Tom wishes to purchase a property that's been valued at  
$
470
,
000
 . He has  
10
%
  of this amount available as a cash deposit and will require a mortgage for the remaining amount. The bank offers him a  
30
 -year mortgage at  
2.75
%
  interest. Calculate the total interest Tom will pay over the lifetime of the loan. 

Round your answer to the nearest dollar.  Do not round until you have calculated the final answer.



To calculate the total interest Tom will pay over the lifetime of the loan, we can break the problem into a few steps.

### Step 1: Calculate the loan amount.
Tom has 10% of the property price as a deposit, so the remaining 90% will be covered by the mortgage.

\[
\text{Loan Amount} = \text{Property Value} - \text{Deposit}
\]
\[
\text{Deposit} = 10\% \times 470,000 = 0.10 \times 470,000 = 47,000
\]
\[
\text{Loan Amount} = 470,000 - 47,000 = 423,000
\]

### Step 2: Calculate the monthly interest rate.
The annual interest rate is 2.75%, so the monthly interest rate is:

\[
\text{Monthly Interest Rate} = \frac{2.75\%}{12} = \frac{0.0275}{12} \approx 0.00229167
\]

### Step 3: Determine the number of monthly payments.
For a 30-year mortgage, the number of monthly payments is:

\[
\text{Number of Payments} = 30 \times 12 = 360
\]

### Step 4: Use the loan amortization formula to calculate the monthly payment.
The formula for the monthly payment is:

\[
M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
\]

Where:
- $ M $ is the monthly payment,
- $ P $ is the loan amount (\$423,000),
- $ r $ is the monthly interest rate (0.00229167),
- $ n $ is the number of payments (360).

Substituting the values:

\[
M = 423,000 \times \frac{0.00229167(1+0.00229167)^{360}}{(1+0.00229167)^{360} - 1}
\]

Let's calculate the monthly payment first.The monthly mortgage payment for Tom is approximately \$1,726.86.

### Step 5: Calculate the total payment over the lifetime of the loan.
To find the total amount Tom will pay over the 30 years, multiply the monthly payment by the number of payments:

\[
\text{Total Payment} = \text{Monthly Payment} \times \text{Number of Payments}
\]

\[
\text{Total Payment} = 1,726.86 \times 360
\]

Let's calculate that.The total amount Tom will pay over the lifetime of the loan is approximately \$621,670.

### Step 6: Calculate the total interest paid.
To find the total interest paid, subtract the original loan amount from the total payment:

\[
\text{Total Interest} = \text{Total Payment} - \text{Loan Amount}
\]

\[
\text{Total Interest} = 621,670 - 423,000
\]

Let's compute that.The total interest Tom will pay over the lifetime of the loan is approximately **\$198,670**.

Would you like further clarification or a breakdown of the calculations? 

Here are five related questions to consider:
1. How would the total interest change if the interest rate increased to 3.5%?
2. What would the monthly payment be if the loan term was 15 years instead of 30 years?
3. How does increasing the deposit amount affect the total interest paid?
4. What happens to the total payment if the loan amount increases by 5%?
5. How does the monthly payment change if Tom pays extra each month?

**Tip:** Reducing the loan term (e.g., 15 years instead of 30) can significantly lower the total interest paid over the lifetime of the loan, even if the monthly payments increase.

Tom wishes to purchase a property that's been valued at $470,000. He has 10% of this amount available as a cash deposit and will require a mortgage for the remaining amount. The bank offers him a 30-year mortgage at 2.75% interest. Calculate the total interest Tom will pay over the lifetime of the loan. Round your answer to the nearest dollar. Do not round until you have calculated the final answer.

Learn how to calculate the total interest paid on a 30-year mortgage for a $470,000 property with a 2.75% interest rate. The problem covers loan amortization, interest calculation, and includes a step-by-step breakdown of the process. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation