If the median of `(a)/(3) , (a)/(2), (a)/(4) , (2a)/(5) , (a)/(6)` is 12, then find the value of `a (a gt 0).`

Options

A. 36
B. 48
C. 30
D. 24

Answer

To solve the problem, we need to find the value of \( a \) given that the median of the numbers \( \frac{a}{3}, \frac{a}{2}, \frac{a}{4}, \frac{2a}{5}, \frac{a}{6} \) is 12.

### Step-by-Step Solution:

1. **List the Numbers**:
We have the following numbers:
\[ \frac{a}{3}, \frac{a}{2}, \frac{a}{4}, \frac{2a}{5}, \frac{a}{6} \]

2. **Find a Common Factor**:
To compare these fractions easily, we can factor out \( a \):
\[ a \left( \frac{1}{3}, \frac{1}{2}, \frac{1}{4}, \frac{2}{5}, \frac{1}{6} \right) \]

3. **Convert to Decimal for Easier Comparison**:
Let's convert these fractions to decimal values:
- \( \frac{1}{3} \approx 0.333 \)
- \( \frac{1}{2} = 0.5 \)
- \( \frac{1}{4} = 0.25 \)
- \( \frac{2}{5} = 0.4 \)
- \( \frac{1}{6} \approx 0.167 \)

4. **Arrange in Ascending Order**:
Now, we can arrange these decimal values:
\[ \frac{a}{6}, \frac{a}{4}, \frac{2a}{5}, \frac{a}{3}, \frac{a}{2} \]

5. **Identify the Median**:
Since there are 5 numbers (an odd count), the median is the middle number, which is the 3rd term in the ordered list:
\[ \text{Median} = \frac{a}{3} \]

6. **Set the Median Equal to 12**:
According to the problem, the median is given as 12:
\[ \frac{a}{3} = 12 \]

7. **Solve for \( a \)**:
To find \( a \), multiply both sides by 3:
\[ a = 12 \times 3 = 36 \]

### Final Answer:
The value of \( a \) is \( 36 \).