To solve the problem, we need to follow a series of steps to find the mean deviation of the numbers given. Let's break it down step-by-step. ### Step 1: Calculate the Mean We are given four numbers: 1. \(0\) 2. \(x + 2\) 3. \(3x + 6\) 4. \(4x + 8\) To find the mean of these numbers, we first need to sum them up and then divide by the number of values (which is 4). **Sum of the numbers:** \[ 0 + (x + 2) + (3x + 6) + (4x + 8) = 0 + x + 2 + 3x + 6 + 4x + 8 \] Combining like terms: \[ (0 + x + 3x + 4x) + (2 + 6 + 8) = 8x + 16 \] **Mean:** \[ \text{Mean} = \frac{\text{Sum of the numbers}}{\text{Number of values}} = \frac{8x + 16}{4} = 2x + 4 \] We are told that this mean equals 4: \[ 2x + 4 = 4 \] **Solving for \(x\):** \[ 2x = 4 - 4 \] \[ 2x = 0 \implies x = 0 \] ### Step 2: Substitute \(x\) Back into the Numbers Now that we have \(x = 0\), we can substitute it back into the original numbers: 1. \(0\) 2. \(x + 2 = 0 + 2 = 2\) 3. \(3x + 6 = 3(0) + 6 = 6\) 4. \(4x + 8 = 4(0) + 8 = 8\) So the numbers are now: - \(0\) - \(2\) - \(6\) - \(8\) ### Step 3: Calculate the Mean Deviation **Mean Deviation** is defined as the average of the absolute differences between each number and the mean of the numbers. **Step 3.1: Calculate the Mean of the New Numbers** \[ \text{Mean} = \frac{0 + 2 + 6 + 8}{4} = \frac{16}{4} = 4 \] **Step 3.2: Calculate the Absolute Deviations** Now we find the absolute deviations from the mean (which is 4): 1. \(|0 - 4| = 4\) 2. \(|2 - 4| = 2\) 3. \(|6 - 4| = 2\) 4. \(|8 - 4| = 4\) **Step 3.3: Sum of Absolute Deviations** \[ \text{Sum of absolute deviations} = 4 + 2 + 2 + 4 = 12 \] **Step 3.4: Mean Deviation** \[ \text{Mean Deviation} = \frac{\text{Sum of absolute deviations}}{\text{Number of values}} = \frac{12}{4} = 3 \] ### Conclusion The mean deviation of the numbers is **3**. ### Explanation of Options - **Option A (0)**: This is incorrect because the mean deviation cannot be zero unless all numbers are the same, which is not the case here. - **Option B (2)**: This is incorrect as the calculated mean deviation is higher than 2. - **Option C (3)**: This is the correct answer, as we calculated the mean deviation to be 3. - **Option D (4)**: This is incorrect because the mean deviation is not equal to the mean itself, and we calculated it to be 3. ### Revision Summary - The mean of the numbers was calculated to be 4, leading to \(x = 0\). - The new numbers were determined to be \(0, 2, 6, 8\). - The mean deviation was calculated as 3 by finding the absolute deviations from the mean. - The correct answer is **C (3)**.