To find three rational numbers between the given pairs, we will follow a systematic approach for each pair.
### (i) Find three rational numbers between \( \frac{1}{2} \) and \( \frac{7}{3} \)
**Step 1:** Convert both fractions to have a common denominator.
- The denominators are 2 and 3. The least common multiple (LCM) of 2 and 3 is 6.
- Convert \( \frac{1}{2} \) to sixths:
\[
\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}
\]
- Convert \( \frac{7}{3} \) to sixths:
\[
\frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6}
\]
**Step 2:** Identify rational numbers between \( \frac{3}{6} \) and \( \frac{14}{6} \).
- The rational numbers between \( \frac{3}{6} \) and \( \frac{14}{6} \) can be found by selecting fractions with the same denominator (6).
- Possible rational numbers include \( \frac{4}{6}, \frac{5}{6}, \frac{6}{6}, \frac{7}{6}, \frac{8}{6}, \frac{9}{6}, \frac{10}{6}, \frac{11}{6}, \frac{12}{6}, \frac{13}{6} \).
**Step 3:** Choose any three rational numbers from the list.
- We can select \( \frac{4}{6}, \frac{5}{6}, \frac{6}{6} \) as our three rational numbers.
### (ii) Find three rational numbers between \( -\frac{3}{5} \) and \( \frac{2}{7} \)
**Step 1:** Convert both fractions to have a common denominator.
- The denominators are 5 and 7. The LCM of 5 and 7 is 35.
- Convert \( -\frac{3}{5} \) to thirty-fifths:
\[
-\frac{3}{5} = \frac{-3 \times 7}{5 \times 7} = -\frac{21}{35}
\]
- Convert \( \frac{2}{7} \) to thirty-fifths:
\[
\frac{2}{7} = \frac{2 \times 5}{7 \times 5} = \frac{10}{35}
\]
**Step 2:** Identify rational numbers between \( -\frac{21}{35} \) and \( \frac{10}{35} \).
- Possible rational numbers include \( -\frac{20}{35}, -\frac{19}{35}, -\frac{18}{35}, \ldots, \frac{9}{35} \).
**Step 3:** Choose any three rational numbers from the list.
- We can select \( -\frac{20}{35}, -\frac{19}{35}, -\frac{18}{35} \) as our three rational numbers.
### (iii) Find three rational numbers between \( \frac{2}{5} \) and \( \frac{8}{5} \)
**Step 1:** Check if the denominators are the same.
- Both fractions have the same denominator (5), so we can directly find rational numbers between them.
**Step 2:** Identify rational numbers between \( \frac{2}{5} \) and \( \frac{8}{5} \).
- Possible rational numbers include \( \frac{3}{5}, \frac{4}{5}, \frac{5}{5}, \frac{6}{5}, \frac{7}{5} \).
**Step 3:** Choose any three rational numbers from the list.
- We can select \( \frac{3}{5}, \frac{4}{5}, \frac{5}{5} \) as our three rational numbers.
### Summary of Answers:
1. Three rational numbers between \( \frac{1}{2} \) and \( \frac{7}{3} \): \( \frac{4}{6}, \frac{5}{6}, \frac{6}{6} \)
2. Three rational numbers between \( -\frac{3}{5} \) and \( \frac{2}{7} \): \( -\frac{20}{35}, -\frac{19}{35}, -\frac{18}{35} \)
3. Three rational numbers between \( \frac{2}{5} \) and \( \frac{8}{5} \): \( \frac{3}{5}, \frac{4}{5}, \frac{5}{5} \)
