Roberto has a mortgage of $$ \$ 89,500 $$ at $$ 8^{\frac{1}{2}} \% $$ for 20 years. Find the total interest. Select one: a. $$ \$ 99,502.31 $$ b. $$ \$ 103,201.28 $$ c. $$ \$ 96,946.40 $$ d. $$ \$ 97,855.68 $$

Question

UpStudy AI · Accepted Answer

To find the total interest paid on the mortgage, we can use the formula for the total amount paid over the life of the loan and then subtract the principal amount. The formula for the monthly payment $$ M $$ on a fixed-rate mortgage is given by:

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

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

### Step 1: Identify the known values
- Principal $$ P = 89500 $$
- Annual interest rate $$ = 8.5\% = 0.085 $$
- Loan term $$ = 20 $$ years $$ = 20 \times 12 = 240 $$ months

### Step 2: Calculate the monthly interest rate
$$
r = \frac{0.085}{12}
$$

### Step 3: Calculate the total number of payments
$$
n = 20 \times 12 = 240
$$

### Step 4: Calculate the monthly payment $$ M $$
Now we can substitute these values into the formula for $$ M $$:

$$
M = 89500 \cdot \frac{\frac{0.085}{12}(1 + \frac{0.085}{12})^{240}}{(1 + \frac{0.085}{12})^{240} - 1}
$$

### Step 5: Calculate the total amount paid over the life of the loan
The total amount paid is:
$$
\text{Total Amount Paid} = M \times n
$$

### Step 6: Calculate the total interest paid
The total interest paid is:
$$
\text{Total Interest} = \text{Total Amount Paid} - P
$$

Now, let's perform the calculations step by step. First, we will calculate $$ r $$ and then $$ M $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.085}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{17}{200}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{17}{200}\times \frac{1}{12}$$
- step3: Multiply the fractions:
  $$\frac{17}{200\times 12}$$
- step4: Multiply:
  $$\frac{17}{2400}$$
Calculate or simplify the expression $$ 89500*(0.085/12)*(1+(0.085/12))^(20*12)/((1+(0.085/12))^(20*12)-1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{89500\left(\frac{0.085}{12}\right)\left(1+\left(\frac{0.085}{12}\right)\right)^{20\times 12}}{\left(\left(1+\left(\frac{0.085}{12}\right)\right)^{20\times 12}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{89500\left(\frac{0.085}{12}\right)\left(1+\left(\frac{0.085}{12}\right)\right)^{20\times 12}}{\left(1+\left(\frac{0.085}{12}\right)\right)^{20\times 12}-1}$$
- step2: Divide the terms:
  $$\frac{89500\left(\frac{0.085}{12}\right)\left(1+\frac{17}{2400}\right)^{20\times 12}}{\left(1+\left(\frac{0.085}{12}\right)\right)^{20\times 12}-1}$$
- step3: Add the numbers:
  $$\frac{89500\left(\frac{0.085}{12}\right)\left(\frac{2417}{2400}\right)^{20\times 12}}{\left(1+\left(\frac{0.085}{12}\right)\right)^{20\times 12}-1}$$
- step4: Divide the terms:
  $$\frac{89500\left(\frac{0.085}{12}\right)\left(\frac{2417}{2400}\right)^{20\times 12}}{\left(1+\frac{17}{2400}\right)^{20\times 12}-1}$$
- step5: Add the numbers:
  $$\frac{89500\left(\frac{0.085}{12}\right)\left(\frac{2417}{2400}\right)^{20\times 12}}{\left(\frac{2417}{2400}\right)^{20\times 12}-1}$$
- step6: Divide the terms:
  $$\frac{89500\times \frac{17}{2400}\left(\frac{2417}{2400}\right)^{20\times 12}}{\left(\frac{2417}{2400}\right)^{20\times 12}-1}$$
- step7: Multiply the numbers:
  $$\frac{89500\times \frac{17}{2400}\left(\frac{2417}{2400}\right)^{240}}{\left(\frac{2417}{2400}\right)^{20\times 12}-1}$$
- step8: Multiply the numbers:
  $$\frac{89500\times \frac{17}{2400}\left(\frac{2417}{2400}\right)^{240}}{\left(\frac{2417}{2400}\right)^{240}-1}$$
- step9: Multiply:
  $$\frac{\frac{3043\times 2417^{240}}{24\times 5^{479}\times 96^{240}}}{\left(\frac{2417}{2400}\right)^{240}-1}$$
- step10: Subtract the numbers:
  $$\frac{\frac{3043\times 2417^{240}}{24\times 5^{479}\times 96^{240}}}{\frac{2417^{240}-2400^{240}}{2400^{240}}}$$
- step11: Multiply by the reciprocal:
  $$\frac{3043\times 2417^{240}}{24\times 5^{479}\times 96^{240}}\times \frac{2400^{240}}{2417^{240}-2400^{240}}$$
- step12: Rewrite the expression:
  $$\frac{3043\times 2417^{240}}{24\times 5^{479}\times 96^{240}}\times \frac{24^{240}\times 100^{240}}{2417^{240}-2400^{240}}$$
- step13: Reduce the numbers:
  $$\frac{3043\times 2417^{240}}{5^{479}\times 96^{240}}\times \frac{24^{239}\times 100^{240}}{2417^{240}-2400^{240}}$$
- step14: Rewrite the expression:
  $$\frac{3043\times 2417^{240}}{5^{479}\times 24^{240}\times 4^{240}}\times \frac{24^{239}\times 100^{240}}{2417^{240}-2400^{240}}$$
- step15: Reduce the numbers:
  $$\frac{3043\times 2417^{240}}{5^{479}\times 24\times 4^{240}}\times \frac{100^{240}}{2417^{240}-2400^{240}}$$
- step16: Rewrite the expression:
  $$\frac{3043\times 2417^{240}}{5^{479}\times 24\times 4^{240}}\times \frac{25^{240}\times 4^{240}}{2417^{240}-2400^{240}}$$
- step17: Rewrite the expression:
  $$\frac{3043\times 2417^{240}}{5^{479}\times 24\times 4^{240}}\times \frac{5^{480}\times 4^{240}}{2417^{240}-2400^{240}}$$
- step18: Reduce the numbers:
  $$\frac{3043\times 2417^{240}}{24\times 4^{240}}\times \frac{5\times 4^{240}}{2417^{240}-2400^{240}}$$
- step19: Rewrite the expression:
  $$\frac{3043\times 2417^{240}}{8\times 3\times 4^{240}}\times \frac{5\times 4^{240}}{2417^{240}-2400^{240}}$$
- step20: Rewrite the expression:
  $$\frac{3043\times 2417^{240}}{2^{3}\times 3\times 4^{240}}\times \frac{5\times 2^{480}}{2417^{240}-2400^{240}}$$
- step21: Reduce the numbers:
  $$\frac{3043\times 2417^{240}}{3\times 4^{240}}\times \frac{5\times 2^{477}}{2417^{240}-2400^{240}}$$
- step22: Rewrite the expression:
  $$\frac{3043\times 2417^{240}}{3\times 2^{480}}\times \frac{5\times 2^{477}}{2417^{240}-2400^{240}}$$
- step23: Reduce the numbers:
  $$\frac{3043\times 2417^{240}}{3\times 2^{3}}\times \frac{5}{2417^{240}-2400^{240}}$$
- step24: Multiply the fractions:
  $$\frac{3043\times 2417^{240}\times 5}{24\left(2417^{240}-2400^{240}\right)}$$
- step25: Multiply:
  $$\frac{15215\times 2417^{240}}{24\times 2417^{240}-24\times 2400^{240}}$$
Calculate or simplify the expression $$ 776.701794 * 240 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$776.701794\times 240$$
- step1: Multiply the numbers:
  $$186408.43056$$
Calculate or simplify the expression $$ 186408.43056 - 89500 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$186408.43056-89500$$
- step1: Subtract the numbers:
  $$96908.43056$$
### Summary of Calculations

1. **Monthly Interest Rate $$ r $$**:
   $$
   r = \frac{0.085}{12} \approx 0.00708333
   $$

2. **Monthly Payment $$ M $$**:
   $$
   M \approx 776.70
   $$

3. **Total Amount Paid**:
   $$
   \text{Total Amount Paid} = M \times n = 776.70 \times 240 \approx 186408.43
   $$

4. **Total Interest Paid**:
   $$
   \text{Total Interest} = \text{Total Amount Paid} - P = 186408.43 - 89500 \approx 96908.43
   $$

### Final Result
The total interest paid is approximately $$ \$ 96,908.43 $$.

Among the options provided, the closest match is:
c. $$ \$ 96,946.40 $$

Thus, the answer is **c. $$ \$ 96,946.40 $$**.
The total interest paid is approximately \$96,946.40, so the correct answer is option c.

Roberto has a mortgage of \( \$ 89,500 \) at \( 8^{\frac{1}{2}} \% \) for 20 years. Find the total interest. Select one: a. \( \$ 99,502.31 \) b. \( \$ 103,201.28 \) c. \( \$ 96,946.40 \) d. \( \$ 97,855.68 \)

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