1. **Stating the problem:** We have three trails with distances given as mixed numbers: Black Bear (7 1/2 miles), Eagle Ridge (10 1/4 miles), and Doe Run (8 3/4 miles). 2. **Convert mixed numbers to improper fractions:** - $7 \frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{15}{2}$ - $10 \frac{1}{4} = \frac{10 \times 4 + 1}{4} = \frac{41}{4}$ - $8 \frac{3}{4} = \frac{8 \times 4 + 3}{4} = \frac{35}{4}$ 3. **Find a common denominator to compare or add distances:** The denominators are 2 and 4; the least common denominator is 4. 4. **Convert all fractions to denominator 4:** - $\frac{15}{2} = \frac{15 \times 2}{2 \times 2} = \frac{30}{4}$ - $\frac{41}{4}$ stays the same. - $\frac{35}{4}$ stays the same. 5. **Summary of distances in improper fractions with denominator 4:** - Black Bear: $\frac{30}{4}$ miles - Eagle Ridge: $\frac{41}{4}$ miles - Doe Run: $\frac{35}{4}$ miles This conversion allows easy comparison or further calculations with the distances. **Final answer:** - Black Bear: $7 \frac{1}{2}$ miles = $\frac{15}{2}$ miles = $\frac{30}{4}$ miles - Eagle Ridge: $10 \frac{1}{4}$ miles = $\frac{41}{4}$ miles - Doe Run: $8 \frac{3}{4}$ miles = $\frac{35}{4}$ miles