634 CHAPTER 10 Consumer Mathematics In Example 3, we use the formula to calculate the monthly principal and interest payment for a mortgage. In the formula, m is the principal and interest payment, p is the amount of the mortgage, r is the interest rate as a decimal number, n is the number of mortgage payments per year, and t is the time in years. Example 3 Using the Principal and Interest Payment Formula Reread Example 2. Use the principal and interest payment formula to calculate the Martins’ monthly principal and interest payment. Recall that the Martins are seeking a 30-year, $211,650 mortgage with an interest rate of 4%. Solution In this example the mortgage amount is $211,650, so p 211,650; = the interest rate is 4% so r 0.04; = the payments will be made monthly, so n 12; = and the length of the mortgage is 30 years, so t 30. = We substitute these values into the principal and interest payment formula. = ⎛ ⎝ ⎞ ⎠ − ⎛ + ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ − ⎛ + ⎝ ⎞ ⎠ ≈ − ≈ − ≈ ≈ − ⋅ − ⋅ − m p r n r n 1 1 211,650 0.04 12 1 1 0.04 12 211,650(0.003333333) 1 (1.003333333) 705.4999295 1 0.301795912 705.4999295 0.6982040988 1010.45 n t ( ) ( 12 30) 360 Thus, the Martins’ monthly principal and interest payment is $1010.45. Notice, this is the same payment that was determined in Example 2 part (b) using Table 10.4. To see how to determine the monthly principal and interest payment on a scientific calculator, or by using Excel, see the Technology Tip on page 618. 7 Now try Exercise 11 Examples 2 and 3 demonstrated that either Table 10.4 or the principal and interest payment formula can be used to calculate the principal and interest payment. The formula’s advantage is that it can be used for any interest rate or any length of mortgage—not just those shown in Table 10.4. Principal and interest payments can also be calculated on many websites, such as Bankrate.com. What is the effect on the total monthly mortgage payments when only the period of time of the mortgage has been changed? The total monthly mortgage payments, including principal, interest, property taxes, and homeowners’ insurance for a 4.0% mortgage for the Martins in Example 2, would be about $2422.85 for 10 years, $1845.55 for 15 years, $1562.56 for 20 years, $1397.17 for 25 years, and $1290.45 for 30 years. (You should verify these numbers for yourself.) Increasing the length of time decreases the monthly payment but increases the total amount of interest paid because MATHEMATICS TODAY Affordable and Overvalued Housing Most Affordable Markets Market Median House Price 1. Harlingen, TX $103,500 2. McAllen, TX $107,500 3. Decatur, IL $110,800 4. Joplin, MO $149,500 5. Topeka, KS $154,800 How much does a house cost? Many factors affect the price of a house, including the city in which the house is located. The tables below show median house prices in the five most affordable markets and median house prices in the five most overvalued markets as of April 2023. The ranking is determined by the percentage of the median household income needed to pay the median mortgage payment. Source: Kiplinger.com Most Overvalued Markets Market Median House Price 1. Manhattan, NY $2,434,977 2. Honolulu, HA $1,606,915 3. San Francisco, CA $1,502,557 4. Brooklyn, NY $1,349,755 5. Orange County, CA $1,248,882 Why This Is Important When purchasing a home, choosing the market in which the home is located is an important decision. The location of a home may impact whether or not you can afford to purchase the home.
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