Find the fourth proportional to :
`(1)/(3), (2)/(5), 6`

Answer

To find the fourth proportional to the numbers \( \frac{1}{3}, \frac{2}{5}, \) and \( 6 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Proportionality**:
The fourth proportional \( x \) to the numbers \( a, b, c \) is defined such that:
\[ \frac{a}{b} = \frac{c}{x} \]
In this case, \( a = \frac{1}{3} \), \( b = \frac{2}{5} \), and \( c = 6 \).

2. **Set Up the Proportion**:
We can write the proportion as:
\[ \frac{\frac{1}{3}}{\frac{2}{5}} = \frac{6}{x} \]

3. **Cross-Multiply**:
Cross-multiplying gives us:
\[ \frac{1}{3} \cdot x = \frac{2}{5} \cdot 6 \]

4. **Calculate the Right Side**:
Calculate \( \frac{2}{5} \cdot 6 \):
\[ \frac{2 \cdot 6}{5} = \frac{12}{5} \]

5. **Substitute Back**:
Now substitute this back into the equation:
\[ \frac{1}{3} \cdot x = \frac{12}{5} \]

6. **Solve for \( x \)**:
To isolate \( x \), multiply both sides by \( 3 \):
\[ x = \frac{12}{5} \cdot 3 \]
This simplifies to:
\[ x = \frac{36}{5} \]

7. **Conclusion**:
Therefore, the fourth proportional is:
\[ x = \frac{36}{5} \]

### Final Answer:
The fourth proportional to \( \frac{1}{3}, \frac{2}{5}, \) and \( 6 \) is \( \frac{36}{5} \).