Find the number of years for $14,100 to grow to $16,600 at 3% compounded semiannually. Round to the nearest tenth of a year.
A) 5.4 yr
B) 6.6 yr
C) 4.6 yr
D) 5.5 yr

Question

Quizplus · Accepted Answer

The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the amount of money accumulated after $n$ years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (decimal), $n$ is the number of times that interest is compounded per year, and $t$ is the time the money is invested for in years. Here, $A = \$16,600$, $P = \$14,100$, $r = 3\% = 0.03$, and $n = 2$ (since the interest is compounded semiannually). We need to solve for $t$.Rearranging the formula to solve for $t$, we get:$$t = \frac{\log(A/P)}{n \cdot \log(1 + r/n)}$$Substituting the given values:$$t = \frac{\log(16600/14100)}{2 \cdot \log(1 + 0.03/2)}$$$$t = \frac{\log(1.1773)}{2 \cdot \log(1.015)}$$$$t \approx \frac{0.0706}{2 \cdot 0.0064}$$$$t \approx \frac{0.0706}{0.0128}$$$$t \approx 5.5156$$Rounded to the nearest tenth of a year, $t \approx 5.5$ years.

Find the Number of Years for $14,100 to Grow to $16,600