Answer :
Certainly! Let's solve the two parts of the question step-by-step.
### Part I: Finding 3 Rational Numbers between [tex]\( \frac{1}{3} \)[/tex] and 12
1. Identify the starting and ending points:
- The starting point is [tex]\( \frac{1}{3} \)[/tex].
- The ending point is 12.
2. Find the total distance between these two points:
- The difference is [tex]\( 12 - \frac{1}{3} \)[/tex].
3. Divide this difference into 4 equal parts to find 3 intermediate values:
- The step size will be [tex]\( \frac{12 - \frac{1}{3}}{4} \)[/tex].
4. Calculate the intermediate rational numbers:
- The first rational number is [tex]\( \frac{1}{3} + \text{step size} \)[/tex].
- The second rational number is [tex]\( \frac{1}{3} + 2 \times \text{step size} \)[/tex].
- The third rational number is [tex]\( \frac{1}{3} + 3 \times \text{step size} \)[/tex].
When we perform these calculations, we get the following rational numbers:
- The first rational number is [tex]\( 3.25 \)[/tex].
- The second rational number is [tex]\( 6.166666666666666 \)[/tex].
- The third rational number is [tex]\( 9.083333333333334 \)[/tex].
### Part II: Finding 3 Rational Numbers between -2 and 0
1. Identify the starting and ending points:
- The starting point is -2.
- The ending point is 0.
2. Find the total distance between these two points:
- The difference is [tex]\( 0 - (-2) \)[/tex].
3. Divide this difference into 4 equal parts to find 3 intermediate values:
- The step size will be [tex]\( \frac{0 - (-2)}{4} \)[/tex].
4. Calculate the intermediate rational numbers:
- The first rational number is [tex]\( -2 + \text{step size} \)[/tex].
- The second rational number is [tex]\( -2 + 2 \times \text{step size} \)[/tex].
- The third rational number is [tex]\( -2 + 3 \times \text{step size} \)[/tex].
When we perform these calculations, we get the following rational numbers:
- The first rational number is [tex]\( -1.5 \)[/tex].
- The second rational number is [tex]\( -1.0 \)[/tex].
- The third rational number is [tex]\( -0.5 \)[/tex].
### Summary
- Three rational numbers between [tex]\( \frac{1}{3} \)[/tex] and 12 are [tex]\( 3.25 \)[/tex], [tex]\( 6.166666666666666 \)[/tex], and [tex]\( 9.083333333333334 \)[/tex].
- Three rational numbers between -2 and 0 are [tex]\( -1.5 \)[/tex], [tex]\( -1.0 \)[/tex], and [tex]\( -0.5 \)[/tex].
### Part I: Finding 3 Rational Numbers between [tex]\( \frac{1}{3} \)[/tex] and 12
1. Identify the starting and ending points:
- The starting point is [tex]\( \frac{1}{3} \)[/tex].
- The ending point is 12.
2. Find the total distance between these two points:
- The difference is [tex]\( 12 - \frac{1}{3} \)[/tex].
3. Divide this difference into 4 equal parts to find 3 intermediate values:
- The step size will be [tex]\( \frac{12 - \frac{1}{3}}{4} \)[/tex].
4. Calculate the intermediate rational numbers:
- The first rational number is [tex]\( \frac{1}{3} + \text{step size} \)[/tex].
- The second rational number is [tex]\( \frac{1}{3} + 2 \times \text{step size} \)[/tex].
- The third rational number is [tex]\( \frac{1}{3} + 3 \times \text{step size} \)[/tex].
When we perform these calculations, we get the following rational numbers:
- The first rational number is [tex]\( 3.25 \)[/tex].
- The second rational number is [tex]\( 6.166666666666666 \)[/tex].
- The third rational number is [tex]\( 9.083333333333334 \)[/tex].
### Part II: Finding 3 Rational Numbers between -2 and 0
1. Identify the starting and ending points:
- The starting point is -2.
- The ending point is 0.
2. Find the total distance between these two points:
- The difference is [tex]\( 0 - (-2) \)[/tex].
3. Divide this difference into 4 equal parts to find 3 intermediate values:
- The step size will be [tex]\( \frac{0 - (-2)}{4} \)[/tex].
4. Calculate the intermediate rational numbers:
- The first rational number is [tex]\( -2 + \text{step size} \)[/tex].
- The second rational number is [tex]\( -2 + 2 \times \text{step size} \)[/tex].
- The third rational number is [tex]\( -2 + 3 \times \text{step size} \)[/tex].
When we perform these calculations, we get the following rational numbers:
- The first rational number is [tex]\( -1.5 \)[/tex].
- The second rational number is [tex]\( -1.0 \)[/tex].
- The third rational number is [tex]\( -0.5 \)[/tex].
### Summary
- Three rational numbers between [tex]\( \frac{1}{3} \)[/tex] and 12 are [tex]\( 3.25 \)[/tex], [tex]\( 6.166666666666666 \)[/tex], and [tex]\( 9.083333333333334 \)[/tex].
- Three rational numbers between -2 and 0 are [tex]\( -1.5 \)[/tex], [tex]\( -1.0 \)[/tex], and [tex]\( -0.5 \)[/tex].
