The numbers 3, 5, 7 and 9 have their respective frequencies x - 2,x+2,x-3andx+3`. If the mean is 6.5 ,then the value of x is

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem, we need to find the value of \( x \) given the numbers and their respective frequencies, as well as the mean.

### Step-by-Step Solution:

1. **Identify the Numbers and Frequencies**:
The numbers are \( 3, 5, 7, 9 \) with respective frequencies:
- For \( 3 \): \( f_1 = x - 2 \)
- For \( 5 \): \( f_2 = x + 2 \)
- For \( 7 \): \( f_3 = x - 3 \)
- For \( 9 \): \( f_4 = x + 3 \)

2. **Calculate the Total Frequency**:
The total frequency \( \Sigma f_i \) is:
\[ \Sigma f_i = (x - 2) + (x + 2) + (x - 3) + (x + 3) \]
Simplifying this:
\[ \Sigma f_i = 4x \]

3. **Calculate the Weighted Sum of the Numbers**:
The weighted sum \( \Sigma (f_i \cdot x_i) \) is:
\[ \Sigma (f_i \cdot x_i) = 3(x - 2) + 5(x + 2) + 7(x - 3) + 9(x + 3) \]
Expanding this:
\[ = 3x - 6 + 5x + 10 + 7x - 21 + 9x + 27 \]
Combining like terms:
\[ = (3x + 5x + 7x + 9x) + (-6 + 10 - 21 + 27) \]
\[ = 24x + 10 \]

4. **Set Up the Mean Equation**:
The mean is given as \( 6.5 \). The formula for the mean is:
\[ \text{Mean} = \frac{\Sigma (f_i \cdot x_i)}{\Sigma f_i} \]
Substituting the values we found:
\[ 6.5 = \frac{24x + 10}{4x} \]

5. **Cross-Multiply to Solve for \( x \)**:
Cross-multiplying gives:
\[ 6.5 \cdot 4x = 24x + 10 \]
Simplifying:
\[ 26x = 24x + 10 \]

6. **Isolate \( x \)**:
Subtract \( 24x \) from both sides:
\[ 2x = 10 \]
Dividing both sides by 2:
\[ x = 5 \]

### Final Answer:
The value of \( x \) is \( 5 \).