1. **Problem 1: Evaluate the expression** $7 \frac{1}{4} - \left[ \frac{5}{6} \div \left\{ \frac{1}{3} - \left( \frac{1}{2} \times \left( \frac{3}{4} - \frac{1}{4} \right) \right) \right\} \right]$. 2. **Step 1: Convert mixed number to improper fraction** $7 \frac{1}{4} = \frac{29}{4}$. 3. **Step 2: Simplify inside the innermost parentheses** $\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$. 4. **Step 3: Multiply** $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$. 5. **Step 4: Subtract inside the braces** $\frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}$. 6. **Step 5: Divide** $\frac{5}{6} \div \frac{1}{12} = \frac{5}{6} \times \frac{12}{1} = \frac{60}{6} = 10$. 7. **Step 6: Subtract from the initial fraction** $\frac{29}{4} - 10 = \frac{29}{4} - \frac{40}{4} = -\frac{11}{4} = -2 \frac{3}{4}$. --- 8. **Problem 2: Find the ratio of software engineers proficient in Python to those proficient in Java.** 9. **Given data:** - Proficient in Python and Java: 50 - Proficient in Python only: 70 - Proficient in Java only: 60 - Proficient in neither: 20 10. **Calculate total proficient in Python:** $50 + 70 = 120$ 11. **Calculate total proficient in Java:** $50 + 60 = 110$ 12. **Ratio of Python to Java proficient engineers:** $\frac{120}{110} = \frac{12}{11}$. **Final answers:** - Value of expression: $-2 \frac{3}{4}$ - Ratio of Python to Java proficient engineers: $\frac{12}{11}$