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 answer is Rs 6400.

Step 1: Define the variables for the deposited amounts
Let the amount deposited by Rohan be Rs. $x$.
Then, the amount deposited by Sameer is Rs. $(x+300)$.
Both deposits are for a duration of 2 years.

Step 2: Calculate Simple Interest earned by Rohan
Rohan deposited his amount at a simple interest rate of $10\%$ per annum for 2 years.
The formula for Simple Interest (SI) is:

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

Substituting the values for Rohan:

$\text{SI (Rohan)}=\frac{x\times 10\times 2}{100}=\frac{20x}{100}=0.20x$

Step 3: Calculate Compound Interest earned by Sameer
Sameer deposited his amount at compound interest (CI) at a rate of $12\frac{1}{2}\%$ per annum compounded annually.

First, convert the rate fraction into a simple fraction:

$12\frac{1}{2}\%=\frac{25}{2}\%=\frac{25}{2\times 100}=\frac{1}{8}$

The formula for Compound Interest (CI) for 2 years is:

$\text{CI}=P{(1+r)}^2-P=P[{(1+r)}^2-1]$

Substituting $P=x+300$ and $r=\frac{1}{8}$:

$\text{CI (Sameer)}=(x+300)[{(1+\frac{1}{8})}^2-1]$

$\text{CI (Sameer)}=(x+300)[{\left(\frac{9}{8}\right)}^2-1]$

$\text{CI (Sameer)}=(x+300)[\frac{81}{64}-1]$

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

Step 4: Set up the equation using the interest difference
We are given that Sameer earned Rs. 480 more interest than Rohan:

$\text{CI (Sameer)}-\text{SI (Rohan)}=480$

$(x+300)\times \frac{17}{64}-0.20x=480$

Converting $\frac{17}{64}$ to its decimal form $0.265625$:

$0.265625(x+300)-0.20x=480$

$0.265625x+79.6875-0.20x=480$

$0.065625x+79.6875=480$

$0.065625x=480-79.6875$

$0.065625x=400.3125$

$x=\frac{400.3125}{0.065625}=6100$

Step 5: Calculate the total amount deposited by Sameer
The total amount deposited by Sameer is:

$\text{Amount deposited by Sameer}=x+300$

$\text{Amount deposited by Sameer}=6100+300=6400$

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