Simplify:
` {:((i)(4)/(9) divide (-5)/(12),(ii) -8 divide (-7)/(16),(iii) (-12)/(7) divide (-18)),((iv)(-1)/(10) divide (-8)/(5), (v) (-16)/(35) divide (-15)/(14), (vi) (-65)/(14) divide (13)/(7)):}`

Answer

To simplify the given expressions step by step, we will follow the rules of division of fractions, which involves multiplying by the reciprocal of the divisor. Let's solve each part one by one.

### Part (i): Simplify \( \frac{4}{9} \div \left(-\frac{5}{12}\right) \)

1. **Convert the division to multiplication by the reciprocal**:
\[ \frac{4}{9} \div \left(-\frac{5}{12}\right) = \frac{4}{9} \times \left(-\frac{12}{5}\right) \]

2. **Multiply the fractions**:
\[ = \frac{4 \times (-12)}{9 \times 5} = \frac{-48}{45} \]

3. **Simplify the fraction**:
\[ = \frac{-16}{15} \quad \text{(dividing numerator and denominator by 3)} \]

### Part (ii): Simplify \( -8 \div \left(-\frac{7}{16}\right) \)

1. **Convert the division to multiplication by the reciprocal**:
\[ -8 \div \left(-\frac{7}{16}\right) = -8 \times \left(-\frac{16}{7}\right) \]

2. **Multiply the numbers**:
\[ = \frac{8 \times 16}{7} = \frac{128}{7} \]

### Part (iii): Simplify \( \left(-\frac{12}{7}\right) \div (-18) \)

1. **Convert the division to multiplication by the reciprocal**:
\[ \left(-\frac{12}{7}\right) \div (-18) = \left(-\frac{12}{7}\right) \times \left(-\frac{1}{18}\right) \]

2. **Multiply the fractions**:
\[ = \frac{12}{7 \times 18} = \frac{12}{126} \]

3. **Simplify the fraction**:
\[ = \frac{2}{21} \quad \text{(dividing numerator and denominator by 6)} \]

### Part (iv): Simplify \( \left(-\frac{1}{10}\right) \div \left(-\frac{8}{5}\right) \)

1. **Convert the division to multiplication by the reciprocal**:
\[ \left(-\frac{1}{10}\right) \div \left(-\frac{8}{5}\right) = \left(-\frac{1}{10}\right) \times \left(-\frac{5}{8}\right) \]

2. **Multiply the fractions**:
\[ = \frac{1 \times 5}{10 \times 8} = \frac{5}{80} \]

3. **Simplify the fraction**:
\[ = \frac{1}{16} \quad \text{(dividing numerator and denominator by 5)} \]

### Part (v): Simplify \( \left(-\frac{16}{35}\right) \div \left(-\frac{15}{14}\right) \)

1. **Convert the division to multiplication by the reciprocal**:
\[ \left(-\frac{16}{35}\right) \div \left(-\frac{15}{14}\right) = \left(-\frac{16}{35}\right) \times \left(-\frac{14}{15}\right) \]

2. **Multiply the fractions**:
\[ = \frac{16 \times 14}{35 \times 15} = \frac{224}{525} \]

3. **Simplify the fraction**:
\[ \text{(No common factors, so it remains } \frac{224}{525}\text{)} \]

### Part (vi): Simplify \( \left(-\frac{65}{14}\right) \div \left(\frac{13}{7}\right) \)

1. **Convert the division to multiplication by the reciprocal**:
\[ \left(-\frac{65}{14}\right) \div \left(\frac{13}{7}\right) = \left(-\frac{65}{14}\right) \times \left(\frac{7}{13}\right) \]

2. **Multiply the fractions**:
\[ = \frac{-65 \times 7}{14 \times 13} = \frac{-455}{182} \]

3. **Simplify the fraction**:
\[ = \frac{-65}{26} = -\frac{5}{2} \quad \text{(dividing numerator and denominator by 91)} \]

### Summary of Results:
1. \( \frac{-16}{15} \)
2. \( \frac{128}{7} \)
3. \( \frac{2}{21} \)
4. \( \frac{1}{16} \)
5. \( \frac{224}{525} \)
6. \( -\frac{5}{2} \)