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 will define the three numbers based on the relationships given in the question.

### Step 1: Define the Variables
Let the second number be \( x \). According to the problem:
- The first number is half of the second number:
\[ \text{First Number} = \frac{x}{2} \]
- The third number is one-third of the second number:
\[ \text{Third Number} = \frac{x}{3} \]

### Step 2: Set Up the Average Equation
The average of the three numbers is given as 44. The average can be calculated as:
\[ \text{Average} = \frac{\text{First Number} + \text{Second Number} + \text{Third Number}}{3} \]
Substituting the values we defined:
\[ \frac{\frac{x}{2} + x + \frac{x}{3}}{3} = 44 \]

### Step 3: Simplify the Equation
To simplify the equation, first combine the terms in the numerator:
\[ \frac{x}{2} + x + \frac{x}{3} = \frac{3x}{6} + \frac{6x}{6} + \frac{2x}{6} = \frac{11x}{6} \]
Now, substitute this back into the average equation:
\[ \frac{\frac{11x}{6}}{3} = 44 \]
Multiply both sides by 3:
\[ \frac{11x}{6} = 132 \]

### Step 4: Solve for \( x \)
Now, multiply both sides by 6:
\[ 11x = 792 \]
Divide by 11:
\[ x = 72 \]

### Step 5: Find the First and Third Numbers
Now that we have \( x \), we can find the first and third numbers:
- First Number:
\[ \text{First Number} = \frac{x}{2} = \frac{72}{2} = 36 \]
- Third Number:
\[ \text{Third Number} = \frac{x}{3} = \frac{72}{3} = 24 \]

### Step 6: Calculate the Difference
Finally, we need to find the difference between the first number and the third number:
\[ \text{Difference} = \text{First Number} - \text{Third Number} = 36 - 24 = 12 \]

### Final Answer
The difference between the first number and the third number is \( \boxed{12} \).

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