A fund is split among three deposits in the ratio 2:3:5. The deposits earn simple interest for 2 years at annual rates of 6%, 8%, and 10%, respectively. If the combined simple interest is Rs. 5,160, what was the original amount of the fund?

The correct option is Rs. 30,000.

Step 1: Understand the ratio and define variables.
The total fund is divided into three deposits in the ratio 2 : 3 : 5.
Let the common multiplier be $x$.
Therefore, the amounts in the three deposits are:
First deposit ($P_1$) = $2x$
Second deposit ($P_2$) = $3x$
Third deposit ($P_3$) = $5x$
The total original fund is equal to the sum of all three deposits:

$\text{Total Fund}=2x+3x+5x=10x$

Step 2: Calculate simple interest for each deposit.
The formula for simple interest (SI) is:

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

where $P$ is the principal amount, $R$ is the annual rate of interest, and $T$ is the time period in years.
Given that the time period $T=2$ years for all deposits:

For the first deposit ($P_1=2x$, rate $R_1=6\%$):

${\text{SI}}_1=\frac{2x\times 6\times 2}{100}=\frac{24x}{100}$

For the second deposit ($P_2=3x$, rate $R_2=8\%$):

${\text{SI}}_2=\frac{3x\times 8\times 2}{100}=\frac{48x}{100}$

For the third deposit ($P_3=5x$, rate $R_3=10\%$):

${\text{SI}}_3=\frac{5x\times 10\times 2}{100}=\frac{100x}{100}$

Step 3: Set up the equation using total interest.
The combined simple interest from all three deposits is Rs. 5,160.

${\text{SI}}_1+{\text{SI}}_2+{\text{SI}}_3=5160$

$\frac{24x}{100}+\frac{48x}{100}+\frac{100x}{100}=5160$

$\frac{172x}{100}=5160$

Step 4: Solve for $x$ and calculate the original fund.

$172x=5160\times 100$

$172x=516000$

$x=\frac{516000}{172}=3000$

Now, substitute the value of $x$ back into the total fund expression ($10x$):

$\text{Original Amount of Fund}=10\times 3000=\text{Rs. 30,000}$

Thus, the original amount of the fund was Rs. 30,000.