To solve the problem of how many oranges the three boys shared, we need to break down the information given step by step. ### Step 1: Understanding the Distribution of Oranges 1. **First Boy's Share**: The first boy received \( \frac{1}{3} \) of the total number of oranges. Let's denote the total number of oranges as \( x \). Therefore, the first boy received: \[ \text{First Boy's Share} = \frac{1}{3}x \] 2. **Remaining Oranges After First Boy**: After the first boy took his share, the remaining oranges can be calculated as: \[ \text{Remaining Oranges} = x - \frac{1}{3}x = \frac{2}{3}x \] 3. **Second Boy's Share**: The second boy received \( \frac{2}{3} \) of the remaining oranges. The remaining oranges after the first boy is \( \frac{2}{3}x \), so the second boy received: \[ \text{Second Boy's Share} = \frac{2}{3} \times \frac{2}{3}x = \frac{4}{9}x \] 4. **Remaining Oranges After Second Boy**: After the second boy took his share, the remaining oranges can be calculated as: \[ \text{Remaining Oranges} = \frac{2}{3}x - \frac{4}{9}x \] To perform this subtraction, we need a common denominator. The common denominator for \( \frac{2}{3} \) and \( \frac{4}{9} \) is 9. We can rewrite \( \frac{2}{3} \) as \( \frac{6}{9} \): \[ \text{Remaining Oranges} = \frac{6}{9}x - \frac{4}{9}x = \frac{2}{9}x \] ### Step 2: Third Boy's Share According to the problem, the third boy received the remaining 12 oranges. Therefore, we can set up the equation: \[ \frac{2}{9}x = 12 \] ### Step 3: Solving for \( x \) To find the total number of oranges \( x \), we can solve the equation: 1. Multiply both sides by 9 to eliminate the fraction: \[ 2x = 12 \times 9 \] \[ 2x = 108 \] 2. Now, divide both sides by 2: \[ x = \frac{108}{2} = 54 \] ### Conclusion The total number of oranges shared by the three boys is **54**. ### Step 4: Analyzing the Options Now, let's review the options provided: - **A. 60**: This is incorrect because it does not satisfy the equation derived from the problem. - **B. 54**: This is correct as we calculated \( x = 54 \). - **C. 48**: This is incorrect for the same reason as option A; it does not satisfy the equation. - **D. 42**: This is also incorrect as it does not satisfy the equation. ### Revision Summary - The first boy received \( \frac{1}{3} \) of the total oranges. - The second boy received \( \frac{2}{3} \) of the remaining oranges after the first boy. - The third boy received the remaining 12 oranges, leading to the equation \( \frac{2}{9}x = 12 \). - Solving for \( x \) gives a total of 54 oranges shared among the boys.