a sum of money is invested at 12% compounded ...
a sum of money is invested at 12% compounded quarterly. about how long will it take for the amount of money to double? compound interest formula: v(t)=p(1 + r/n)^nt t = years since initial deposit n = number of times compounded per year r = annual interest rate (as a decimal) p = initial (principal) investment v(t) = value of investment after t years 23.4 years 6.1 years 24.5 years 5.9 years
Answer
# Explanation:
## Step1: Identify values
Given $r = 0.12$ (12% as a decimal), $n = 4$ (compounded quarterly), and since the money doubles, $V(t)=2P$. Substitute into $V(t)=P(1 +\frac{r}{n})^{nt}$.
$2P=P(1+\frac{0.12}{4})^{4t}$
## Step2: Simplify equation
Divide both sides by $P$ (since $P\neq0$), we get $2=(1 + 0.03)^{4t}$, that is $2 = 1.03^{4t}$.
## Step3: Take logarithms
Take the natural - logarithm of both sides: $\ln(2)=\ln(1.03^{4t})$.
Using the property $\ln(a^b)=b\ln(a)$, we have $\ln(2)=4t\ln(1.03)$.
## Step4: Solve for $t$
$t=\frac{\ln(2)}{4\ln(1.03)}$.
Calculate $\ln(2)\approx0.6931$ and $\ln(1.03)\approx0.0296$.
$t=\frac{0.6931}{4\times0.0296}=\frac{0.6931}{0.1184}\approx5.9$
# Answer:
5.9 years