1. **State the problem:** We have two similar triangles, B and C. Triangle B has sides 6 \frac{1}{4} and 2 \frac{1}{2}, and triangle C has sides a and 1 \frac{3}{4}. We need to find the value of $a$. 2. **Recall the property of similar triangles:** Corresponding sides of similar triangles are proportional. This means: $$\frac{\text{side of triangle C}}{\text{corresponding side of triangle B}} = \text{constant scale factor}$$ 3. **Convert mixed numbers to improper fractions:** - $6 \frac{1}{4} = \frac{25}{4}$ - $2 \frac{1}{2} = \frac{5}{2}$ - $1 \frac{3}{4} = \frac{7}{4}$ 4. **Set up the proportion using the known sides:** $$\frac{1 \frac{3}{4}}{2 \frac{1}{2}} = \frac{a}{6 \frac{1}{4}}$$ Substitute improper fractions: $$\frac{\frac{7}{4}}{\frac{5}{2}} = \frac{a}{\frac{25}{4}}$$ 5. **Simplify the left side:** $$\frac{\frac{7}{4}}{\frac{5}{2}} = \frac{7}{4} \times \frac{2}{5} = \frac{14}{20} = \frac{7}{10}$$ 6. **Write the equation:** $$\frac{7}{10} = \frac{a}{\frac{25}{4}}$$ 7. **Solve for $a$ by cross-multiplying:** $$a = \frac{7}{10} \times \frac{25}{4}$$ 8. **Multiply fractions:** $$a = \frac{7 \times 25}{10 \times 4} = \frac{175}{40}$$ 9. **Simplify the fraction:** $$a = \frac{\cancel{175}^{\times 7 \times 25}}{\cancel{40}^{\times 10 \times 4}} = \frac{35}{8}$$ 10. **Convert to mixed number:** $$\frac{35}{8} = 4 \frac{3}{8}$$ **Final answer:** $$a = 4 \frac{3}{8}$$