How long did it take for $1635 to increase to $2310 if the investment earned interest at a rate of 8.08% compounded quarterly (give your final answer in years and months, e.g., 3 years and 7.23 months)?
A) 7 years 7.69 months
B) 4 years .64 months
C) 3 years 4.79 months
D) 4 years 3.84 months
E) 8 years 1.69 months

Question

Quizplus · Accepted Answer

We can use the formula $A=P\left(1+\frac{r}{n}ight)^{nt}$, where $A$ is the final amount, $P$ is the initial amount, $r$ is the annual interest rate, $n$ is the number of times the interest is compounded in a year, and $t$ is the time in years. We are given $P=1635$, $A=2310$, $r=8.08\%$, and $n=4$ since the interest is compounded quarterly. We can solve for $t$:

\begin{align*}
2310 &= 1635\left(1+\frac{0.0808}{4}ight)^{4t}\
\frac{2310}{1635} &= 1.0202^{4t}\
\ln\left(\frac{2310}{1635}ight) &= \ln\left(1.0202^{4t}ight)\
\ln\left(\frac{2310}{1635}ight) &= 4t\ln(1.0202)\
t &= \frac{\ln\left(\frac{2310}{1635}ight)}{4\ln(1.0202)} \approx 4.81 	ext{ quarters}
\end{align*}

Note that we converted the time to quarters since the interest is compounded quarterly. To convert quarters to years and months, we divide by 4 and use the fact that there are 12 months in a year.

$t\approx 4.81	ext{ quarters} \approx 1.20	ext{ years} \approx 1 	ext{ year and } 2.41 	ext{ months}.$

Therefore, the answer is D) 4 years 3.84 months.

How Long Did It Take for $1635 to Increase to $2310