If the median of `(x)/(6),(x)/(2),(x)/(4),(3x)/(5)` and `(7x)/(10)` is 3, then the mean of the given observations is :

Options

A. `2.5`
B. `3.06`
C. `2.16`
D. `2.66`

Answer

To solve the given problem, we need to follow these steps:

### Step 1: Identify the observations
The observations given are:
1. \( \frac{x}{6} \)
2. \( \frac{x}{2} \)
3. \( \frac{x}{4} \)
4. \( \frac{3x}{5} \)
5. \( \frac{7x}{10} \)

### Step 2: Arrange the observations in ascending order
To find the median, we first need to arrange these observations in ascending order. To do this, we can evaluate the expressions in terms of \( x \):

- \( \frac{x}{6} \) = 0.1667x
- \( \frac{x}{4} \) = 0.25x
- \( \frac{x}{2} \) = 0.5x
- \( \frac{3x}{5} \) = 0.6x
- \( \frac{7x}{10} \) = 0.7x

Arranging these in ascending order gives us:
1. \( \frac{x}{6} \)
2. \( \frac{x}{4} \)
3. \( \frac{x}{2} \)
4. \( \frac{3x}{5} \)
5. \( \frac{7x}{10} \)

### Step 3: Find the median
Since there are 5 observations (an odd number), the median is the middle value. The median is the 3rd term in the ordered list:
- Median = \( \frac{x}{2} \)

According to the problem, the median is given as 3:
\[ \frac{x}{2} = 3 \]

### Step 4: Solve for \( x \)
To find \( x \), we multiply both sides by 2:
\[ x = 6 \]

### Step 5: Calculate the mean of the observations
Now, we will calculate the mean of the observations:
Mean = \( \frac{\text{Sum of Observations}}{\text{Number of Observations}} \)

Substituting the values of \( x \):
\[ \text{Sum of Observations} = \frac{6}{6} + \frac{6}{2} + \frac{6}{4} + \frac{3 \cdot 6}{5} + \frac{7 \cdot 6}{10} \]

Calculating each term:
1. \( \frac{6}{6} = 1 \)
2. \( \frac{6}{2} = 3 \)
3. \( \frac{6}{4} = 1.5 \)
4. \( \frac{18}{5} = 3.6 \)
5. \( \frac{42}{10} = 4.2 \)

Now, summing these values:
\[ 1 + 3 + 1.5 + 3.6 + 4.2 = 13.3 \]

### Step 6: Calculate the mean
Now, divide the sum by the number of observations (which is 5):
\[ \text{Mean} = \frac{13.3}{5} = 2.66 \]

### Final Answer
The mean of the given observations is \( 2.66 \).

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