Answer :
To find three rational numbers between [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] using the method of finding the mean of two numbers, we will perform the following steps:
1. Compute the First Rational Number (Mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex])
First, we add [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] and then divide by 2 to find their mean.
[tex]\[ \text{Mean}_1 = \frac{\frac{-2}{3} + \frac{1}{6}}{2} \][/tex]
We need a common denominator to add these fractions, which is 6 in this case:
[tex]\[ \frac{-2}{3} = \frac{-2 \times 2}{3 \times 2} = \frac{-4}{6} \][/tex]
Now add the two fractions:
[tex]\[ \frac{-4}{6} + \frac{1}{6} = \frac{-4+1}{6} = \frac{-3}{6} = \frac{-1}{2} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_1 = \frac{\frac{-1}{2}}{2} = \frac{-1}{4} = -0.25 \][/tex]
So, the first rational number is [tex]\(-0.25\)[/tex].
2. Compute the Second Rational Number (Mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(-0.25\)[/tex])
Find the mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(-0.25\)[/tex]:
[tex]\[ \text{Mean}_2 = \frac{\frac{-2}{3} + (-0.25)}{2} \][/tex]
Convert [tex]\(-0.25\)[/tex] into fraction form:
[tex]\[ -0.25 = \frac{-1}{4} \][/tex]
Find a common denominator (12 in this case):
[tex]\[ \frac{-2}{3} = \frac{-2 \times 4}{3 \times 4} = \frac{-8}{12}, \quad \frac{-1}{4} = \frac{-1 \times 3}{4 \times 3} = \frac{-3}{12} \][/tex]
Add the fractions:
[tex]\[ \frac{-8}{12} + \frac{-3}{12} = \frac{-11}{12} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_2 = \frac{\frac{-11}{12}}{2} = \frac{-11}{24} \approx -0.4583333333333333 \][/tex]
So, the second rational number is approximately [tex]\(-0.4583\)[/tex].
3. Compute the Third Rational Number (Mean of [tex]\(-0.25\)[/tex] and [tex]\(\frac{1}{6}\)[/tex])
Find the mean of [tex]\(-0.25\)[/tex] and [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[ \text{Mean}_3 = \frac{-0.25 + \frac{1}{6}}{2} \][/tex]
Convert [tex]\(-0.25\)[/tex] to fraction form:
[tex]\[ -0.25 = \frac{-1}{4} \][/tex]
Find a common denominator (12 in this case):
[tex]\[ \frac{-1}{4} = \frac{-1 \times 3}{4 \times 3} = \frac{-3}{12}, \quad \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \][/tex]
Add the fractions:
[tex]\[ \frac{-3}{12} + \frac{2}{12} = \frac{-1}{12} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_3 = \frac{\frac{-1}{12}}{2} = \frac{-1}{24} \approx -0.04166666666666667 \][/tex]
So, the third rational number is approximately [tex]\(-0.0417\)[/tex].
Finally, the three rational numbers between [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] are approximately:
- [tex]\(-0.25\)[/tex]
- [tex]\(-0.4583\)[/tex]
- [tex]\(-0.0417\)[/tex]
### Representing the Numbers on the Number Line
To represent these numbers on the number line, we mark:
1. [tex]\(\frac{-2}{3} \approx -0.6667\)[/tex]
2. [tex]\(\frac{1}{6} \approx 0.1667\)[/tex]
3. [tex]\(-0.25\)[/tex]
4. [tex]\(-0.4583\)[/tex]
5. [tex]\(-0.0417\)[/tex]
Visualise plotting these points on the number line from left to right:
[tex]\[ -0.6667 \quad -0.4583 \quad -0.25 \quad -0.0417 \quad 0.1667 \][/tex]
These points fall between [tex]\(\frac{-2}{3}\)[/tex] (approximately [tex]\(-0.6667\)[/tex]) and [tex]\(\frac{1}{6}\)[/tex] (approximately [tex]\(0.1667\)[/tex]), successfully identifying three rational numbers between the two given numbers.
1. Compute the First Rational Number (Mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex])
First, we add [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] and then divide by 2 to find their mean.
[tex]\[ \text{Mean}_1 = \frac{\frac{-2}{3} + \frac{1}{6}}{2} \][/tex]
We need a common denominator to add these fractions, which is 6 in this case:
[tex]\[ \frac{-2}{3} = \frac{-2 \times 2}{3 \times 2} = \frac{-4}{6} \][/tex]
Now add the two fractions:
[tex]\[ \frac{-4}{6} + \frac{1}{6} = \frac{-4+1}{6} = \frac{-3}{6} = \frac{-1}{2} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_1 = \frac{\frac{-1}{2}}{2} = \frac{-1}{4} = -0.25 \][/tex]
So, the first rational number is [tex]\(-0.25\)[/tex].
2. Compute the Second Rational Number (Mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(-0.25\)[/tex])
Find the mean of [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(-0.25\)[/tex]:
[tex]\[ \text{Mean}_2 = \frac{\frac{-2}{3} + (-0.25)}{2} \][/tex]
Convert [tex]\(-0.25\)[/tex] into fraction form:
[tex]\[ -0.25 = \frac{-1}{4} \][/tex]
Find a common denominator (12 in this case):
[tex]\[ \frac{-2}{3} = \frac{-2 \times 4}{3 \times 4} = \frac{-8}{12}, \quad \frac{-1}{4} = \frac{-1 \times 3}{4 \times 3} = \frac{-3}{12} \][/tex]
Add the fractions:
[tex]\[ \frac{-8}{12} + \frac{-3}{12} = \frac{-11}{12} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_2 = \frac{\frac{-11}{12}}{2} = \frac{-11}{24} \approx -0.4583333333333333 \][/tex]
So, the second rational number is approximately [tex]\(-0.4583\)[/tex].
3. Compute the Third Rational Number (Mean of [tex]\(-0.25\)[/tex] and [tex]\(\frac{1}{6}\)[/tex])
Find the mean of [tex]\(-0.25\)[/tex] and [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[ \text{Mean}_3 = \frac{-0.25 + \frac{1}{6}}{2} \][/tex]
Convert [tex]\(-0.25\)[/tex] to fraction form:
[tex]\[ -0.25 = \frac{-1}{4} \][/tex]
Find a common denominator (12 in this case):
[tex]\[ \frac{-1}{4} = \frac{-1 \times 3}{4 \times 3} = \frac{-3}{12}, \quad \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \][/tex]
Add the fractions:
[tex]\[ \frac{-3}{12} + \frac{2}{12} = \frac{-1}{12} \][/tex]
Then find the mean:
[tex]\[ \text{Mean}_3 = \frac{\frac{-1}{12}}{2} = \frac{-1}{24} \approx -0.04166666666666667 \][/tex]
So, the third rational number is approximately [tex]\(-0.0417\)[/tex].
Finally, the three rational numbers between [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] are approximately:
- [tex]\(-0.25\)[/tex]
- [tex]\(-0.4583\)[/tex]
- [tex]\(-0.0417\)[/tex]
### Representing the Numbers on the Number Line
To represent these numbers on the number line, we mark:
1. [tex]\(\frac{-2}{3} \approx -0.6667\)[/tex]
2. [tex]\(\frac{1}{6} \approx 0.1667\)[/tex]
3. [tex]\(-0.25\)[/tex]
4. [tex]\(-0.4583\)[/tex]
5. [tex]\(-0.0417\)[/tex]
Visualise plotting these points on the number line from left to right:
[tex]\[ -0.6667 \quad -0.4583 \quad -0.25 \quad -0.0417 \quad 0.1667 \][/tex]
These points fall between [tex]\(\frac{-2}{3}\)[/tex] (approximately [tex]\(-0.6667\)[/tex]) and [tex]\(\frac{1}{6}\)[/tex] (approximately [tex]\(0.1667\)[/tex]), successfully identifying three rational numbers between the two given numbers.
