A sum becomes ₹7,200 in 2 years and ₹8,640 in 3 years at compound interest. What is the original principal?

The correct option is ₹5,000.

Step-by-Step Explanation:

Step 1: Understand the given data

Let the principal amount be $P$ and the rate of interest per annum be $r$.

The amount after 2 years ($A_2$) = ₹7,200

The amount after 3 years ($A_3$) = ₹8,640

Step 2: Find the rate of interest ($r$)

In compound interest, the amount at the end of a year acts as the principal for the next year.

Therefore, the interest earned in the 3rd year is the difference between the amounts at the end of the 3rd year and the 2nd year:

$\text{Interest for the 3rd year}=\text{₹8,640}-\text{₹7,200}=\text{₹1,440}$

This interest of ₹1,440 is earned on ₹7,200 in 1 year. So, we can find the rate of interest ($r$) using the simple interest formula for 1 year:

$r=\left(\frac{1440}{7200}\right)\times 100$

$r=20\%$

Step 3: Calculate the original principal ($P$)

The formula for compound interest amount after $n$ years is:

$A=P{(1+\frac{r}{100})}^n$

Substituting $A_2=7200$, $r=20$, and $n=2$:

$7200=P{(1+\frac{20}{100})}^2$

$7200=P{(1+\frac{1}{5})}^2$

$7200=P{\left(\frac{6}{5}\right)}^2$

$7200=P\times \frac{36}{25}$

Now, solving for $P$:

$P=\frac{7200\times 25}{36}$

$P=200\times 25$

$P=5000$

Thus, the original principal is ₹5,000.