To find the range of a set of numbers, we need to determine the difference between the maximum and minimum values in that set. Let's go through the steps to find the range of the numbers provided: \( \frac{1}{6}, \frac{1}{3}, \frac{3}{2}, \frac{2}{3}, \frac{8}{9}, \frac{4}{3} \). ### Step 1: Identify the Maximum and Minimum Values 1. **Convert all fractions to a common denominator** (if necessary) to easily compare their values. The denominators of the fractions are 6, 3, 2, and 9. The least common multiple (LCM) of these denominators is 18. We will convert each fraction: - \( \frac{1}{6} = \frac{3}{18} \) - \( \frac{1}{3} = \frac{6}{18} \) - \( \frac{3}{2} = \frac{27}{18} \) - \( \frac{2}{3} = \frac{12}{18} \) - \( \frac{8}{9} = \frac{16}{18} \) - \( \frac{4}{3} = \frac{24}{18} \) 2. **List the converted fractions**: - \( \frac{1}{6} = \frac{3}{18} \) - \( \frac{1}{3} = \frac{6}{18} \) - \( \frac{2}{3} = \frac{12}{18} \) - \( \frac{8}{9} = \frac{16}{18} \) - \( \frac{4}{3} = \frac{24}{18} \) - \( \frac{3}{2} = \frac{27}{18} \) 3. **Identify the maximum and minimum**: - The **maximum value** is \( \frac{3}{2} = \frac{27}{18} \). - The **minimum value** is \( \frac{1}{6} = \frac{3}{18} \). ### Step 2: Calculate the Range The range is calculated using the formula: \[ \text{Range} = \text{Maximum} - \text{Minimum} \] Substituting the values we found: \[ \text{Range} = \frac{3}{2} - \frac{1}{6} \] ### Step 3: Find a Common Denominator for the Calculation To perform the subtraction, we need a common denominator. The LCM of 2 and 6 is 6. We convert \( \frac{3}{2} \) to have a denominator of 6: \[ \frac{3}{2} = \frac{9}{6} \] Now we can perform the subtraction: \[ \text{Range} = \frac{9}{6} - \frac{1}{6} = \frac{9 - 1}{6} = \frac{8}{6} = \frac{4}{3} \] ### Conclusion The range of the numbers \( \frac{1}{6}, \frac{1}{3}, \frac{3}{2}, \frac{2}{3}, \frac{8}{9}, \frac{4}{3} \) is \( \frac{4}{3} \). ### Explanation of Options - **Option A: \( \frac{3}{4} \)** - This is incorrect because it does not represent the difference between the maximum and minimum values. - **Option B: \( \frac{5}{6} \)** - This is also incorrect for the same reason; it does not reflect the calculated range. - **Option C: \( \frac{7}{6} \)** - This is incorrect as it is not the difference between the maximum and minimum values. - **Option D: \( \frac{4}{3} \)** - This is the correct answer, as we calculated the range to be \( \frac{4}{3} \). ### Revision Summary - The range is calculated as the difference between the maximum and minimum values in a set of numbers. - Convert fractions to a common denominator for easier comparison. - The maximum value in the set is \( \frac{3}{2} \) and the minimum is \( \frac{1}{6} \). - The final range is \( \frac{4}{3} \), which corresponds to option D.