Of the three numbers, the second is twice the first and it is also thrice the third. If the average of three numbers is 44, the difference of the first number and the third number is :

Options

A. 24
B. 18
C. 12
D. 6

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the Variables
Let the first number be \( x \). According to the problem:
- The second number is twice the first number: \( 2x \)
- The third number is one-third of the second number: \( \frac{2x}{3} \)

### Step 2: Set Up the Average Equation
The average of the three numbers is given as 44. Therefore, we can express this mathematically:
\[ \text{Average} = \frac{x + 2x + \frac{2x}{3}}{3} = 44 \]

### Step 3: Simplify the Equation
Combine the terms in the numerator:
\[ \frac{3x + \frac{2x}{3}}{3} = 44 \]
To combine \( 3x \) and \( \frac{2x}{3} \), we convert \( 3x \) to a fraction:
\[ 3x = \frac{9x}{3} \]
Thus, the equation becomes:
\[ \frac{\frac{9x + 2x}{3}}{3} = 44 \]
This simplifies to:
\[ \frac{11x}{3} = 44 \]

### Step 4: Solve for \( x \)
Multiply both sides by 3 to eliminate the fraction:
\[ 11x = 132 \]
Now, divide by 11:
\[ x = 12 \]

### Step 5: Find the Second and Third Numbers
Now that we have \( x \):
- The first number \( x = 12 \)
- The second number \( 2x = 2 \times 12 = 24 \)
- The third number \( \frac{2x}{3} = \frac{24}{3} = 8 \)

### Step 6: Calculate the Difference
The difference between the first number and the third number is:
\[ x - \frac{2x}{3} = 12 - 8 = 4 \]

### Final Answer
The difference between the first number and the third number is \( 4 \).

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