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: 1. Start with the equation: \[ x + y = 2(x - y) \] 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 \( x \) in Terms of \( y \) From the equation \( 3y = x \), we can express \( x \) in terms of \( y \): \[ x = 3y \] ### Step 4: Relate \( y \) to \( P \) We know from the problem statement that the difference of the numbers is \( P \): \[ x - y = P \] Substituting \( x = 3y \) into the difference equation: \[ 3y - y = P \] \[ 2y = P \] ### Step 5: Solve for \( y \) Now, we can solve for \( y \): \[ y = \frac{P}{2} \] ### Step 6: Find \( x \) Now that we have \( y \), we can find \( x \): \[ x = 3y = 3\left(\frac{P}{2}\right) = \frac{3P}{2} \] ### Conclusion: Identify the Larger Number Since \( x \) is the larger number, we conclude that: \[ \text{The larger of the two numbers is } \frac{3P}{2}. \] ### Answer Thus, the correct option 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 correspond to either number based on our derived equations. - **Option D: \( 3P \)**: This is also not derived from our equations and does not represent either number. ### Revision Summary - The sum of two numbers is twice their difference, leading to the equation \( x + y = 2(x - y) \). - We derived that \( x = 3y \) and \( y = \frac{P}{2} \). - The larger number \( x \) is found to be \( \frac{3P}{2} \). - The correct answer is option B, \( \frac{3P}{2} \).