Answer:

36. To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.

So, \( \frac{4}{5} \div \frac{3}{10} = \frac{4}{5} \times \frac{10}{3} \).

Cancelling out common factors, we get \( \frac{4}{5} \times \frac{10}{3} = \frac{4 \times 10}{5 \times 3} \).

Simplifying further, we have \( \frac{40}{15} \).

To simplify fractions, we divide both the numerator and denominator by their greatest common divisor, which is 5 in this case.

Hence, \( \frac{40}{15} = \frac{8}{3} \).

Therefore, \( \frac{4}{5} \div \frac{3}{10} = \frac{8}{3} \).

The correct answer is \( \frac{8}{3} \).

37. To divide algebraic fractions, we use the same approach as in the previous question.

So, \( \frac{12 a^{2}}{18 b^{2}} \div \frac{4 a^{2}}{3 b} = \frac{12 a^{2}}{18 b^{2}} \times \frac{3 b}{4 a^{2}} \).

Cancelling out common factors, we get \( \frac{12 a^{2}}{18 b^{2}} \times \frac{3 b}{4 a^{2}} = \frac{12}{18} \times \frac{a^{2}}{b^{2}} \times \frac{3 b}{4} \).

Simplifying further, we have \( \frac{12}{18} \times \frac{3}{4} \times \frac{a^{2}}{a^{2}} \times \frac{b}{b} \).

This simplifies to \( \frac{2}{3} \times \frac{3}{4} \times \frac{b}{1} \).

Finally, \( \frac{2}{3} \times \frac{3}{4} \times b = \frac{2}{4} \times b \).

Simplifying, we have \( \frac{1}{2} \times b = \frac{b}{2} \).

Hence, \( \frac{12 a^{2}}{18 b^{2}} \div \frac{4 a^{2}}{3 b} = \frac{b}{2} \).

The correct answer is \( \frac{b}{2} \).

38. Similar to the previous questions, we will divide the algebraic fractions by multiplying by the reciprocal.

So, \( \frac{6 c^{2}}{(4 c)^{3}} \div \frac{90 c^{5}}{16 c^{6}} = \frac{6 c^{2}}{(4 c)^{3}} \times \frac{16 c^{6}}{90 c^{5}} \).

Cancelling out common factors, we get \( \frac{6}{64} \times \frac{c^{2}}{c^{3}} \times \frac{c^{6}}{c^{5}} \times \frac{16}{90} \).

Simplifying further, we have \( \frac{3}{32} \times \frac{c^{6 - 3}}{c^{5 - 3}} \times \frac{16}{10} \).

This simplifies to \( \frac{3}{32} \times \frac{c^{3}}{c^{2}} \times \frac{16}{10} \).

Finally, \( \frac{3}{32} \times \frac{c^{3}}{c^{2}} \times \frac{16}{10} = \frac{3}{32} \times \frac{16}{10} \times \frac{c^{3}}{c^{2}} \).

Simplifying, we have \( \frac{12}{20} \times c = \frac{3}{5} \times c \).

Hence, \( \frac{6 c^{2}}{(4 c)^{3}} \div \frac{90 c^{5}}{16 c^{6}} = \frac{3}{5} \times c \).

The correct answer is \( \frac{3}{5} \times c \).

39. For this question, we simply divide the numerical part of the fraction by the whole number.

So, \( \frac{20d^{2}}{4} \div 5d = \frac{20d^{2}}{4} \times \frac{1}{5d