1. The problem asks us to divide fractions and express the answers in lowest terms. 2. Recall the rule for dividing fractions: \n\n$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$\n\nThis means we multiply by the reciprocal of the divisor. 3. Let's solve each part step-by-step: (a) $4 \div \frac{1}{5}$ Rewrite 4 as $\frac{4}{1}$: $$\frac{4}{1} \div \frac{1}{5} = \frac{4}{1} \times \frac{5}{1} = \frac{4 \times 5}{1 \times 1} = \frac{20}{1} = 20$$ (b) $4 \div \frac{1}{4}$ Rewrite 4 as $\frac{4}{1}$: $$\frac{4}{1} \div \frac{1}{4} = \frac{4}{1} \times \frac{4}{1} = \frac{4 \times 4}{1 \times 1} = \frac{16}{1} = 16$$ (c) $\frac{1}{3} \div 5$ Rewrite 5 as $\frac{5}{1}$: $$\frac{1}{3} \div \frac{5}{1} = \frac{1}{3} \times \frac{1}{5} = \frac{1 \times 1}{3 \times 5} = \frac{1}{15}$$ (d) $\frac{1}{4} \div 8$ Rewrite 8 as $\frac{8}{1}$: $$\frac{1}{4} \div \frac{8}{1} = \frac{1}{4} \times \frac{1}{8} = \frac{1 \times 1}{4 \times 8} = \frac{1}{32}$$ 4. Final answers in lowest terms: (a) 20 (b) 16 (c) $\frac{1}{15}$ (d) $\frac{1}{32}$