1. **State the problem:** We need to revise the hat size table so that it includes all possible head sizes measured to the nearest 1/8 inch between 21 inches and 24 inches. 2. **Understand the measurement precision:** Measuring to the nearest 1/8 inch means head sizes can be 21, 21 \frac{1}{8}, 21 \frac{2}{8} (which is 21 \frac{1}{4}), 21 \frac{3}{8}, 21 \frac{4}{8} (21 \frac{1}{2}), and so on, increasing by increments of 1/8 inch. 3. **Calculate the total number of increments:** From 21 to 24 inches is 3 inches. Since each increment is 1/8 inch, the number of increments is $$\frac{3}{\frac{1}{8}} = 3 \times 8 = 24.$$ So, there are 25 possible head sizes including both 21 and 24 inches. 4. **Assign ranges to hat sizes:** The original table has 4 hat sizes: Small, Medium, Large, Extra Large. To cover all 25 increments, divide the range 21 to 24 inches into 4 intervals, each covering about $$\frac{3}{4} = 0.75$$ inches or 6 increments (since 6 increments \times \frac{1}{8} = \frac{6}{8} = 0.75$). 5. **Define the ranges:** - Small: 21 to 21 \frac{6}{8} (21 \frac{3}{4}) inches - Medium: 21 \frac{7}{8} to 22 \frac{3}{8} inches - Large: 22 \frac{1}{2} to 23 \frac{1}{8} inches - Extra Large: 23 \frac{1}{4} to 24 inches 6. **Express the revised table:** | Hat Size | Head Size Range (inches) | |------------|----------------------------------| | Small | 21 to 21 \frac{3}{4} | | Medium | 21 \frac{7}{8} to 22 \frac{3}{8} | | Large | 22 \frac{1}{2} to 23 \frac{1}{8} | | Extra Large| 23 \frac{1}{4} to 24 | 7. **Summary:** This revised table includes every possible head size measured to the nearest 1/8 inch between 21 and 24 inches, with no gaps or overlaps, ensuring customers can find the correct hat size.