Step 1: Understand the Condition for Doubling

When a sum of money doubles itself, the final amount $A$ becomes twice the principal $P$. Therefore, the Simple Interest ($\text{SI}$) earned is equal to the principal itself.

$\text{SI} = A - P = 2P - P = P$

Step 2: Apply the Simple Interest Formula

The formula for Simple Interest is $\text{SI} = \frac{P \times R \times T}{100}$. Rearranging this formula to solve for time $T$, we get:

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

Step 3: Substitute the Given Values

Substitute $\text{SI} = P$ and Rate $R = 10\%$ into the rearranged formula:

$T = \frac{100 \times P}{P \times 10} = \frac{100}{10} = 10 \text{ years}$