To solve the problem of finding the sum of the range and the mode of the given set of numbers, we will follow these steps: ### Step 1: Identify the Set of Numbers The set of numbers provided is: \[ 10, 9, 10, 9, 8, 7, 7, 10, 8, 10, 8, 4, 6, 9, 10, 9, 10, 9, 7, 10, 6, 5 \] ### Step 2: Calculate the Mode The mode is the number that appears most frequently in a data set. 1. **Count the frequency of each number:** - 4 appears 1 time - 5 appears 1 time - 6 appears 2 times - 7 appears 3 times - 8 appears 3 times - 9 appears 4 times - 10 appears 6 times 2. **Identify the mode:** The number that appears the most is 10, which appears 6 times. Therefore, the mode is: \[ \text{Mode} = 10 \] ### Step 3: Calculate the Range The range is the difference between the maximum and minimum values in the data set. 1. **Identify the maximum and minimum values:** - Maximum value = 10 - Minimum value = 4 2. **Calculate the range:** \[ \text{Range} = \text{Maximum} - \text{Minimum} = 10 - 4 = 6 \] ### Step 4: Sum the Mode and the Range Now that we have both the mode and the range, we can find their sum: \[ \text{Sum} = \text{Mode} + \text{Range} = 10 + 6 = 16 \] ### Conclusion The final answer is: \[ \text{Correct Option: A. 16} \] ### Explanation of Other Options - **Option B (14):** This option is incorrect because it does not account for the correct mode (10) and range (6). The sum of these values is 16, not 14. - **Option C (12):** This option is also incorrect for the same reason. The calculations show that the sum of the mode and range is 16, not 12. - **Option D (10):** This option is incorrect as it only considers the mode without adding the range. The mode is 10, but the range must also be included in the sum. ### Revision Summary - The **mode** of the set is the most frequently occurring number, which is 10. - The **range** is calculated as the difference between the maximum and minimum values, which is 6. - The **sum** of the mode and range is 16. - The correct answer is **Option A: 16**.