2^{1078}\times 3^{360}}\times \frac{25^{2}\times 2^{1083}\times 3^{361}}{7\times 1207^{360}}$$
- step21: Reduce the numbers:
  $$\left(23\times 1207^{360}-23\times 1200^{360}\right)\times \frac{25^{2}\times 2^{5}\times 3}{7\times 1207^{360}}$$
- step22: Multiply the fractions:
  $$\frac{\left(23\times 1207^{360}-23\times 1200^{360}\right)\times 60000}{7\times 1207^{360}}$$
- step23: Multiply:
  $$\frac{1380000\times 1207^{360}-1380000\times 1200^{360}}{7\times 1207^{360}}$$
Calculate or simplify the expression $$ 414000 - 172853.70314 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$414000-172853.70314$$
- step1: Subtract the numbers:
  $$241146.29686$$
### a) How big of a loan can you afford?
Using the formula, we calculated the principal $$ P $$:

$$
P \approx 172854
$$

So, you can afford a loan of approximately **$172,854**.

### b) How much total money will you pay the loan company?
The total amount paid over the life of the loan is:

$$
\text{Total Paid} = 1150 \times (30 \times 12) = 1150 \times 360 = 414000
$$

So, the total amount you will pay the loan company is **$414,000**.

### c) How much of that money is interest?
The total interest paid is calculated as follows:

$$
\text{Total Interest} = \text{Total Paid} - P = 414000 - 172854 \approx 241146
$$

Thus, the total interest paid is approximately **$241,146**.

### Summary of Answers:
- a) Loan amount: **$172,854**
- b) Total payment: **$414,000**
- c) Total interest: **$241,146**
You can afford a loan of approximately $172,854. Over 30 years, you'll pay a total of $414,000, with $241,146 of that being interest.

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