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Chapter 3: Problem 77

The total interest \(u\) paid on a home mortgage of \(P\) dollars at interest rate \(r\) for \(t\) years is \(u=P\left[\frac{r t}{1-\left(\frac{1}{1+r / 12}\right)^{12 t}}-1\right]\) Consider a $$\$ 120,000$$ home mortgage at \(7 \frac{1}{2} \%\). (a) Use a graphing utility to graph the total interest function. (b) Approximate the length of the mortgage for which the total interest paid is the same as the size of the mortgage. Is it possible that some people are paying twice as much in interest charges as the size of the mortgage?

Short Answer

Expert verified
The length of the mortgage for which the total interest paid matches the size of the mortgage, and whether some individuals pay twice as much interest as their mortgage's size would require numerical solutions which are beyond the scope of this analysis. It is, however, technically feasible, particularly for long-term, high-interest loans.

Step by step solution

01

Input the given values

First, substitute the given values into the equation: \(P = \$120,000\) (the amount of the home mortgage) and \(r = 0.075\) (the annual interest rate, converted from percentage to decimal form).
02

Approximating the length of the mortgage where total interest equals mortgage size

This is the case when \(u = P\). Therefore, set \( u = P \) in the equation. Doing so gives us: \( \$120,000 = $120,000 \left[ \frac{0.075t}{1-\left (\frac{1}{1+0.075/12}\right )^{12t}} - 1 \right ]\). Solving this equation numerically, (for example, using a programming language or a graphing calculator), would give an approximation for \(t\), the length of the mortgage where total interest paid is the same as the mortgage size.
03

Investigating if total interest could be twice the size of the mortgage

This case happens when \(u = 2P\). As such, we would set \( u = 2P \) in the equation, giving us: \( \$240,000 = \$120,000 \left[ \frac{0.075t}{1 - \left (\frac{1}{1+0.075/12}\right ) ^{12t}} - 1 \right ]\). Again, solving this equation numerically would give us a value for \(t\), the length of the mortgage where total interest paid is twice the mortgage size. If such a \(t\) exists, then yes, some people are indeed paying twice as much in interest charges as the size of the mortgage.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Total Interest Formula
Understanding the total interest formula for a home mortgage is critical in financial planning. This formula calculates the total amount of interest you will pay over the life of the mortgage. Given by

\( u = P\left[\frac{r t}{1-\left(\frac{1}{1+r / 12}\right)^{12 t}}-1\right] \),

where \( u \) is the total interest, \( P \) is the principal amount (the initial loan), \( r \) is the annual interest rate, and \( t \) is the time in years that the mortgage will be held. To break it down, this formula takes into account how interest compounds monthly and how that accumulation translates over the years. Understanding this concept can help prevent being caught off guard by the total amount paid in interest which can sometimes equal or even surpass the actual mortgage size.
Mortgage Size Comparison
Comparing the total interest to the size of the mortgage is an eye-opening exercise. For instance, when dealing with a $120,000 mortgage at a 7.5% interest rate, one might find that over a certain period, the interest accrued can equal or even double the initial mortgage size.

Through calculation and graphing utilities, one can find that there's a specific time \( t \) where \( u = P \), i.e., the total interest equals the mortgage size. This can serve as a powerful illustration of how interest can significantly add to the cost of borrowing. It also emphasizes the importance of considering the interest rate and the mortgage term when taking out a loan, as extending the term can sometimes lead to paying more in interest than the amount borrowed.
Amortization of Mortgage
Amortization refers to the process of spreading out a loan (in this case, a mortgage) into a series of fixed payments over time. These payments cover both the principal and the interest on the mortgage, with the goal of completely paying off the loan by the end of the term.

The amortization schedule is pivotal as it helps both lenders and borrowers understand the timeline of payments, how much of each payment goes towards interest versus principal, and how the balance of the mortgage decreases over time. Earlier payments in an amortization schedule are typically more heavily weighted towards interest, while later payments are more directed towards reducing the principal. Understanding this can lead to strategies such as making extra principal payments early on to reduce the total interest paid over the life of the mortgage.

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$6 \log _{3}(0.5 x)=11$$

Determine whether the statement is true or false. Justify your answer. The domain of a logistic growth function cannot be the set of real numbers.

A $$\$ 120,000$$ home mortgage for 30 years at \(7 \frac{1}{2} \%\) has a monthly payment of $$\$ 839.06 .$$ Part of the monthly payment is paid toward the interest charge on the unpaid balance, and the remainder of the payment is used to reduce the principal. The amount that is paid toward the interest is \(u=M-\left(M-\frac{P r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}\) and the amount that is paid toward the reduction of the principal is \(v=\left(M-\frac{P r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}\) In these formulas, \(P\) is the size of the mortgage, \(r\) is the interest rate, \(M\) is the monthly payment, and \(t\) is the time (in years). (a) Use a graphing utility to graph each function in the same viewing window. (The viewing window should show all 30 years of mortgage payments.) (b) In the early years of the mortgage, is the larger part of the monthly payment paid toward the interest or the principal? Approximate the time when the monthly payment is evenly divided between interest and principal reduction. (c) Repeat parts (a) and (b) for a repayment period of 20 years \((M=\$ 966.71) .\) What can you conclude?

The table shows the time \(t\) (in seconds) required for a car to attain a speed of \(s\) miles per hour from a standing start. $$ \begin{array}{|c|c|} \hline \text { Speed, } s & \text { Time, } t \\ \hline 30 & 3.4 \\ 40 & 5.0 \\ 50 & 7.0 \\ 60 & 9.3 \\ 70 & 12.0 \\ 80 & 15.8 \\ 90 & 20.0 \\ \hline \end{array} $$ Two models for these data are as follows. \(t_{1}=40.757+0.556 s-15.817 \ln s\) \(t_{2}=1.2259+0.0023 s^{2}\) (a) Use the regression feature of a graphing utility to find a linear model \(t_{3}\) and an exponential model \(t_{4}\) for the data. (b) Use a graphing utility to graph the data and each model in the same viewing window. (c) Create a table comparing the data with estimates obtained from each model. (d) Use the results of part (c) to find the sum of the absolute values of the differences between the data and the estimated values given by each model. Based on the four sums, which model do you think best fits the data? Explain.

The amount of time (in hours per week) a student utilizes a math-tutoring center roughly follows the normal distribution \(y=0.7979 e^{-(x-5.4)^{2} / 0.5},\) \(4 \leq x \leq 7,\) where \(x\) is the number of hours. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average number of hours per week a student uses the tutoring center.
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