1. 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 numbe line to model the problem.

To solve the problem, let's break it down step by step. ### **Problem Statement** - **Point \( X \)** is located at \( \frac{2}{3} \) on a number line. - **Point \( Y \)** is the same distance from 0 as point \( X \) but has a numerator of 8. We need to find the denominator of the fraction representing point \( Y \). ### **Understanding the Problem** 1. **Distance from 0:** - Point \( X \) is at \( \frac{2}{3} \), which means it is \( \frac{2}{3} \) units away from 0. - Since point \( Y \) is the same distance from 0 as \( X \), it can be either: - \( \frac{2}{3} \) (same side as \( X \)) - \( -\frac{2}{3} \) (opposite side of \( X \)) 2. **Fraction Representation for \( Y \):** - Given that the numerator of \( Y \) is 8, we represent \( Y \) as \( \frac{8}{d} \), where \( d \) is the denominator we need to find. ### **Setting Up the Equation** Since \( Y \) is the same distance from 0 as \( X \): \[ \left| \frac{8}{d} \right| = \left| \frac{2}{3} \right| \] This leads to two possibilities: 1. \( \frac{8}{d} = \frac{2}{3} \) 2. \( \frac{8}{d} = -\frac{2}{3} \) Solving for \( d \): 1. **First Case:** \[ \frac{8}{d} = \frac{2}{3} \implies d = \frac{8 \times 3}{2} = 12 \] 2. **Second Case:** \[ \frac{8}{d} = -\frac{2}{3} \implies d = \frac{8 \times (-3)}{2} = -12 \] **Since denominators are typically positive in fraction representations, we take \( d = 12 \).** ### **Conclusion** The denominator of the fraction at point \( Y \) is **12**. ### **Number Line Representation** Below is a simple representation of the number line with points \( X \) and \( Y \): ``` <---|---------|---------|---------|---------|---> -1 -2/3 0 2/3 1 Point Y at -2/3 (represented as 8/12) Point X at 2/3 ``` - **Point \( X \)** is at \( \frac{2}{3} \). - **Point \( Y \)** is at \( -\frac{2}{3} \) (which is equivalent to \( \frac{8}{12} \)). This visual helps illustrate that \( Y \) is indeed the same distance from 0 as \( X \) but on the opposite side of the number line.