Question

Anna wishes to invest some amount in a scheme. If the scheme awards her an annual compound interest of 25%, how long will it take for her investment to double?

• A. More than 3 years
• B. More than 4 years
• C. More than 5 years
• D. More than 2 years

Answer 1

The correct option is A More than 3 years

$Annualcompoundinterest=R=25\%$

Any principal amount (P) that earns an annual compound interest will grow in 'n' years:

$P\left(n\right)=P(1+R{)}^n$

Where

$P\left(n\right)=Principalafter{}^{\prime }{n}^{\prime }yearsP=InitialprincipalamountinvestedR=Percentageofannualcompoundinterestn=Numberofyears$

We want to find how long it will take for the investment to double.

$\therefore P\left(n\right)=2\times P$
$⟹2\times P=P(1+R{)}^n$
$⟹2=(1+R{)}^n$

In this case, the annual compound interest is 25%.

$⟹2=(1+25\%{)}^n$
$⟹2=(1+\frac{25}{100}{)}^n$
$⟹2=(1+\frac{1}{4}{)}^n$
$⟹2=(\frac{5}{4}{)}^n$

$\text{We examine the value of}(\frac{5}{4}{)}^n\text{for different}\text{natural number values of n.}$

$Case1:n=1$
$(\frac{5}{4}{)}^n=(\frac{5}{4}{)}^1=\frac{5}{4}<2$

$Case2:n=2$
$(\frac{5}{4}{)}^n=(\frac{5}{4}{)}^2=\frac{25}{16}<2$

$Case3:n=3$
$(\frac{5}{4}{)}^n=(\frac{5}{4}{)}^3=\frac{125}{64}<2$

$Case4:n=4$
$(\frac{5}{4}{)}^n=(\frac{5}{4}{)}^4=\frac{625}{256}>2$

$⟹3<n<4$

Hence, it will take somewhere between 3 and 4 years for the invested amount to double.