micha (2) Reason Quantitatively Algebra Write a numerator that makes the statement true. 11. \( \frac{1}{2}<\frac{}{10}<\frac{4}{5} \) 12. \( \frac{1}{4}<\frac{5}{12}<\frac{}{6} \) 13. \( \frac{}{8}<\frac{3}{4}<\frac{7}{8} \) Chapter \( 6 \cdot \) Lesson 8269

To solve the inequalities given in the problems, we need to find numerators that make the statements true. Let's analyze each inequality step by step. ### Problem 11: \[ \frac{1}{2} < \frac{x}{10} < \frac{4}{5} \] 1. **Finding the left inequality**: \[ \frac{1}{2} < \frac{x}{10} \] Multiply both sides by 10: \[ 5 < x \quad \Rightarrow \quad x > 5 \] 2. **Finding the right inequality**: \[ \frac{x}{10} < \frac{4}{5} \] Multiply both sides by 10: \[ x < 8 \] 3. **Combining the results**: \[ 5 < x < 8 \] A suitable integer for \( x \) could be 6 or 7. ### Problem 12: \[ \frac{1}{4} < \frac{5}{12} < \frac{y}{6} \] 1. **Finding the left inequality**: \[ \frac{1}{4} < \frac{5}{12} \] This is true since \( \frac{1}{4} = \frac{3}{12} \) and \( \frac{3}{12} < \frac{5}{12} \). 2. **Finding the right inequality**: \[ \frac{5}{12} < \frac{y}{6} \] Multiply both sides by 6: \[ \frac{5 \cdot 6}{12} < y \quad \Rightarrow \quad \frac{30}{12} < y \quad \Rightarrow \quad 2.5 < y \] 3. **Combining the results**: \[ y > 2.5 \] A suitable integer for \( y \) could be 3. ### Problem 13: \[ \frac{z}{8} < \frac{3}{4} < \frac{7}{8} \] 1. **Finding the left inequality**: \[ \frac{z}{8} < \frac{3}{4} \] Multiply both sides by 8: \[ z < 6 \] 2. **Finding the right inequality**: \[ \frac{3}{4} < \frac{7}{8} \] This is true since \( \frac{3}{4} = \frac{6}{8} \) and \( \frac{6}{8} < \frac{7}{8} \). 3. **Combining the results**: \[ z < 6 \] A suitable integer for \( z \) could be 5. ### Summary of Results: - For Problem 11, \( x \) could be 6 or 7. - For Problem 12, \( y \) could be 3. - For Problem 13, \( z \) could be 5. Thus, the numerators that make the statements true are: - 11: 6 or 7 - 12: 3 - 13: 5