(x + 2), x and (x - 1) are the frequencies of the numbers 12, 15 and 20 respectively. If the mean of the distribution is 14.5, the value of x is

Options

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

Answer

To solve the problem step by step, we will first define the frequencies and then use the formula for the mean of a frequency distribution.

### Step 1: Define the Frequencies
Given:
- For the number 12, the frequency is \( f_1 = x + 2 \)
- For the number 15, the frequency is \( f_2 = x \)
- For the number 20, the frequency is \( f_3 = x - 1 \)

### Step 2: Calculate the Total Frequency
The total frequency \( \Sigma f_i \) is:
\[ \Sigma f_i = f_1 + f_2 + f_3 = (x + 2) + x + (x - 1) = 3x + 1 \]

### Step 3: Calculate the Weighted Sum of Frequencies
Now, we calculate \( \Sigma (x_i f_i) \):
\[ \Sigma (x_i f_i) = (12)(x + 2) + (15)(x) + (20)(x - 1) \]
Calculating each term:
- For 12: \( 12(x + 2) = 12x + 24 \)
- For 15: \( 15(x) = 15x \)
- For 20: \( 20(x - 1) = 20x - 20 \)

Adding these together:
\[ \Sigma (x_i f_i) = 12x + 24 + 15x + 20x - 20 = 47x + 4 \]

### Step 4: Set Up the Mean Equation
The mean of the distribution is given as 14.5. The formula for the mean is:
\[ \text{Mean} = \frac{\Sigma (x_i f_i)}{\Sigma f_i} \]
Substituting the known values:
\[ 14.5 = \frac{47x + 4}{3x + 1} \]

### Step 5: Cross-Multiply to Solve for x
Cross-multiplying gives:
\[ 14.5(3x + 1) = 47x + 4 \]
Expanding the left side:
\[ 43.5x + 14.5 = 47x + 4 \]

### Step 6: Rearranging the Equation
Rearranging the equation:
\[ 43.5x + 14.5 - 4 = 47x \]
\[ 43.5x + 10.5 = 47x \]
\[ 10.5 = 47x - 43.5x \]
\[ 10.5 = 3.5x \]

### Step 7: Solve for x
Now, divide both sides by 3.5:
\[ x = \frac{10.5}{3.5} = 3 \]

### Final Answer
The value of \( x \) is \( 3 \).

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