ANSWER
22 years
EXPLANATION
The balance in a savings account with compound interest is,
[tex]A=P\mleft(1+\frac{r}{n}\mright)^{nt}[/tex]Where:
• P is the principal amount
,• r is the interest rate
,• n is the number of times the interest is compounded per year
,• t is the time in years
,• A is the balance in the account after t years
In this case, we have that the interest rate is r = 0.05, the interest is compounded quarterly so n = 4, and we have to find t for A = 3P,
[tex]3P=P\mleft(1+\frac{0.05}{4}\mright)^{4t}[/tex]To solve, divide both sides by P,
[tex]\begin{gathered} 3=\mleft(1+0.0125\mright)^{4t} \\ 3=(1.0125)^{4t} \end{gathered}[/tex]Then, take the logarithm to both sides of the equation. This way we apply the property of the logarithm of a power,
[tex]\begin{gathered} \log 3=\log (1.0125^{4t}) \\ \log 3=4t\log (1.0125) \end{gathered}[/tex]Divide both sides by 4*log(1.0125) and solve,
[tex]t=\frac{\log 3}{4\log (1.0125)}\approx22.11[/tex]Hence, it will take approximately 22 years for the balance to triple.
