Find a rational number between :
`(i)(3)/(7)and(5)/(14)""(ii)(2)/(5)and-(1)/(3)""(iii)-(1)/(3)and-(1)/(2)`

Explanation

To find a rational number between the given pairs of fractions, we can follow these steps:

### (i) Find a rational number between \( \frac{3}{7} \) and \( \frac{5}{14} \)

1. **Convert the fractions to have a common denominator.**
- The least common multiple (LCM) of 7 and 14 is 14.
- Convert \( \frac{3}{7} \) to have a denominator of 14:
\[ \frac{3}{7} = \frac{3 \times 2}{7 \times 2} = \frac{6}{14} \]
- The second fraction \( \frac{5}{14} \) remains the same.

2. **Identify the two fractions:**
- Now we have \( \frac{6}{14} \) and \( \frac{5}{14} \).
- Since \( \frac{6}{14} > \frac{5}{14} \), we can identify the range as \( \frac{5}{14} < \frac{6}{14} \).

3. **Find a rational number between them.**
- A simple rational number between \( \frac{5}{14} \) and \( \frac{6}{14} \) is:
\[ \frac{5 + 6}{2 \times 14} = \frac{11}{28} \]

### (ii) Find a rational number between \( \frac{2}{5} \) and \( -\frac{1}{3} \)

1. **Convert the fractions to have a common denominator.**
- The LCM of 5 and 3 is 15.
- Convert \( \frac{2}{5} \):
\[ \frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15} \]
- Convert \( -\frac{1}{3} \):
\[ -\frac{1}{3} = \frac{-1 \times 5}{3 \times 5} = \frac{-5}{15} \]

2. **Identify the two fractions:**
- Now we have \( \frac{6}{15} \) and \( \frac{-5}{15} \).
- The range is \( \frac{-5}{15} < \frac{6}{15} \).

3. **Find a rational number between them.**
- A rational number between \( \frac{-5}{15} \) and \( \frac{6}{15} \) can be:
\[ \frac{-5 + 6}{2 \times 15} = \frac{1}{30} \]

### (iii) Find a rational number between \( -\frac{1}{3} \) and \( -\frac{1}{2} \)

1. **Convert the fractions to have a common denominator.**
- The LCM of 3 and 2 is 6.
- Convert \( -\frac{1}{3} \):
\[ -\frac{1}{3} = \frac{-1 \times 2}{3 \times 2} = \frac{-2}{6} \]
- Convert \( -\frac{1}{2} \):
\[ -\frac{1}{2} = \frac{-1 \times 3}{2 \times 3} = \frac{-3}{6} \]

2. **Identify the two fractions:**
- Now we have \( \frac{-2}{6} \) and \( \frac{-3}{6} \).
- The range is \( \frac{-3}{6} < \frac{-2}{6} \).

3. **Find a rational number between them.**
- A rational number between \( \frac{-3}{6} \) and \( \frac{-2}{6} \) can be:
\[ \frac{-3 + (-2)}{2 \times 6} = \frac{-5}{12} \]

### Summary of Answers:
1. Between \( \frac{3}{7} \) and \( \frac{5}{14} \): \( \frac{11}{28} \)
2. Between \( \frac{2}{5} \) and \( -\frac{1}{3} \): \( \frac{1}{30} \)
3. Between \( -\frac{1}{3} \) and \( -\frac{1}{2} \): \( \frac{-5}{12} \)