An initial capital of $x yields a total matured amount of $20,300 at 9% per annum simple interest over the exact same time frame in which a principal of $15,000 matures to $19,500 at 6% per annum simple interest. Find the value of x.

The correct option is $14,000.

To find the value of the initial capital x, we first calculate the time period T using the second investment scenario, and then use that time frame to solve for x in the first scenario.

Step 1: Find the time period (T) using the second investment scenario

Given details for the second scenario:
Principal:
$P_2=15000$
Matured Amount:
$A_2=19500$
Rate of Interest:
$R_2=6\%$

Calculate the simple interest earned:
${\mathrm{SI}}_2=A_2-P_2$
${\mathrm{SI}}_2=19500-15000=4500$

Using the simple interest formula:
${\mathrm{SI}}_2=\frac{P_2\times R_2\times T}{100}$
$4500=\frac{15000\times 6\times T}{100}$
$4500=900\times T$
$T=\frac{4500}{900}=5\text{years}$

Step 2: Calculate the initial capital (x) for the first scenario

Given details for the first scenario:
Matured Amount:
$A_1=20300$
Rate of Interest:
$R_1=9\%$
Time Period:
$T=5\text{years}$

The formula relating matured amount and principal under simple interest is:
$A_1=x+\frac{x\times R_1\times T}{100}$
$A_1=x\left(1+\frac{R_1\times T}{100}\right)$

Substitute the given values into the formula:
$20300=x\left(1+\frac{9\times 5}{100}\right)$
$20300=x\left(1+\frac{45}{100}\right)$
$20300=x\left(\frac{145}{100}\right)$
$20300=1.45x$

Solving for x:
$x=\frac{20300}{1.45}$
$x=14000$

Thus, the value of x is $14,000.