18.9669 years
Explanation:principal = $1600
future value = $7600
rate = 8.3% = 0.083
n = number of times compounded = quarterly
n = 4
time = ?
To determine the time it will take, we will apply the compound interest formula:
[tex]FV\text{ = P(1 +}\frac{r}{n})^{nt}[/tex]substitute the values into the formula:
[tex]\begin{gathered} 7600\text{ = 1600(1 +}\frac{0.083}{4})^{4\times t} \\ 7600=1600(1+0.02075)^{4t} \\ \\ \text{divide both sides by 1600:} \\ \frac{7600}{1600}=\frac{1600(1+0.02075)^{4t}}{1600} \\ 4.75\text{ = }(1+0.02075)^{4t} \\ \end{gathered}[/tex][tex]\begin{gathered} 4.75\text{ = }(1.02075)^{4t} \\ \text{take log of both sides:} \\ \log 4.75\text{ = log }(1.02075)^{4t} \\ \log 4.75\text{ = 4t log }(1.02075) \\ \\ \text{divide both sides by log }(1.02075)\colon \\ \frac{\log 4.75\text{ }}{\text{ log }(1.02075)}\text{=}\frac{\text{ 4t log }(1.02075)}{\text{ log }(1.02075)} \\ 75.8677\text{ = 4t} \end{gathered}[/tex][tex]\begin{gathered} \text{divide both sides by 4:} \\ \frac{75.8677}{4}\text{ = }\frac{4t}{4} \\ t\text{ = 18.9669} \end{gathered}[/tex]It will take 18.9669 years (4 decimal place)
