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, follow these steps:
1. Step 1: Calculate the first mean (First intermediate rational number)
The first mean is the arithmetic mean (average) of the given two numbers [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex].
[tex]\[ \text{First mean} = \frac{\frac{-2}{3} + \frac{1}{6}}{2} \][/tex]
To add the fractions, find a common denominator. The common denominator of 3 and 6 is 6. So,
[tex]\[ \frac{-2}{3} = \frac{-4}{6} \][/tex]
Now, add the fractions:
[tex]\[ \frac{-4}{6} + \frac{1}{6} = \frac{-4 + 1}{6} = \frac{-3}{6} = -\frac{1}{2} \][/tex]
Then, take the mean:
[tex]\[ \text{First mean} = \frac{-\frac{1}{2}}{2} = -\frac{1}{4} \][/tex]
Hence, the first intermediate rational number is [tex]\(-0.25\)[/tex].
2. Step 2: Calculate the second mean (Second intermediate rational number)
The second mean is the arithmetic mean of the first mean and the first given number.
[tex]\[ \text{Second mean} = \frac{\frac{-2}{3} + \left(-\frac{1}{4}\right)}{2} \][/tex]
Convert [tex]\(-\frac{1}{4}\)[/tex] to have a common denominator with [tex]\(\frac{-2}{3}\)[/tex]. The common denominator is 12:
[tex]\[ \frac{-2}{3} = \frac{-8}{12}, \quad \frac{-1}{4} = \frac{-3}{12} \][/tex]
Now, add the fractions:
[tex]\[ \frac{-8}{12} + \frac{-3}{12} = \frac{-11}{12} \][/tex]
Then, take the mean:
[tex]\[ \text{Second mean} = \frac{\frac{-11}{12}}{2} = \frac{-11}{24} \][/tex]
Hence, the second intermediate rational number is approximately [tex]\(-0.4583\)[/tex].
3. Step 3: Calculate the third mean (Third intermediate rational number)
The third mean is the arithmetic mean of the first mean and the second given number.
[tex]\[ \text{Third mean} = \frac{\left(-\frac{1}{4}\right) + \frac{1}{6}}{2} \][/tex]
Convert [tex]\(-\frac{1}{4}\)[/tex] to have a common denominator with [tex]\(\frac{1}{6}\)[/tex]. The common denominator is 12:
[tex]\[ -\frac{1}{4} = \frac{-3}{12}, \quad \frac{1}{6} = \frac{2}{12} \][/tex]
Now, add the fractions:
[tex]\[ \frac{-3}{12} + \frac{2}{12} = \frac{-1}{12} \][/tex]
Then, take the mean:
[tex]\[ \text{Third mean} = \frac{\frac{-1}{12}}{2} = \frac{-1}{24} \][/tex]
Hence, the third intermediate rational number is approximately [tex]\(-0.0417\)[/tex].
Result
The three rational numbers between [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] are:
- [tex]\(-0.25\)[/tex]
- [tex]\(-0.4583\)[/tex]
- [tex]\(-0.0417\)[/tex]
Number Line Representation
To represent these points on the number line, mark the values:
- [tex]\(\frac{-2}{3} = -0.6667\)[/tex]
- [tex]\(-0.4583\)[/tex]
- [tex]\(-0.25\)[/tex]
- [tex]\(-0.0417\)[/tex]
- [tex]\(\frac{1}{6} = 0.1667\)[/tex]
The number line will look like this:
```
<--|----|----|----|----|----|----|----|----|----|----|----|----|----|-->
-2/3 -0.4583 -0.25 -0.0417 1/6
```
This clearly shows that the three numbers [tex]\(-0.4583\)[/tex], [tex]\(-0.25\)[/tex], and [tex]\(-0.0417\)[/tex] lie between [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex].
1. Step 1: Calculate the first mean (First intermediate rational number)
The first mean is the arithmetic mean (average) of the given two numbers [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex].
[tex]\[ \text{First mean} = \frac{\frac{-2}{3} + \frac{1}{6}}{2} \][/tex]
To add the fractions, find a common denominator. The common denominator of 3 and 6 is 6. So,
[tex]\[ \frac{-2}{3} = \frac{-4}{6} \][/tex]
Now, add the fractions:
[tex]\[ \frac{-4}{6} + \frac{1}{6} = \frac{-4 + 1}{6} = \frac{-3}{6} = -\frac{1}{2} \][/tex]
Then, take the mean:
[tex]\[ \text{First mean} = \frac{-\frac{1}{2}}{2} = -\frac{1}{4} \][/tex]
Hence, the first intermediate rational number is [tex]\(-0.25\)[/tex].
2. Step 2: Calculate the second mean (Second intermediate rational number)
The second mean is the arithmetic mean of the first mean and the first given number.
[tex]\[ \text{Second mean} = \frac{\frac{-2}{3} + \left(-\frac{1}{4}\right)}{2} \][/tex]
Convert [tex]\(-\frac{1}{4}\)[/tex] to have a common denominator with [tex]\(\frac{-2}{3}\)[/tex]. The common denominator is 12:
[tex]\[ \frac{-2}{3} = \frac{-8}{12}, \quad \frac{-1}{4} = \frac{-3}{12} \][/tex]
Now, add the fractions:
[tex]\[ \frac{-8}{12} + \frac{-3}{12} = \frac{-11}{12} \][/tex]
Then, take the mean:
[tex]\[ \text{Second mean} = \frac{\frac{-11}{12}}{2} = \frac{-11}{24} \][/tex]
Hence, the second intermediate rational number is approximately [tex]\(-0.4583\)[/tex].
3. Step 3: Calculate the third mean (Third intermediate rational number)
The third mean is the arithmetic mean of the first mean and the second given number.
[tex]\[ \text{Third mean} = \frac{\left(-\frac{1}{4}\right) + \frac{1}{6}}{2} \][/tex]
Convert [tex]\(-\frac{1}{4}\)[/tex] to have a common denominator with [tex]\(\frac{1}{6}\)[/tex]. The common denominator is 12:
[tex]\[ -\frac{1}{4} = \frac{-3}{12}, \quad \frac{1}{6} = \frac{2}{12} \][/tex]
Now, add the fractions:
[tex]\[ \frac{-3}{12} + \frac{2}{12} = \frac{-1}{12} \][/tex]
Then, take the mean:
[tex]\[ \text{Third mean} = \frac{\frac{-1}{12}}{2} = \frac{-1}{24} \][/tex]
Hence, the third intermediate rational number is approximately [tex]\(-0.0417\)[/tex].
Result
The three rational numbers between [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] are:
- [tex]\(-0.25\)[/tex]
- [tex]\(-0.4583\)[/tex]
- [tex]\(-0.0417\)[/tex]
Number Line Representation
To represent these points on the number line, mark the values:
- [tex]\(\frac{-2}{3} = -0.6667\)[/tex]
- [tex]\(-0.4583\)[/tex]
- [tex]\(-0.25\)[/tex]
- [tex]\(-0.0417\)[/tex]
- [tex]\(\frac{1}{6} = 0.1667\)[/tex]
The number line will look like this:
```
<--|----|----|----|----|----|----|----|----|----|----|----|----|----|-->
-2/3 -0.4583 -0.25 -0.0417 1/6
```
This clearly shows that the three numbers [tex]\(-0.4583\)[/tex], [tex]\(-0.25\)[/tex], and [tex]\(-0.0417\)[/tex] lie between [tex]\(\frac{-2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex].
