If at the same rate of interest, in 2 years, the simple interest is ₹48 and compound interest, compounded annually, is ₹54, then what is the principal (in ₹)?8460

The correct option is 96.

Step 1: Understand the given data
Let the principal amount be $P$ and the annual rate of interest be $r$%.
Time period ($t$) = 2 years.
Simple Interest ($\mathrm{SI}$) for 2 years = ₹48.
Compound Interest ($\mathrm{CI}$) for 2 years = ₹54.

Step 2: Find the rate of interest ($r$)
Simple interest is equal for each year.
Therefore, the simple interest for 1 year =

$\frac{48}{2}=\text{₹}24$

For the first year, both simple interest and compound interest are equal (₹24).
In the second year, compound interest includes the interest earned on the first year's interest.
The difference between Compound Interest and Simple Interest for 2 years is:

$\mathrm{CI}-\mathrm{SI}=54-48=\text{₹}6$

This difference of ₹6 is the interest earned on the first year's interest of ₹24.
So, the rate of interest ($r$) can be calculated as:

$r=\left(\frac{6}{24}\right)\×100=25\%$

Step 3: Calculate the Principal ($P$)
Using the Simple Interest formula for 2 years:

$\mathrm{SI}=\frac{P\×r\×t}{100}$

Substitute the known values:

$48=\frac{P\×25\×2}{100}$

$48=\frac{50P}{100}$

$48=\frac{P}{2}$

$P=48\×2=96$

Thus, the principal is ₹96.