To solve the given problem, which is to find a rational number between \(\frac{4}{5}\) and \(\frac{5}{9}\) and then arrange these numbers in ascending order, follow the steps below:
1. Identify the given fractions:
- Fraction 1: \(\frac{4}{5}\)
- Fraction 2: \(\frac{5}{9}\)
2. Convert the fractions to decimal form for easier comparison and calculation:
- \(\frac{4}{5} = 0.8\)
- \(\frac{5}{9} \approx 0.5555555555555556\)
3. Find a rational number between \(\frac{4}{5}\) and \(\frac{5}{9}\):
- To find a rational number between the two given fractions, we can average them.
- The average of \(0.8\) and \(0.5555555555555556\) is:
[tex]\[
\text{Average} = \frac{0.8 + 0.5555555555555556}{2} = \frac{1.3555555555555556}{2} = 0.6777777777777778
\][/tex]
4. Verify that \(0.6777777777777778\) lies between \(0.8\) and \(0.5555555555555556\):
- Clearly, \(0.6777777777777778\) is greater than \(0.5555555555555556\) and less than \(0.8\).
5. Arrange the numbers in ascending order:
- Arrange \(0.5555555555555556\) (which is \(\frac{5}{9}\)), \(0.6777777777777778\) (the average), and \(0.8\) (\(\frac{4}{5}\)) in ascending order:
[tex]\[
0.5555555555555556 \ (\frac{5}{9}), \ 0.6777777777777778, \ 0.8 \ (\frac{4}{5})
\][/tex]
Thus, the rational number we have inserted between \(\frac{4}{5}\) and \(\frac{5}{9}\) is \(0.6777777777777778\). Arranged in ascending order, the numbers are:
[tex]\[
0.5555555555555556 \ (\frac{5}{9}), \ 0.6777777777777778, \ 0.8 \ (\frac{4}{5})
\][/tex]
Therefore, the fractions in ascending order along with the rational number inserted between them are [tex]\((\frac{5}{9}, 0.6777777777777778, \frac{4}{5})\)[/tex].
