11. Point \( X \) is at \( \frac{2}{3} \) on a number line. On the same number line, point \( Y \) is the same distance from 0 as point \( X \), but it has a numerator of 8 . What is the denominator of the fraction at point \( Y \) ? Draw a number

Let's analyze the problem step by step. 1. **Position of Point X:** - Point \( X \) is located at \( \frac{2}{3} \) on the number line. 2. **Position of Point Y:** - Point \( Y \) is the same distance from 0 as point \( X \). This means \( Y \) is either at \( \frac{2}{3} \) or \( -\frac{2}{3} \). - However, it's given that the fraction representing \( Y \) has a numerator of 8. Therefore, \( Y \) must be at \( \frac{8}{d} \), where \( d \) is the denominator we need to find. 3. **Setting Up the Equation:** - Since \( Y \) is the same distance from 0 as \( X \), we have: \[ \left| \frac{8}{d} \right| = \left| \frac{2}{3} \right| \] - Simplifying the absolute values: \[ \frac{8}{|d|} = \frac{2}{3} \] 4. **Solving for \( d \):** - Cross-multiplying gives: \[ 8 \times 3 = 2 \times |d| \] \[ 24 = 2|d| \] \[ |d| = 12 \] - Since the denominator is a positive number, \( d = 12 \). **Answer:** The denominator of the fraction at point \( Y \) is **12**. **Visualization:** Here’s a simple number line showing the positions of points \( X \) and \( Y \): ``` ---|-----|-----|-----|-----|-----|--- -1 -2/3 0 2/3 1 4/3 Y X (-8/12) (2/3) ``` - Point \( X \) is at \( \frac{2}{3} \). - Point \( Y \) is at \( -\frac{8}{12} \), which simplifies to \( -\frac{2}{3} \), maintaining the same distance from 0 as \( X \).