The numbers 3,5,7,4 have frequencies `x,x+4,x-3,x+8`. If their arithmetic mean is 4, the value of x is

Options

A. `5/3`
B. `7/4`
C. `2/3`
D. none of these

Answer

To solve the problem, we need to find the value of \( x \) given that the numbers 3, 5, 7, and 4 have frequencies \( x, x+4, x-3, x+8 \) respectively, and their arithmetic mean is 4.

### Step-by-Step Solution:

1. **Identify the numbers and their frequencies**:
- Numbers: 3, 5, 7, 4
- Frequencies: \( x, x+4, x-3, x+8 \)

2. **Calculate the total frequency**:
\[ \text{Total frequency} = x + (x + 4) + (x - 3) + (x + 8) = 4x + 9 \]

3. **Calculate the weighted sum of the numbers**:
\[ \text{Weighted sum} = 3x + 5(x + 4) + 7(x - 3) + 4(x + 8) \]
Expanding this:
\[ = 3x + 5x + 20 + 7x - 21 + 4x + 32 \]
Combining like terms:
\[ = (3x + 5x + 7x + 4x) + (20 - 21 + 32) = 19x + 31 \]

4. **Set up the equation for the arithmetic mean**:
The arithmetic mean is given by:
\[ \text{Mean} = \frac{\text{Weighted sum}}{\text{Total frequency}} = 4 \]
Therefore, we have:
\[ \frac{19x + 31}{4x + 9} = 4 \]

5. **Cross-multiply to eliminate the fraction**:
\[ 19x + 31 = 4(4x + 9) \]
Expanding the right side:
\[ 19x + 31 = 16x + 36 \]

6. **Rearranging the equation**:
\[ 19x - 16x = 36 - 31 \]
Simplifying:
\[ 3x = 5 \]

7. **Solve for \( x \)**:
\[ x = \frac{5}{3} \]

### Final Answer:
The value of \( x \) is \( \frac{5}{3} \).