Sure, let me guide you through the process of finding 10 rational numbers between [tex]\(\frac{1}{5}\)[/tex] and 2.
1. Understanding the Range:
The range you are given is between [tex]\(\frac{1}{5}\)[/tex] (which is equivalent to 0.2) and 2.
2. Dividing the Range:
To find 10 rational numbers between these two endpoints, you can divide the interval [tex]\((0.2, 2)\)[/tex] into 11 equal parts. When you divide a range into [tex]\(n\)[/tex] parts, you get [tex]\(n - 1\)[/tex] internal points.
3. Calculating the Step Size:
The step size can be determined by subtracting the starting point from the ending point and then dividing by the number of parts.
[tex]\[
\text{Step size} = \frac{2 - 0.2}{11} = \frac{1.8}{11} \approx 0.16363636363636365
\][/tex]
4. Generating the Rational Numbers:
Starting from [tex]\(\frac{1}{5}\)[/tex] (0.2), you then add the step size to generate the sequence:
[tex]\[
\text{First point: } 0.2 + 0.16363636363636365 \approx 0.36363636363636365
\][/tex]
[tex]\[
\text{Second point: } 0.2 + 2 \times 0.16363636363636365 \approx 0.5272727272727273
\][/tex]
[tex]\[
\text{Third point: } 0.2 + 3 \times 0.16363636363636365 \approx 0.6909090909090909
\][/tex]
[tex]\[
\text{Fourth point: } 0.2 + 4 \times 0.16363636363636365 \approx 0.8545454545454545
\][/tex]
[tex]\[
\text{Fifth point: } 0.2 + 5 \times 0.16363636363636365 \approx 1.018181818181818
\][/tex]
[tex]\[
\text{Sixth point: } 0.2 + 6 \times 0.16363636363636365 \approx 1.1818181818181819
\][/tex]
[tex]\[
\text{Seventh point: } 0.2 + 7 \times 0.16363636363636365 \approx 1.3454545454545455
\][/tex]
[tex]\[
\text{Eighth point: } 0.2 + 8 \times 0.16363636363636365 \approx 1.509090909090909
\][/tex]
[tex]\[
\text{Ninth point: } 0.2 + 9 \times 0.16363636363636365 \approx 1.6727272727272726
\][/tex]
[tex]\[
\text{Tenth point: } 0.2 + 10 \times 0.16363636363636365 \approx 1.8363636363636362
\][/tex]
5. Final List of Rational Numbers:
The 10 rational numbers between [tex]\(\frac{1}{5}\)[/tex] and 2 are:
[tex]\[
\begin{align*}
0.36363636363636365, \\
0.5272727272727273, \\
0.6909090909090909, \\
0.8545454545454545, \\
1.018181818181818, \\
1.1818181818181819, \\
1.3454545454545455, \\
1.509090909090909, \\
1.6727272727272726, \\
1.8363636363636362
\end{align*}
\][/tex]
These values effectively partition the range from [tex]\(\frac{1}{5}\)[/tex] to 2 into ten equal intervals, providing you with 10 rational numbers within the specified range.
