Which of the two rational numbers is greater in the given pair?
`{:((i) (-4)/(3) or (-8)/(7) , (ii) (7)/(-9) or (-5)/(8),(iii) (-1)/(3) or (4)/(-5)),((iv) (9)/(-13) or (7)/(-12),(v) (4)/(-5) or (-7)/(10) , (vi) (-12)/(5) or -3 ):}`

Answer

To determine which of the two rational numbers is greater in each pair, we will follow a systematic approach by finding a common denominator for the fractions and then comparing their values. Let's solve each part step by step.

### Part (i): Compare \(-\frac{4}{3}\) and \(-\frac{8}{7}\)

1. **Find a common denominator**: The denominators are 3 and 7. The least common multiple (LCM) of 3 and 7 is 21.
2. **Convert fractions**:
- For \(-\frac{4}{3}\): Multiply numerator and denominator by 7:
\(-\frac{4 \times 7}{3 \times 7} = -\frac{28}{21}\)
- For \(-\frac{8}{7}\): Multiply numerator and denominator by 3:
\(-\frac{8 \times 3}{7 \times 3} = -\frac{24}{21}\)
3. **Compare**:
\(-\frac{28}{21}\) and \(-\frac{24}{21}\). Since -24 is greater than -28, we conclude:
\(-\frac{8}{7}\) is greater than \(-\frac{4}{3}\).

### Part (ii): Compare \(\frac{7}{-9}\) and \(-\frac{5}{8}\)

1. **Find a common denominator**: The denominators are -9 and 8. The LCM is 72.
2. **Convert fractions**:
- For \(\frac{7}{-9}\): Multiply numerator and denominator by 8:
\(\frac{7 \times 8}{-9 \times 8} = \frac{56}{-72}\)
- For \(-\frac{5}{8}\): Multiply numerator and denominator by -9:
\(-\frac{5 \times -9}{8 \times -9} = \frac{45}{-72}\)
3. **Compare**:
\(\frac{56}{-72}\) and \(\frac{45}{-72}\). Since 56 is less than 45 (in negative terms), we conclude:
\(-\frac{5}{8}\) is greater than \(\frac{7}{-9}\).

### Part (iii): Compare \(-\frac{1}{3}\) and \(\frac{4}{-5}\)

1. **Find a common denominator**: The denominators are 3 and -5. The LCM is 15.
2. **Convert fractions**:
- For \(-\frac{1}{3}\): Multiply numerator and denominator by 5:
\(-\frac{1 \times 5}{3 \times 5} = -\frac{5}{15}\)
- For \(\frac{4}{-5}\): Multiply numerator and denominator by 3:
\(\frac{4 \times 3}{-5 \times 3} = -\frac{12}{15}\)
3. **Compare**:
\(-\frac{5}{15}\) and \(-\frac{12}{15}\). Since -5 is greater than -12, we conclude:
\(-\frac{1}{3}\) is greater than \(\frac{4}{-5}\).

### Part (iv): Compare \(\frac{9}{-13}\) and \(\frac{7}{-12}\)

1. **Find a common denominator**: The denominators are -13 and -12. The LCM is 156.
2. **Convert fractions**:
- For \(\frac{9}{-13}\): Multiply numerator and denominator by 12:
\(\frac{9 \times 12}{-13 \times 12} = \frac{108}{-156}\)
- For \(\frac{7}{-12}\): Multiply numerator and denominator by 13:
\(\frac{7 \times 13}{-12 \times 13} = \frac{91}{-156}\)
3. **Compare**:
\(\frac{108}{-156}\) and \(\frac{91}{-156}\). Since 108 is less than 91, we conclude:
\(\frac{7}{-12}\) is greater than \(\frac{9}{-13}\).

### Part (v): Compare \(\frac{4}{-5}\) and \(-\frac{7}{10}\)

1. **Find a common denominator**: The denominators are -5 and 10. The LCM is 10.
2. **Convert fractions**:
- For \(\frac{4}{-5}\): Multiply numerator and denominator by -2:
\(\frac{4 \times -2}{-5 \times -2} = \frac{-8}{10}\)
- For \(-\frac{7}{10}\): It remains \(-\frac{7}{10}\).
3. **Compare**:
\(-\frac{8}{10}\) and \(-\frac{7}{10}\). Since -8 is less than -7, we conclude:
\(-\frac{7}{10}\) is greater than \(\frac{4}{-5}\).

### Part (vi): Compare \(-\frac{12}{5}\) and -3

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

### Summary of Results:
1. \(-\frac{8}{7}\) is greater than \(-\frac{4}{3}\).
2. \(-\frac{5}{8}\) is greater than \(\frac{7}{-9}\).
3. \(-\frac{1}{3}\) is greater than \(\frac{4}{-5}\).
4. \(\frac{7}{-12}\) is greater than \(\frac{9}{-13}\).
5. \(-\frac{7}{10}\) is greater than \(\frac{4}{-5}\).
6. \(-\frac{12}{5}\) is greater than -3.