Certainly! Let's tackle each part of the question step by step.
### 2.2: Write 5 rational numbers between [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{7}{9}\)[/tex]
1. Identify the bounds:
- The lower bound is [tex]\(\frac{5}{7}\)[/tex].
- The upper bound is [tex]\(\frac{7}{9}\)[/tex].
2. Calculations to find 5 rational numbers in between:
- To find 5 equally spaced rational numbers between [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{7}{9}\)[/tex], we need to divide the interval into 6 equal parts. This is because dividing into '6' parts will give us 5 intervals (and thus 5 numbers between the boundaries).
3. Rational numbers between [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{7}{9}\)[/tex]:
- After calculating the step size and adding it incrementally to the lower bound [tex]\( \frac{5}{7}\)[/tex], we get:
```
0.7248677248677249
0.7354497354497355
0.746031746031746
0.7566137566137566
0.7671957671957672
```
So, the 5 rational numbers between [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{7}{9}\)[/tex] are approximately:
[tex]\[
0.7248677248677249, \quad 0.7354497354497355, \quad 0.746031746031746, \quad 0.7566137566137566, \quad 0.7671957671957672
\][/tex]
### 43: Write 5 rational numbers between [tex]\(\frac{-3}{8}\)[/tex] and [tex]\(\frac{-5}{9}\)[/tex]
1. Identify the bounds:
- The lower bound is [tex]\(\frac{-3}{8}\)[/tex].
- The upper bound is [tex]\(\frac{-5}{9}\)[/tex].
2. Calculations to find 5 rational numbers in between:
- Similarly, as in the previous part, we will divide the interval into 6 equal parts to find 5 rational numbers in between.
3. Rational numbers between [tex]\(\frac{-3}{8}\)[/tex] and [tex]\(\frac{-5}{9}\)[/tex]:
- After calculating the step size and adding it incrementally to the upper bound [tex]\( \frac{-5}{9}\)[/tex], we get:
```
-0.525462962962963
-0.4953703703703704
-0.4652777777777778
-0.4351851851851852
-0.40509259259259256
```
So, the 5 rational numbers between [tex]\(\frac{-3}{8}\)[/tex] and [tex]\(\frac{-5}{9}\)[/tex] are approximately:
[tex]\[
-0.525462962962963, \quad -0.4953703703703704, \quad -0.4652777777777778, \quad -0.4351851851851852, \quad -0.40509259259259256
\][/tex]
In summary:
1. For the interval between [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{7}{9}\)[/tex]:
[tex]\[
0.7248677248677249, \quad 0.7354497354497355, \quad 0.746031746031746, \quad 0.7566137566137566, \quad 0.7671957671957672
\][/tex]
2. For the interval between [tex]\(\frac{-3}{8}\)[/tex] and [tex]\(\frac{-5}{9}\)[/tex]:
[tex]\[
-0.525462962962963, \quad -0.4953703703703704, \quad -0.4652777777777778, \quad -0.4351851851851852, \quad -0.40509259259259256
\][/tex]
