The numbers 4 and 9 have frequencies x and (x - 1) respectively. If their arithmetic mean is 6, then x is equal to :

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to find the value of \( x \) given that the numbers 4 and 9 have frequencies \( x \) and \( (x - 1) \) respectively, and their arithmetic mean is 6.

### Step 1: Understand the formula for the arithmetic mean
The arithmetic mean (average) of a set of numbers is calculated as:
\[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of values}} \]

### Step 2: Set up the equation for the arithmetic mean
In this case, the numbers are 4 and 9, with frequencies \( x \) and \( (x - 1) \). The total number of values is:
\[ x + (x - 1) = 2x - 1 \]
The sum of all values is:
\[ 4x + 9(x - 1) = 4x + 9x - 9 = 13x - 9 \]
Thus, the arithmetic mean can be expressed as:
\[ \text{Mean} = \frac{13x - 9}{2x - 1} \]

### Step 3: Set the mean equal to 6
According to the problem, the arithmetic mean is 6. Therefore, we can set up the equation:
\[ \frac{13x - 9}{2x - 1} = 6 \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 13x - 9 = 6(2x - 1) \]
Expanding the right side:
\[ 13x - 9 = 12x - 6 \]

### Step 5: Rearranging the equation
Now, rearranging the equation to isolate \( x \):
\[ 13x - 12x = -6 + 9 \]
This simplifies to:
\[ x = 3 \]

### Conclusion
Thus, the value of \( x \) is \( 3 \).