Out of three numbers , the first is twice the second and is half of the third .If the average of three numbers is 56 , then the difference of the first and third is

Options

A. 12
B. 20
C. 24
D. 48

Answer

To solve the problem step by step, let's denote the three numbers based on the information given:

1. **Let the first number be \( x \)**.
2. **The second number** is twice the first number, so:
\[ \text{Second number} = \frac{x}{2} \]
3. **The third number** is half of the first number, so:
\[ \text{Third number} = 2x \]

Now, we know that the average of these three numbers is 56. The formula for the average of three numbers is:
\[ \text{Average} = \frac{\text{First number} + \text{Second number} + \text{Third number}}{3} \]

Substituting the values we have:
\[ 56 = \frac{x + \frac{x}{2} + 2x}{3} \]

To eliminate the fraction, we can multiply both sides by 3:
\[ 168 = x + \frac{x}{2} + 2x \]

Now, let's combine the terms on the right side. First, we need a common denominator to add the fractions. The common denominator for \( x \) and \( \frac{x}{2} \) is 2:
\[ 168 = \frac{2x}{2} + \frac{x}{2} + \frac{4x}{2} \]
This simplifies to:
\[ 168 = \frac{2x + x + 4x}{2} = \frac{7x}{2} \]

Now, multiply both sides by 2 to eliminate the fraction:
\[ 336 = 7x \]

Next, divide both sides by 7 to solve for \( x \):
\[ x = \frac{336}{7} = 48 \]

Now that we have \( x \), we can find the three numbers:
- **First number**: \( x = 48 \)
- **Second number**: \( \frac{x}{2} = \frac{48}{2} = 24 \)
- **Third number**: \( 2x = 2 \times 48 = 96 \)

Finally, we need to find the difference between the first and third numbers:
\[ \text{Difference} = \text{Third number} - \text{First number} = 96 - 48 = 48 \]

Thus, the difference between the first and third numbers is:
\[ \boxed{48} \]