Which of the rational number is greater in each of the following pairs?
(i) `frac(5)(9)("or")frac(-3)(-8)`
(ii) `frac(4)(-3)("or")frac(-8)(7)`
(iii) `frac(-12)(5)("or")-3`
(iv) `frac(7)(-9)("or")frac(-5)(8)`
(v) `frac(4)(-5)("or")frac(-7)(8)`
(vi) `frac(9)(-13)("or")frac(7)(-12)`

Answer

To determine which rational number is greater in each of the given pairs, we will convert the fractions to a common denominator and compare their values. Let's solve each part step by step.

### (i) Compare \( \frac{5}{9} \) and \( \frac{-3}{-8} \)

1. **Convert \( \frac{-3}{-8} \) to \( \frac{3}{8} \)** (since a negative divided by a negative is positive).
2. **Find a common denominator**: The LCM of 9 and 8 is 72.
3. **Convert \( \frac{5}{9} \)**:
\[ \frac{5}{9} = \frac{5 \times 8}{9 \times 8} = \frac{40}{72} \]
4. **Convert \( \frac{3}{8} \)**:
\[ \frac{3}{8} = \frac{3 \times 9}{8 \times 9} = \frac{27}{72} \]
5. **Compare**: \( \frac{40}{72} \) is greater than \( \frac{27}{72} \).

**Answer**: \( \frac{5}{9} \) is greater.

### (ii) Compare \( \frac{4}{-3} \) and \( \frac{-8}{7} \)

1. **Convert \( \frac{4}{-3} \) to \( -\frac{4}{3} \)**.
2. **Find a common denominator**: The LCM of 3 and 7 is 21.
3. **Convert \( -\frac{4}{3} \)**:
\[ -\frac{4}{3} = -\frac{4 \times 7}{3 \times 7} = -\frac{28}{21} \]
4. **Convert \( \frac{-8}{7} \)**:
\[ \frac{-8}{7} = \frac{-8 \times 3}{7 \times 3} = -\frac{24}{21} \]
5. **Compare**: \( -\frac{28}{21} \) is less than \( -\frac{24}{21} \).

**Answer**: \( \frac{-8}{7} \) is greater.

### (iii) Compare \( \frac{-12}{5} \) and \( -3 \)

1. **Convert \( -3 \) to a fraction**: \( -3 = \frac{-15}{5} \).
2. **Compare**: \( \frac{-12}{5} \) and \( \frac{-15}{5} \).
3. **Since \(-12\) is greater than \(-15\)**, we have \( \frac{-12}{5} > -3 \).

**Answer**: \( -3 \) is greater.

### (iv) Compare \( \frac{7}{-9} \) and \( \frac{-5}{8} \)

1. **Convert \( \frac{7}{-9} \) to \( -\frac{7}{9} \)**.
2. **Find a common denominator**: The LCM of 9 and 8 is 72.
3. **Convert \( -\frac{7}{9} \)**:
\[ -\frac{7}{9} = -\frac{7 \times 8}{9 \times 8} = -\frac{56}{72} \]
4. **Convert \( \frac{-5}{8} \)**:
\[ \frac{-5}{8} = \frac{-5 \times 9}{8 \times 9} = -\frac{45}{72} \]
5. **Compare**: \( -\frac{56}{72} < -\frac{45}{72} \).

**Answer**: \( \frac{-5}{8} \) is greater.

### (v) Compare \( \frac{4}{-5} \) and \( \frac{-7}{8} \)

1. **Convert \( \frac{4}{-5} \) to \( -\frac{4}{5} \)**.
2. **Find a common denominator**: The LCM of 5 and 8 is 40.
3. **Convert \( -\frac{4}{5} \)**:
\[ -\frac{4}{5} = -\frac{4 \times 8}{5 \times 8} = -\frac{32}{40} \]
4. **Convert \( \frac{-7}{8} \)**:
\[ \frac{-7}{8} = \frac{-7 \times 5}{8 \times 5} = -\frac{35}{40} \]
5. **Compare**: \( -\frac{32}{40} < -\frac{35}{40} \).

**Answer**: \( \frac{-7}{8} \) is greater.

### (vi) Compare \( \frac{9}{-13} \) and \( \frac{7}{-12} \)

1. **Convert \( \frac{9}{-13} \) to \( -\frac{9}{13} \)**.
2. **Find a common denominator**: The LCM of 13 and 12 is 156.
3. **Convert \( -\frac{9}{13} \)**:
\[ -\frac{9}{13} = -\frac{9 \times 12}{13 \times 12} = -\frac{108}{156} \]
4. **Convert \( \frac{7}{-12} \)**:
\[ \frac{7}{-12} = -\frac{7 \times 13}{12 \times 13} = -\frac{91}{156} \]
5. **Compare**: \( -\frac{108}{156} < -\frac{91}{156} \).

**Answer**: \( \frac{7}{-12} \) is greater.

### Summary of Answers:
- (i) \( \frac{5}{9} \)
- (ii) \( \frac{-8}{7} \)
- (iii) \( -3 \)
- (iv) \( \frac{-5}{8} \)
- (v) \( \frac{-7}{8} \)
- (vi) \( \frac{7}{-12} \)