Find three rational number between :
`(i)(1)/(2)and(7)/(3)""(ii)-(3)/(5)and(2)/(7)""(iii)(2)/(5)and(8)/(5)`

Explanation

To find three rational numbers between the given pairs of fractions, we can follow a systematic approach. Let's solve each part step by step.

### (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 a fraction with a denominator of 6:
\[ \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \]
- Convert \( \frac{7}{3} \) to a fraction with a denominator of 6:
\[ \frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6} \]

**Step 2:** Identify three rational numbers between \( \frac{3}{6} \) and \( \frac{14}{6} \).

- The rational numbers can be found by choosing fractions with the same denominator. We can select:
\[ \frac{4}{6}, \frac{5}{6}, \frac{6}{6} \]

### (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 a fraction with a denominator of 35:
\[ -\frac{3}{5} = -\frac{3 \times 7}{5 \times 7} = -\frac{21}{35} \]
- Convert \( \frac{2}{7} \) to a fraction with a denominator of 35:
\[ \frac{2}{7} = \frac{2 \times 5}{7 \times 5} = \frac{10}{35} \]

**Step 2:** Identify three rational numbers between \( -\frac{21}{35} \) and \( \frac{10}{35} \).

- The rational numbers can be found by selecting fractions with the same denominator. We can choose:
\[ -\frac{20}{35}, -\frac{19}{35}, -\frac{18}{35} \]

### (iii) Find three rational numbers between \( \frac{2}{5} \) and \( \frac{8}{5} \).

**Step 1:** The denominators are the same (5), so we can directly find numbers between the numerators.

- The fractions are \( \frac{2}{5} \) and \( \frac{8}{5} \), which means we can look at the numerators 2 and 8.
- The rational numbers can be:
\[ \frac{3}{5}, \frac{4}{5}, \frac{5}{5} \]

### Summary of Rational Numbers Found:
1. Between \( \frac{1}{2} \) and \( \frac{7}{3} \): \( \frac{4}{6}, \frac{5}{6}, \frac{6}{6} \)
2. Between \( -\frac{3}{5} \) and \( \frac{2}{7} \): \( -\frac{20}{35}, -\frac{19}{35}, -\frac{18}{35} \)
3. Between \( \frac{2}{5} \) and \( \frac{8}{5} \): \( \frac{3}{5}, \frac{4}{5}, \frac{5}{5} \)