Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 8 - Personal Finance - 8.7 The Cost of Home Ownership - Exercise Set 8.7: 5

Answer

It is more economical to have a 20-year mortgage at a rate of 7.5%. The amount of interest that is saved is $\$106,440$.

Work Step by Step

We can use this formula to calculate the payments for a mortgage: $PMT = \frac{P~(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}~]}$ $PMT$ is the amount of the regular payment $P$ is the price of the home $r$ is the interest rate $n$ is the number of payments per year $t$ is the number of years We can calculate the monthly payments for a 30-year mortgage at 8%. $PMT = \frac{P~(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}~]}$ $PMT = \frac{(\$150,000)~(\frac{0.08}{12})}{[1-(1+\frac{0.08}{12})^{-(12)(30)}~]}$ $PMT = \$1101$ The monthly payments are $\$1101$ We can find the total amount paid. $\$1101 \times 360 = \$396,360$ The interest is the difference between the total amount paid and the price of the home. $I = \$396,360 - \$150,000 = \$246,360$ The interest is $\$246,300$ We can calculate the monthly payments for a 20-year mortgage at 7.5%. $PMT = \frac{P~(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}~]}$ $PMT = \frac{(\$150,000)~(\frac{0.075}{12})}{[1-(1+\frac{0.075}{12})^{-(12)(20)}~]}$ $PMT = \$1208$ The monthly payments are $\$1208$ We can find the total amount paid. $\$1208 \times 240 = \$289,920$ The interest is the difference between the total amount paid and the price of the home. $I = \$289,920 - \$150,000 = \$139,920$ The interest is $\$139,920$ We can calculate the difference in interest paid. $\$246,360-\$139,920 = \$106,440$ It is more economical to have a 20-year mortgage at a rate of 7.5%. The amount of interest that is saved is $\$106,440$
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