Median of `(x)/(5), x, (x)/(4), (x)/(2), (x)/(3)` is 8. If `x gt 0` then value of x is

Options

A. 24
B. 18
C. 27
D. 51

Answer

To find the value of \( x \) given that the median of the numbers \( \frac{x}{5}, x, \frac{x}{4}, \frac{x}{2}, \frac{x}{3} \) is 8, we can follow these steps:

### Step 1: Arrange the terms in ascending order
Given the terms \( \frac{x}{5}, x, \frac{x}{4}, \frac{x}{2}, \frac{x}{3} \), we need to arrange them in ascending order. Since \( x > 0 \), we can compare the fractions:

1. \( \frac{x}{5} \) is the smallest.
2. \( \frac{x}{4} \) is next.
3. \( \frac{x}{3} \) follows.
4. \( \frac{x}{2} \) comes next.
5. Finally, \( x \) is the largest.

Thus, the ascending order is:
\[ \frac{x}{5}, \frac{x}{4}, \frac{x}{3}, \frac{x}{2}, x \]

### Step 2: Identify the median
Since there are 5 observations (which is an odd number), the median is the middle term. The median can be calculated using the formula:
\[ \text{Median} = \frac{n + 1}{2} \text{th term \]
where \( n \) is the number of observations. Here, \( n = 5 \):
\[ \text{Median} = \frac{5 + 1}{2} = 3 \text{rd term} \]
The 3rd term in our ordered list is \( \frac{x}{3} \).

### Step 3: Set the median equal to 8
According to the problem, the median is given as 8. Therefore, we can set up the equation:
\[ \frac{x}{3} = 8 \]

### Step 4: Solve for \( x \)
To find \( x \), we multiply both sides of the equation by 3:
\[ x = 8 \times 3 \]
\[ x = 24 \]

### Conclusion
The value of \( x \) is \( 24 \).

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