a sum of money is invested at 12% compounded quarterly. about how long will it take for the amount of money…

a sum of money is invested at 12% compounded quarterly. about how long will it take for the amount of money to double?\ncompound interest formula: $v(t)=p(1 + \\frac{r}{n})^{nt}$\n$t$ = years since initial deposit\n$n$ = number of times compounded per year\n$r$ = annual interest rate (as a decimal)\n$p$ = initial (principal) investment\n$v(t)$ = value of investment after $t$ years\no 5.9 years\no 6.1 years\no 23.4 years\no 24.5 years

a sum of money is invested at 12% compounded quarterly. about how long will it take for the amount of money to double?\ncompound interest formula: $v(t)=p(1 + \\frac{r}{n})^{nt}$\n$t$ = years since initial deposit\n$n$ = number of times compounded per year\n$r$ = annual interest rate (as a decimal)\n$p$ = initial (principal) investment\n$v(t)$ = value of investment after $t$ years\no 5.9 years\no 6.1 years\no 23.4 years\no 24.5 years

Answer

Explanation:

Step1: Identify the values

We know that $V(t) = 2P$ (since the money doubles), $r=0.12$ (12% as a decimal), and $n = 4$ (compounded quarterly). Substitute into the compound - interest formula $V(t)=P(1 +\frac{r}{n})^{nt}$. $2P=P(1+\frac{0.12}{4})^{4t}$

Step2: Simplify the equation

Divide both sides of the equation by $P$ (since $P\neq0$). We get $2=(1 + 0.03)^{4t}$, which simplifies to $2=(1.03)^{4t}$.

Step3: Take the natural logarithm of both sides

$\ln(2)=\ln((1.03)^{4t})$. Using the property of logarithms $\ln(a^{b})=b\ln(a)$, we have $\ln(2)=4t\ln(1.03)$.

Step4: Solve for $t$

First, isolate $t$. $t=\frac{\ln(2)}{4\ln(1.03)}$. We know that $\ln(2)\approx0.6931$ and $\ln(1.03)\approx0.0296$. $t=\frac{0.6931}{4\times0.0296}=\frac{0.6931}{0.1184}\approx5.9$

Answer:

5.9 years