The numbers 3, 5, 6 and 4 have frequencies of x, x + 2, x -8 and x+ 6, respectively। If their mean is 4 then the value of x is

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem step by step, we need to find the value of \( x \) given the numbers and their corresponding frequencies, and that the mean is 4.

### Step 1: Define the numbers and their frequencies
The numbers are:
- 3 with frequency \( x \)
- 5 with frequency \( x + 2 \)
- 6 with frequency \( x - 8 \)
- 4 with frequency \( x + 6 \)

### Step 2: Calculate the total frequency
The total frequency \( \Sigma f \) can be calculated as:
\[ \Sigma f = x + (x + 2) + (x - 8) + (x + 6) \]
Simplifying this gives:
\[ \Sigma f = x + x + 2 + x - 8 + x + 6 = 4x \]

### Step 3: Calculate the sum of the products of numbers and their frequencies
The sum of the products of the numbers and their frequencies \( \Sigma f \cdot x_i \) is calculated as follows:
\[ \Sigma f \cdot x_i = (3 \cdot x) + (5 \cdot (x + 2)) + (6 \cdot (x - 8)) + (4 \cdot (x + 6)) \]
Calculating each term:
- \( 3x \)
- \( 5(x + 2) = 5x + 10 \)
- \( 6(x - 8) = 6x - 48 \)
- \( 4(x + 6) = 4x + 24 \)

Now, summing these gives:
\[ \Sigma f \cdot x_i = 3x + 5x + 10 + 6x - 48 + 4x + 24 \]
Combining like terms:
\[ \Sigma f \cdot x_i = (3x + 5x + 6x + 4x) + (10 - 48 + 24) = 18x - 14 \]

### Step 4: Set up the equation for the mean
The mean is given by the formula:
\[ \text{Mean} = \frac{\Sigma f \cdot x_i}{\Sigma f} \]
Substituting the known values:
\[ 4 = \frac{18x - 14}{4x} \]

### Step 5: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 4 \cdot 4x = 18x - 14 \]
This simplifies to:
\[ 16x = 18x - 14 \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 16x - 18x = -14 \]
\[ -2x = -14 \]

### Step 7: Solve for \( x \)
Dividing both sides by -2:
\[ x = \frac{14}{2} = 7 \]

### Conclusion
The value of \( x \) is \( 7 \).