Evaluate :

(i) $\dfrac{7}{12} ÷ \dfrac{-4}{3}$

(ii) $\dfrac{-12}{25} ÷ \dfrac{-5}{6}$

(iii) $\dfrac{-27}{32} ÷ \dfrac{-9}{16}$

(iv) $-2\dfrac{4}{7} ÷ \dfrac{6}{35}$

(v) $26 ÷ \dfrac{-1}{13}$

(vi) $\dfrac{1}{25} ÷ -5$

(i) $\dfrac{7}{12} ÷ \dfrac{-4}{3}$

We have:

$\dfrac{7}{12} \div \dfrac{-4}{3} \\[1em] = \dfrac{7}{12} \times \dfrac{3}{-4} \quad \left[\text{Reciprocal of } \dfrac{-4}{3} \text{ is } \dfrac{3}{-4}\right] \\[1em] =\dfrac{7}{12} \times \dfrac{3 \times (-1)}{-4 \times (-1)} \quad \text{[Making denominator positive]} \\[1em] = \dfrac{7}{12} \times \dfrac{-3}{4} \\[1em] = \dfrac{7 \times (-3)}{12 \times 4} \\[1em] = \dfrac{7 \times (-1)}{4 \times 4} \\[1em] = \dfrac{-7}{16}$

Hence, the answer is $\dfrac{-7}{16}$

(ii) $\dfrac{-12}{25} ÷ \dfrac{-5}{6}$

We have:

$\dfrac{-12}{25} \div \dfrac{-5}{6} \\[1em] = \dfrac{-12}{25} \times \dfrac{6}{-5} \quad \left[\text{Reciprocal of } \dfrac{-5}{6} \text{ is } \dfrac{6}{-5}\right] \\[1em] = \dfrac{-12}{25} \times \dfrac{6 \times (-1)}{-5 \times (-1)} \quad \text{[Making denominator positive]} \\[1em] = \dfrac{-12}{25} \times \dfrac{-6}{5} \\[1em] = \dfrac{(-12) \times (-6)}{25 \times 5} \\[1em] = \dfrac{72}{125}$

Hence, the answer is $\dfrac{72}{125}$

(iii) $\dfrac{-27}{32} ÷ \dfrac{-9}{16}$

We have:

$\dfrac{-27}{32} \div \dfrac{-9}{16} \\[1em] = \dfrac{-27}{32} \times \dfrac{16}{-9} \quad \left[\text{Reciprocal of } \dfrac{-9}{16} \text{ is } \dfrac{16}{-9}\right] \\[1em] = \dfrac{-27}{32} \times \dfrac{16 \times (-1)}{-9 \times (-1)} \quad \text{[Making denominator positive]} \\[1em] = \dfrac{-27}{32} \times \dfrac{-16}{9} \\[1em] = \dfrac{-3}{32} \times \dfrac{-16}{1} \\[1em] = \dfrac{-3}{2} \times \dfrac{-1}{1} \\[1em] = \dfrac{(-3) \times (-1)}{2 \times 1} \\[1em] = \dfrac{3}{2}$

Hence, the answer is $\dfrac{3}{2}$

(iv) $-2\dfrac{4}{7} ÷ \dfrac{6}{35}$

We have:

$\phantom{=} -2\dfrac{4}{7} ÷ \dfrac{6}{35} \\[1em] = -\dfrac{18}{7} ÷ \dfrac{6}{35} \\[1em] = \dfrac{-18}{7} \times \dfrac{35}{6} \quad \left[\text{Reciprocal of } \dfrac{6}{35} \text{ is } \dfrac{35}{6}\right] \\[1em] = \dfrac{-3}{7} \times \dfrac{35}{1} \\[1em] = \dfrac{-3}{1} \times \dfrac{5}{1} \\[1em] = \dfrac{-3 \times 5}{1 \times 1} \\[1em] = \dfrac{-15}{1} \\[1em] = -15$

Hence, the answer is -15

(v) $26 ÷ \dfrac{-1}{13}$

Express 26 as $\dfrac{26}{1}$

We have:

$\dfrac{26}{1} \div \dfrac{-1}{13} \\[1em] = \dfrac{26}{1} \times \dfrac{13}{-1} \quad \left[\text{Reciprocal of } \dfrac{-1}{13} \text{ is } \dfrac{13}{-1}\right] \\[1em] = \dfrac{26}{1} \times \dfrac{13 \times (-1)}{-1 \times (-1)} \quad \text{[Making denominator positive]} \\[1em] = \dfrac{26}{1} \times \dfrac{-13}{1} \\[1em] = \dfrac{26 \times (-13)}{1 \times 1} \\[1em] = \dfrac{-338}{1} \\[1em] = -338$

Hence, the answer is -338

(vi) $\dfrac{1}{25} ÷ -5$

Express -5 as $\dfrac{-5}{1}$

$\dfrac{1}{25} \div \dfrac{-5}{1} \\[1em] = \dfrac{1}{25} \times \dfrac{1}{-5} \quad \left[\text{Reciprocal of } \dfrac{-5}{1} \text{ is } \dfrac{1}{-5}\right] \\[1em] = \dfrac{1}{25} \times \dfrac{1 \times (-1)}{-5 \times (-1)} \quad \text{[Making denominator positive]} \\[1em] = \dfrac{1}{25} \times \dfrac{-1}{5} \\[1em] = \dfrac{1 \times (-1)}{25 \times (5)} \\[1em] = \dfrac{-1}{125} \\[1em]$

Hence, the answer is $\dfrac{-1}{125}$