If the median of `(a)/(2), a , (a)/(3) , (a)/(5) and (a)/4` is 6, then find the value of a `(a gt 0).`

Answer

To solve the problem, we need to find the value of \( a \) given that the median of the numbers \( \frac{a}{2}, a, \frac{a}{3}, \frac{a}{5}, \frac{a}{4} \) is 6.

### Step-by-Step Solution:

1. **List the Numbers**: The numbers given are \( \frac{a}{2}, a, \frac{a}{3}, \frac{a}{5}, \frac{a}{4} \).

2. **Order the Numbers**: To find the median, we first need to arrange these numbers in ascending order. We will compare the fractions by factoring out \( a \) (since \( a > 0 \)):
- \( \frac{a}{5} < \frac{a}{4} < \frac{a}{3} < \frac{a}{2} < a \)

Thus, the ordered list is:
\[ \frac{a}{5}, \frac{a}{4}, \frac{a}{3}, \frac{a}{2}, a \]

3. **Determine the Median**: Since there are 5 numbers (an odd count), the median is the middle number. The middle number (3rd term) in our ordered list is \( \frac{a}{3} \).

4. **Set the Median Equal to 6**: According to the problem, the median is given as 6. Therefore, we set up the equation:
\[ \frac{a}{3} = 6 \]

5. **Solve for \( a \)**: To find \( a \), we multiply both sides of the equation by 3:
\[ a = 6 \times 3 \]
\[ a = 18 \]

### Final Answer:
The value of \( a \) is \( 18 \).