Given that the barrel is initially $\(\frac{1}{5}\)$ full, adding $k$ liters makes it $\(\frac{2}{3}\)$ full. We want to express the barrel's total capacity $C$ liters in terms of $k$.

Step 1: Define the variables
Let:
- $C=$ total capacity of the barrel (liters)
- Initially, liquid in barrel $\(=\frac{1}{5} C\)$
- After adding $k$ liters, liquid in barrel $\(=\frac{2}{3} C\)$

Step 2: Write the equation for the final amount of liquid

$$
\(\frac{1}{5} C+k=\frac{2}{3} C\)
$$


Step 3: Solve for $C$
Rearranging,

$$
\(k=\frac{2}{3} C-\frac{1}{5} C=\left(\frac{2}{3}-\frac{1}{5}\right) C\)
$$


Find common denominator for the fractions:

$$
\(\frac{2}{3}=\frac{10}{15}, \quad \frac{1}{5}=\frac{3}{15}\)
$$


So:

$$
\(k=\left(\frac{10}{15}-\frac{3}{15}\right) C=\frac{7}{15} C\)
$$


Step 4: Express $C$ in terms of $k$

$$
\(C=\frac{15}{7} k\)
$$

Step 5: Match with options
Option D) $\(\frac{15}{7} K\)$ matches the expression for $C$.
Final answer: D) $\(\frac{15}{7} K\)$