To insert five rational numbers between [tex]\(\frac{-1}{5}\)[/tex] and [tex]\(\frac{-1}{7}\)[/tex], let's break the process down step by step.
1. Convert the fractions to decimals:
- [tex]\(\frac{-1}{5} = -0.20\)[/tex]
- [tex]\(\frac{-1}{7} \approx -0.142857\)[/tex]
2. Identify the start and end values:
- Start value: [tex]\(-0.20\)[/tex]
- End value: [tex]\(-0.142857\)[/tex]
3. Calculate the difference between the start and end values:
- Difference: [tex]\(-0.142857 - (-0.20) = 0.057143\)[/tex]
4. Determine the step size to generate five numbers between the start and end values:
- Since we need to insert five numbers, we need six equal segments from [tex]\(-0.20\)[/tex] to [tex]\(-0.142857\)[/tex].
- Step size: [tex]\(\frac{0.057143}{6} \approx 0.009524\)[/tex]
5. Generate the five rational numbers by progressively adding the step size:
- [tex]\(-0.20 + 1 \cdot 0.009524 = -0.190476\)[/tex]
- [tex]\(-0.20 + 2 \cdot 0.009524 = -0.180952\)[/tex]
- [tex]\(-0.20 + 3 \cdot 0.009524 = -0.171429\)[/tex]
- [tex]\(-0.20 + 4 \cdot 0.009524 = -0.161905\)[/tex]
- [tex]\(-0.20 + 5 \cdot 0.009524 = -0.152381\)[/tex]
The five rational numbers between [tex]\(\frac{-1}{5}\)[/tex] and [tex]\(\frac{-1}{7}\)[/tex] are:
[tex]\[
-0.190476, \quad -0.180952, \quad -0.171429, \quad -0.161905, \quad -0.152381
\][/tex]
