A family has a $ 113,294​ 15​-year mortgage at 7.5 % compounded monthly. Suppose the family decides to add an extra​ $100 to its mortgage payment each month starting with the very first payment. How long will it take the family to pay off the​ mortgage? How much interest will the family​ save? 

To determine how long it will take the family to pay off the mortgage with the additional $100 monthly payment and how much interest they will save, we need to follow these steps:

1. **Calculate the original monthly payment without extra payments.**
2. **Determine the new monthly payment with the additional $100.**
3. **Calculate the number of payments needed to pay off the mortgage with the new monthly payment.**
4. **Compare the total interest paid in both scenarios to find the savings.**

### Step 1: Calculate the Original Monthly Payment

The formula for the monthly mortgage payment $M$ is:

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

where:
- $ P $ is the loan amount ($113,294)
- $ r $ is the monthly interest rate (annual rate / 12 = 7.5% / 12 = 0.625% = 0.00625)
- $ n $ is the number of payments (15 years * 12 months/year = 180 months)

Let's plug in the values:

\[ M = 113294 \frac{0.00625(1+0.00625)^{180}}{(1+0.00625)^{180} - 1} \]

First, calculate $(1 + 0.00625)^{180}$:

\[ (1 + 0.00625)^{180} \approx 3.057 \]

Then, the numerator:

\[ 0.00625 \times 3.057 \approx 0.01910625 \]

Now, the denominator:

\[ 3.057 - 1 = 2.057 \]

Thus, the monthly payment $M$ is:

\[ M = 113294 \frac{0.01910625}{2.057} \approx 1050.98 \]

### Step 2: Determine the New Monthly Payment

The new monthly payment with the additional $100 is:

\[ M_{new} = 1050.98 + 100 = 1150.98 \]

### Step 3: Calculate the Number of Payments with the New Monthly Payment

We use the formula for the number of payments to pay off a loan:

\[ n = \frac{\log \left(\frac{M}{M - Pr}\right)}{\log (1 + r)} \]

Here, $ M $ is the new monthly payment:

\[ n = \frac{\log \left(\frac{1150.98}{1150.98 - 113294 \times 0.00625}\right)}{\log (1 + 0.00625)} \]

First, calculate $ 113294 \times 0.00625 $:

\[ 113294 \times 0.00625 = 708.0875 \]

Then, $ 1150.98 - 708.0875 = 442.8925 $, and:

\[ \frac{1150.98}{442.8925} \approx 2.598 \]

So:

\[ n = \frac{\log(2.598)}{\log(1.00625)} \approx \frac{0.414}{0.00271} \approx 152.77 \]

This means the mortgage will be paid off in approximately 153 months (or 12 years and 9 months).

### Step 4: Compare Total Interest Paid

**Original Total Interest:**

\[ \text{Total paid} = 1050.98 \times 180 = 189,176.40 \]
\[ \text{Total interest} = 189,176.40 - 113,294 = 75,882.40 \]

**New Total Interest:**

\[ \text{Total paid} = 1150.98 \times 153 = 176,822.94 \]
\[ \text{Total interest} = 176,822.94 - 113,294 = 63,528.94 \]

**Interest Savings:**

\[ \text{Interest savings} = 75,882.40 - 63,528.94 = 12,353.46 \]

### Summary

1. **Time to pay off the mortgage with extra $100 monthly: ~153 months (12 years and 9 months)**
2. **Interest saved: $12,353.46**

Would you like more details or have any questions?

### Follow-Up Questions
1. How does the monthly interest rate affect the total interest paid on a mortgage?
2. What would be the impact of making a one-time extra payment towards the mortgage principal?
3. How would changes in the interest rate influence the mortgage duration and total interest paid?
4. What is the effect of increasing the monthly extra payment to $200 instead of $100?
5. How can refinancing a mortgage impact the overall cost and duration of the loan?

### Tip
Making extra payments on your mortgage not only reduces the principal faster but also significantly lowers the total interest paid over the life of the loan.

A family has a $113,294 15-year mortgage at 7.5% compounded monthly. Suppose the family decides to add an extra $100 to its mortgage payment each month starting with the very first payment. How long will it take the family to pay off the mortgage? How much interest will the family save?

Learn how to calculate the mortgage payoff time with an extra $100 monthly payment for a $113,294 loan at 7.5% interest. This solution covers key concepts like mortgage amortization, compound interest, and loan payment formulas. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation