william invested $550 in an account paying an interest rate of $2\frac{1}{8}%$ compounded monthly. violet…

Explanation:

Step1: Write the compound - interest formula for William

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. William's principal $P = 550$, $A = 3\times550=1650$, $r = 2\frac{1}{8}%=0.02125$, and $n = 12$ (compounded monthly). Substitute into the formula: $1650 = 550(1+\frac{0.02125}{12})^{12t}$. First, divide both sides by 550: $3=(1 +\frac{0.02125}{12})^{12t}$. Take the natural logarithm of both sides: $\ln(3)=\ln((1+\frac{0.02125}{12})^{12t})$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(3)=12t\ln(1+\frac{0.02125}{12})$. Solve for $t$: $t_{1}=\frac{\ln(3)}{12\ln(1+\frac{0.02125}{12})}$. Calculate $1+\frac{0.02125}{12}\approx1.001770833$, $\ln(1.001770833)\approx0.001769$, $12\ln(1.001770833)\approx0.021228$, $\ln(3)\approx1.098612$. $t_{1}=\frac{1.098612}{0.021228}\approx51.75$ years.

Step2: Write the continuous - compounding formula for Violet

The continuous - compounding formula is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal), and $t$ is the number of years. Violet's principal $P = 550$, $A = 3\times550 = 1650$, and $r=2\frac{5}{8}%=0.02625$. Substitute into the formula: $1650 = 550e^{0.02625t}$. Divide both sides by 550: $3=e^{0.02625t}$. Take the natural logarithm of both sides: $\ln(3)=\ln(e^{0.02625t})$. Using the property $\ln(e^x)=x$, we get $\ln(3)=0.02625t$. Solve for $t$: $t_{2}=\frac{\ln(3)}{0.02625}$. Since $\ln(3)\approx1.098612$, $t_{2}=\frac{1.098612}{0.02625}\approx41.85$ years.

Step3: Find the difference in time

The difference $\Delta t=t_{1}-t_{2}$. $\Delta t\approx51.75 - 41.85=9.90$ years.

Answer:

$9.90$