Rohan and Sameer deposited Rs. 'x' and Rs. 'x + 300' for 2 years respectively. Rohan deposited his amount on simple interest at the rate of 10% per annum while Sameer deposited his amount on compound interest at $12\frac{1}{2}\%$ per annum compounding annually. If Sameer earned Rs. 480 more than Rohan as interest, then find the total amount deposited by Sameer?

The correct option is Rs 6400.

Step 1: Understand the given information

Amount deposited by Rohan = Rs. $x$

Amount deposited by Sameer = Rs. $(x+300)$

Time period ($t$) = 2 years

Rohan's rate of Simple Interest ($R_1$) = 10% per annum

Sameer's rate of Compound Interest ($R_2$) = $12\frac{1}{2}\%=12.5\%=\frac{1}{8}$ per annum

Step 2: Calculate the Simple Interest earned by Rohan

Simple Interest ($SI$) formula is given by:

$SI=\frac{P\times R\times T}{100}$

For Rohan:

$SI=\frac{x\times 10\times 2}{100}=\frac{20x}{100}=0.20x$

Step 3: Calculate the Compound Interest earned by Sameer

Using the fractional rate method, since $12.5\%=\frac{1}{8}$, the net compound interest rate for 2 years can be calculated as:

$CI=P\times \left[{\left(1+\frac{1}{8}\right)}^2-1\right]$

$CI=P\times \left[{\left(\frac{9}{8}\right)}^2-1\right]=P\times \left(\frac{81}{64}-1\right)=P\times \frac{17}{64}$

Substituting Sameer's Principal $P=x+300$:

$CI=\frac{17}{64}(x+300)$

Step 4: Set up the equation according to the problem statement

It is given that Sameer earned Rs. 480 more interest than Rohan:

$CI-SI=480$

$\frac{17}{64}(x+300)-\frac{x}{5}=480$

Multiply the entire equation by 320 (the LCM of 64 and 5) to clear the fractions:

$5\times 17(x+300)-64x=480\times 320$

$85(x+300)-64x=153600$

$85x+25500-64x=153600$

$21x=153600-25500$

$21x=128100$

$x=\frac{128100}{21}=6100$

Step 5: Find the total amount deposited by Sameer

Amount deposited by Sameer = $x+300$

$\text{Sameer's Deposit}=6100+300=6400$

Thus, the total amount deposited by Sameer is Rs 6400.