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: 6

Answer

It is more economical to have a 15-year mortgage at a rate of 7.5%. The amount of interest that is saved is $\$87,480$.

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%. $PMT = \frac{P~(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}~]}$ $PMT = \frac{(\$90,000)~(\frac{0.08}{12})}{[1-(1+\frac{0.08}{12})^{-(12)(30)}~]}$ $PMT = \$660$ The monthly payments are $\$660$ We can find the total amount paid. $\$660 \times 360 = \$237,600$ The interest is the difference between the total amount paid and the price of the home. $I = \$237,600 - \$90,000 = \$147,600$ The interest is $\$147,600$ We can calculate the monthly payments for a 15-year mortgage at 7.5%. $PMT = \frac{P~(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}~]}$ $PMT = \frac{(\$90,000)~(\frac{0.075}{12})}{[1-(1+\frac{0.075}{12})^{-(12)(15)}~]}$ $PMT = \$834$ The monthly payments are $\$834$ We can find the total amount paid. $\$834 \times 180 = \$150,120$ The interest is the difference between the total amount paid and the price of the home. $I = \$150,120 - \$90,000 = \$60,120$ The interest is $\$60,120$ We can calculate the difference in interest paid. $\$147,600-\$60,120 = \$87,480$ It is more economical to have a 15-year mortgage at a rate of 7.5%. The amount of interest that is saved is $\$87,480$
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