Fill in the blanks with the correct symbol out of `gt, =` and `lt:`
(i) `frac(-3)(7)....frac(6)(-13)`
(ii) `frac(5)(-13)....frac(-35)(91)`
(iii) `-2....frac(-13)(5)`
(iv) `frac(-2)(3)....frac(5)(-8)`
(v) `0....frac(-3)(-5)`
(vi) `frac(-8)(9)....frac(-9)(10)`

Answer

To solve the problem, we will compare the given rational numbers step by step and fill in the blanks with the correct symbols (greater than `gt`, equal to `=`, or less than `lt`).

### Step-by-Step Solution:

1. **Part (i):** Compare \(-\frac{3}{7}\) and \(\frac{6}{-13}\).
- Convert both fractions to have a common denominator:
- \(-\frac{3}{7} = -\frac{39}{91}\) (by multiplying numerator and denominator by 13)
- \(\frac{6}{-13} = -\frac{42}{91}\) (by multiplying numerator and denominator by 7)
- Now compare \(-\frac{39}{91}\) and \(-\frac{42}{91}\):
- Since \(-39 > -42\), we have \(-\frac{3}{7} > \frac{6}{-13}\).
- **Answer:** \(-\frac{3}{7} \gt \frac{6}{-13}\)

2. **Part (ii):** Compare \(\frac{5}{-13}\) and \(-\frac{35}{91}\).
- Convert \(\frac{5}{-13}\) to a common denominator:
- \(\frac{5}{-13} = -\frac{35}{91}\) (by multiplying numerator and denominator by 7)
- Now compare \(-\frac{35}{91}\) and \(-\frac{35}{91}\):
- They are equal.
- **Answer:** \(\frac{5}{-13} = -\frac{35}{91}\)

3. **Part (iii):** Compare \(-2\) and \(-\frac{13}{5}\).
- Convert \(-2\) to a fraction: \(-2 = -\frac{10}{5}\).
- Now compare \(-\frac{10}{5}\) and \(-\frac{13}{5}\):
- Since \(-10 > -13\), we have \(-2 > -\frac{13}{5}\).
- **Answer:** \(-2 \gt -\frac{13}{5}\)

4. **Part (iv):** Compare \(-\frac{2}{3}\) and \(-\frac{5}{8}\).
- Convert both fractions to a common denominator (24):
- \(-\frac{2}{3} = -\frac{16}{24}\) (by multiplying numerator and denominator by 8)
- \(-\frac{5}{8} = -\frac{15}{24}\) (by multiplying numerator and denominator by 3)
- Now compare \(-\frac{16}{24}\) and \(-\frac{15}{24}\):
- Since \(-16 < -15\), we have \(-\frac{2}{3} < -\frac{5}{8}\).
- **Answer:** \(-\frac{2}{3} \lt -\frac{5}{8}\)

5. **Part (v):** Compare \(0\) and \(\frac{-3}{-5}\).
- Simplify \(\frac{-3}{-5} = \frac{3}{5}\).
- Now compare \(0\) and \(\frac{3}{5}\):
- Since \(0 < \frac{3}{5}\), we have \(0 < \frac{-3}{-5}\).
- **Answer:** \(0 \lt \frac{-3}{-5}\)

6. **Part (vi):** Compare \(-\frac{8}{9}\) and \(-\frac{9}{10}\).
- Convert both fractions to a common denominator (90):
- \(-\frac{8}{9} = -\frac{80}{90}\) (by multiplying numerator and denominator by 10)
- \(-\frac{9}{10} = -\frac{81}{90}\) (by multiplying numerator and denominator by 9)
- Now compare \(-\frac{80}{90}\) and \(-\frac{81}{90}\):
- Since \(-80 > -81\), we have \(-\frac{8}{9} > -\frac{9}{10}\).
- **Answer:** \(-\frac{8}{9} \gt -\frac{9}{10}\)

### Final Answers:
(i) \(-\frac{3}{7} \gt \frac{6}{-13}\)
(ii) \(\frac{5}{-13} = -\frac{35}{91}\)
(iii) \(-2 \gt -\frac{13}{5}\)
(iv) \(-\frac{2}{3} \lt -\frac{5}{8}\)
(v) \(0 \lt \frac{-3}{-5}\)
(vi) \(-\frac{8}{9} \gt -\frac{9}{10}\)