+253 Find the time required for an investment of $5000 to grow to $7000 at an interest rate of 7.5% per year, compounded quarterly.
+118608 $$\\i=0.075/4=0.01875\\
n \;\;quarters\\
\begin{array}{rlll}
FV&=&P(1+i)^n&\\\\
7000&=&5000(1.01875)^n&\\\\
7000/5000&=&(1.01875)^n&\\\\
7/5&=&(1.01875)^n&\\\\
log(7/5)&=&log(1.01875)^n&\\\\
log(7/5)&=&nlog(1.01875)&\\\\
log(7/5)log(1.01875)&=&n&\\\\
n&\approx &18.11\;\;quarters&\\\\
n&\approx &18.11/4\;\;years\\\
n&\approx &4.528\;\;years\\\\
\end{array}$$
Since the payments are made in quarters, the best answer is probably 19 quaters which is 4 and 3/4 years.
+118608 $$\\i=0.075/4=0.01875\\
n \;\;quarters\\
\begin{array}{rlll}
FV&=&P(1+i)^n&\\\\
7000&=&5000(1.01875)^n&\\\\
7000/5000&=&(1.01875)^n&\\\\
7/5&=&(1.01875)^n&\\\\
log(7/5)&=&log(1.01875)^n&\\\\
log(7/5)&=&nlog(1.01875)&\\\\
log(7/5)log(1.01875)&=&n&\\\\
n&\approx &18.11\;\;quarters&\\\\
n&\approx &18.11/4\;\;years\\\
n&\approx &4.528\;\;years\\\\
\end{array}$$
Since the payments are made in quarters, the best answer is probably 19 quaters which is 4 and 3/4 years.
