Under compound interest compounded annually, a sum of money grows to Rs. 24,200 at the end of the 2nd year and Rs. 26,620 at the end of the 3rd year. Find the original principal amount.

Correct Option: Rs. 20,000

Step-by-step Explanation:

Let the original principal amount be $P$ and the annual rate of interest be $r$.

The formula for compound interest compounded annually is:

$A=P{(1+r)}^n$

where $A$ is the amount accumulated after $n$ years.

Given data:

Amount at the end of the 2nd year ($A_2$) = Rs. 24,200
Amount at the end of the 3rd year ($A_3$) = Rs. 26,620

We can express these amounts in terms of $P$ and $r$:

$A_2=P{(1+r)}^2=\mathrm{24,200}$

$A_3=P{(1+r)}^3=\mathrm{26,620}$

Step 1: Calculate the annual interest rate ($r$)

Under compound interest compounded annually, the amount at the end of the 2nd year serves as the principal for the 3rd year. Therefore, the interest earned during the 3rd year is the difference between $A_3$ and $A_2$:

$\text{Interest in 3rd year}=A_3-A_2$

$\text{Interest in 3rd year}=\mathrm{26,620}-\mathrm{24,200}=\mathrm{2,420}$

This interest of Rs. 2,420 is generated on the amount of Rs. 24,200 over 1 year. The rate of interest is calculated as:

$r=\frac{\mathrm{2,420}}{\mathrm{24,200}}\times 100\%=10\%$

Step 2: Calculate the original principal amount ($P$)

With an annual interest rate of 10% ($r=0.10$), the factor $1+r$ is:

$1+r=1+0.10=1.1$

Substituting $1+r=1.1$ into the equation for the 2nd year amount:

$\mathrm{24,200}=P{\left(1.1\right)}^2$

$\mathrm{24,200}=P\times 1.21$

Solving for $P$:

$P=\frac{\mathrm{24,200}}{1.21}=\mathrm{20,000}$

Therefore, the original principal amount is Rs. 20,000.