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

Answer

To solve the question, we need to compare the given rational numbers and fill in the blanks with the correct symbols: greater than (gt), equal to (=), or less than (lt). Let's go through each part step by step.

### Part (i): \((-3/7) \quad ? \quad (6/-13)\)

1. **Find a common denominator**: The denominators are 7 and -13. The LCM of 7 and 13 is 91.
2. **Convert to a common denominator**:
- For \(-3/7\):
\[ -3/7 = (-3 \times 13)/ (7 \times 13) = -39/91 \]
- For \(6/-13\):
\[ 6/-13 = (6 \times 7)/(-13 \times 7) = -42/91 \]
3. **Compare the two fractions**:
\[ -39/91 \quad ? \quad -42/91 \]
Since \(-39\) is greater than \(-42\), we have:
\[ -39/91 > -42/91 \]
So, the answer is:
\[ (-3/7) \gt (6/-13) \]

### Part (ii): \((5/-13) \quad ? \quad (-35/91)\)

1. **Find a common denominator**: The denominators are -13 and 91. The LCM of 13 and 91 is 91.
2. **Convert to a common denominator**:
- For \(5/-13\):
\[ 5/-13 = (5 \times 7)/(-13 \times 7) = -35/91 \]
- The second fraction is already \(-35/91\).
3. **Compare the two fractions**:
\[ -35/91 \quad ? \quad -35/91 \]
Since both are equal:
\[ (5/-13) = (-35/91) \]

### Part (iii): \(-2 \quad ? \quad (-13/5)\)

1. **Convert -2 to a fraction**:
\[ -2 = -2/1 \]
2. **Find a common denominator**: The denominators are 1 and 5. The LCM is 5.
3. **Convert to a common denominator**:
- For \(-2\):
\[ -2/1 = (-2 \times 5)/(1 \times 5) = -10/5 \]
4. **Compare the two fractions**:
\[ -10/5 \quad ? \quad -13/5 \]
Since \(-10\) is greater than \(-13\):
\[ -10/5 > -13/5 \]
So, the answer is:
\[ -2 \gt (-13/5) \]

### Part (iv): \((-2/3) \quad ? \quad (5/-8)\)

1. **Find a common denominator**: The denominators are 3 and -8. The LCM is 24.
2. **Convert to a common denominator**:
- For \(-2/3\):
\[ -2/3 = (-2 \times 8)/(3 \times 8) = -16/24 \]
- For \(5/-8\):
\[ 5/-8 = (5 \times 3)/(-8 \times 3) = -15/24 \]
3. **Compare the two fractions**:
\[ -16/24 \quad ? \quad -15/24 \]
Since \(-16\) is less than \(-15\):
\[ -16/24 < -15/24 \]
So, the answer is:
\[ (-2/3) < (5/-8) \]

### Part (v): \(0 \quad ? \quad (-3/-5)\)

1. **Convert \(-3/-5\)**:
\[ -3/-5 = 3/5 \]
2. **Compare**:
Since \(0\) is less than \(3/5\):
\[ 0 < 3/5 \]
So, the answer is:
\[ 0 < (-3/-5) \]

### Part (vi): \((-8/9) \quad ? \quad (-9/10)\)

1. **Find a common denominator**: The denominators are 9 and 10. The LCM is 90.
2. **Convert to a common denominator**:
- For \(-8/9\):
\[ -8/9 = (-8 \times 10)/(9 \times 10) = -80/90 \]
- For \(-9/10\):
\[ -9/10 = (-9 \times 9)/(10 \times 9) = -81/90 \]
3. **Compare the two fractions**:
\[ -80/90 \quad ? \quad -81/90 \]
Since \(-80\) is greater than \(-81\):
\[ -80/90 > -81/90 \]
So, the answer is:
\[ (-8/9) > (-9/10) \]

### Final Answers:
1. (i) \((-3/7) \gt (6/-13)\)
2. (ii) \((5/-13) = (-35/91)\)
3. (iii) \(-2 \gt (-13/5)\)
4. (iv) \((-2/3) < (5/-8)\)
5. (v) \(0 < (-3/-5)\)
6. (vi) \((-8/9) > (-9/10)\)