1. The problem asks for the quotient of $\frac{1}{2}$ divided by $\frac{2}{5}$. We need to find $\frac{1}{2} \div \frac{2}{5}$.\n\n2. Recall the rule for dividing fractions: \n$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$\nThis means we multiply the first fraction by the reciprocal of the second fraction.\n\n3. Applying the rule: \n$$\frac{1}{2} \div \frac{2}{5} = \frac{1}{2} \times \frac{5}{2}$$\n\n4. Multiply the numerators and denominators: \n$$\frac{1 \times 5}{2 \times 2} = \frac{5}{4}$$\n\n5. The fraction $\frac{5}{4}$ is an improper fraction and is already in simplest form because 5 and 4 have no common factors other than 1.\n\nFinal answer: $$\frac{5}{4}$$