To solve the problem, we need to analyze the relationship between the two numbers based on the information given. Let's denote the two numbers as \( x \) and \( y \), where we assume \( x \) is the larger number and \( y \) is the smaller number. ### Step 1: Set Up the Equations 1. **Sum of the Numbers**: The sum of the two numbers is given by: \[ x + y \] 2. **Difference of the Numbers**: The difference of the two numbers is given by: \[ x - y \] 3. **Given Condition**: According to the problem, the sum of the two numbers is twice their difference: \[ x + y = 2(x - y) \] ### Step 2: Simplify the Equation Now, let's simplify the equation we derived from the given condition: 1. Start with: \[ x + y = 2(x - y) \] 2. Distribute the 2 on the right side: \[ x + y = 2x - 2y \] 3. Rearranging the equation to isolate terms involving \( x \) and \( y \): \[ x + y + 2y = 2x \] \[ x + 3y = 2x \] 4. Subtract \( x \) from both sides: \[ 3y = 2x - x \] \[ 3y = x \] ### Step 3: Express the Larger Number in Terms of P We know from the problem that the difference of the numbers is \( P \): \[ x - y = P \] Now, we can substitute \( x \) from the previous equation into this difference equation: 1. Substitute \( x = 3y \) into \( x - y = P \): \[ 3y - y = P \] \[ 2y = P \] 2. Solve for \( y \): \[ y = \frac{P}{2} \] 3. Now, substitute \( y \) back into the equation for \( x \): \[ x = 3y = 3\left(\frac{P}{2}\right) = \frac{3P}{2} \] ### Conclusion: Identify the Larger Number From our calculations, we have found that the larger number \( x \) is: \[ x = \frac{3P}{2} \] ### Final Answer Thus, the larger of the two numbers is: **B. \( \frac{3P}{2} \)** ### Explanation of Other Options - **Option A: \( \frac{P}{2} \)**: This represents the smaller number \( y \), not the larger number \( x \). - **Option C: \( \frac{5P}{2} \)**: This value does not satisfy the conditions set by the problem and is not derived from our equations. - **Option D: \( 3P \)**: This is also not derived from our equations and does not represent either number correctly. ### Revision Summary - The sum of two numbers is twice their difference, leading to the equation \( x + y = 2(x - y) \). - By expressing the larger number \( x \) in terms of the smaller number \( y \), we found \( x = 3y \). - The difference \( P \) allowed us to express \( y \) as \( \frac{P}{2} \) and subsequently find \( x = \frac{3P}{2} \). - The correct answer is \( \frac{3P}{2} \), which corresponds to option B.