Chapter 2.4, Problem 31E

Home Equity When you purchase a home by securing a mortgage, the total paid toward the principle is your equity in the home. If your mortgage is for $P$ dollars, and if the term of the mortgage is $t$ months, then your equity $E$ in dollars, after $k$ monthly payments is given by

$E=P\times \frac{{\left(1+r\right)}^k-1}{{\left(1+r\right)}^t-1}$

Here $r$ is the monthly interest rate as a decimal, with $r=APR/12$

Suppose you have a mortgage of $\$425,000$ at an $APR$ of $9\%$ and a term of $30$ years. How long does it take for your equity to reach half of the amount of the original mortgage? (Round $r$ to four decimal places.)