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.

Question

UpStudy AI · Accepted Answer

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.
The denominator of the fraction at point $$ Y $$ is **12**.

Point is at on a number line. On the
same number line, point is the same
distance from 0 as point , but it has a
numerator of 8 . What is the denominator
of the fraction at point ? Draw a numbe
line to model the problem.

Upstudy AI Solution

Answer

Solution

Problem Statement

Understanding the Problem

Setting Up the Equation

Conclusion

Number Line Representation

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Point is at on a number line. On the same number line, point is the same distance from 0 as point , but it has a numerator of 8 . What is the denominator of the fraction at point ? Draw a numbe line to model the problem.