Let's tackle the problem step-by-step for each pair of numbers separately.
### Part (i): Insert six rational numbers between [tex]\(-\frac{1}{4}\)[/tex] and [tex]\(-\frac{2}{5}\)[/tex]
We need to find six rational numbers that lie between [tex]\(-\frac{1}{4}\)[/tex] and [tex]\(-\frac{2}{5}\)[/tex]. First, let's convert these fractions to decimal form for simplicity:
[tex]\[
-\frac{1}{4} = -0.25
\][/tex]
[tex]\[
-\frac{2}{5} = -0.4
\][/tex]
Now, we need to insert six rational numbers evenly spaced between these two values.
The resulting rational numbers between [tex]\(-0.25\)[/tex] and [tex]\(-0.4\)[/tex] are:
[tex]\[
-0.2714285714285714, -0.2928571428571428, -0.31428571428571433, -0.33571428571428574, -0.35714285714285715, -0.3785714285714286
\][/tex]
### Part (ii): Insert six rational numbers between [tex]\(\frac{21}{12}\)[/tex] and [tex]\(\frac{12}{21}\)[/tex]
We need to find six rational numbers that lie between [tex]\(\frac{21}{12}\)[/tex] and [tex]\(\frac{12}{21}\)[/tex]. First, let's convert these fractions to decimal form for simplicity:
[tex]\[
\frac{21}{12} \approx 1.75
\][/tex]
[tex]\[
\frac{12}{21} \approx 0.5714285714285714
\][/tex]
Now, we need to insert six rational numbers evenly spaced between these two values.
The resulting rational numbers between [tex]\(1.75\)[/tex] and [tex]\(0.5714285714285714\)[/tex] are:
[tex]\[
1.5816326530612244, 1.413265306122449, 1.2448979591836733, 1.0765306122448979, 0.9081632653061223, 0.7397959183673469
\][/tex]
### Summary:
Six rational numbers between [tex]\(-\frac{1}{4}\)[/tex] and [tex]\(-\frac{2}{5}\)[/tex]:
[tex]\[
-0.2714285714285714, -0.2928571428571428, -0.31428571428571433, -0.33571428571428574, -0.35714285714285715, -0.3785714285714286
\][/tex]
Six rational numbers between [tex]\(\frac{21}{12}\)[/tex] and [tex]\(\frac{12}{21}\)[/tex]:
[tex]\[
1.5816326530612244, 1.413265306122449, 1.2448979591836733, 1.0765306122448979, 0.9081632653061223, 0.7397959183673469
\][/tex]
By carefully calculating and inserting these rational numbers, we ensure that they lie evenly between the given pairs.
