To determine the ratio of the $$\mathrm{R}_f$$ values of sample $$\mathrm{A}$$ and sample $$\mathrm{C}$$, follow these steps:

1. Understanding the $$\mathrm{R}_f$$ Value:

The $$\mathrm{R}_f$$ value (Retention factor) is calculated using the formula:

$$\mathrm{R}_{\mathrm{f}} = \frac{\text{Distance travelled by the substance from the base line }(\mathrm{x})}{\text{Distance travelled by the solvent from the base line }(\mathrm{y})}$$

1. Calculate $$\mathrm{R}_f$$ for Sample $$A$$:

• Distance travelled by substance $$A$$ (from the figure): 5 cm


• Distance travelled by the solvent: 12.5 cm


• So, $$\left(R_f\right)_A = \frac{5}{12.5}$$

1. Calculate $$\mathrm{R}_f$$ for Sample $$C$$:

• Distance travelled by substance $$C$$ (from the figure): 10 cm


• Distance travelled by the solvent: 12.5 cm


• So, $$\left(R_f\right)_C = \frac{10}{12.5}$$

1. Ratio of $$\mathrm{R}_f$$ values for $$A$$ and $$C$$:

• The ratio is given by:

$$\frac{\left(R_f\right)_A}{\left(R_f\right)_C} = \frac{\frac{5}{12.5}}{\frac{10}{12.5}}$$

• Simplify the ratio:

$$\frac{\left(R_f\right)_A}{\left(R_f\right)_C} = \frac{5}{12.5} \times \frac{12.5}{10} = \frac{5}{10} = 0.5$$

1. Converting to the given form:

• The simplified ratio (0.5) should be expressed in the form $$x \times 10^{-2}$$.


• Since $$0.5 = 50 \times 10^{-2}$$,

we can identify $$x = 50$$.

Therefore, the value of $$x$$ is 50.