Question : At a certain rate of interest per annum, compounded annually, a certain sum of money amounts to two times itself in 11 years. In how many years will the sum of money amount to four times itself at the previous rate of interest per annum, also compounded annually?
Option 1: 20 years
Option 2: 5.5 years
Option 3: 22 years
Option 4: 33 years
Correct Answer: 22 years
Solution :
Present value = $P(1+\frac{r}{100})^n$, where $P$ is the principal, $r$ is the annual interest rate and $n$ is the time in years.
Let the sum of money be $P$ and the interest rate is $r$.
The amount after 11 years is,
$2P=P(1+\frac{r}{100})^{11}$
⇒ $(1+\frac{r}{100})=2^{\frac{1}{11}}$
Let the amount be increased to four times after n years.
⇒ $4P=P(1+\frac{r}{100})^n$
⇒ $4=(2^{\frac{1}{11}})^n$
⇒ $2^2=2^{\frac{n}{11}}$
⇒ $2=\frac{n}{11}$
⇒ $n=22$
Hence, the correct answer is 22 years.
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