The retention factor in chromatography is defined as $R_f = \frac{\text{distance travelled by solute spot}}{\text{distance travelled by solvent front}}.$

Since samples $A$ and $C$ are on the same chromatogram, the solvent front distance is the same for both. Therefore, $\frac{R_f(A)}{R_f(C)} = \frac{\text{distance travelled by }A}{\text{distance travelled by }C}.$

From the given figure, the spot of sample $A$ has travelled half the distance travelled by sample $C$. Hence, $\frac{R_f(A)}{R_f(C)} = \frac{1}{2} = 0.5.$

The question states that this ratio is $x \times 10^{-2}$. So, $x \times 10^{-2} = 0.5$ $x = 0.5 \times 10^2 = 50.$

Therefore, the required value is $\boxed{50}.$