A millionaire bought a lot of hats $\frac{1}{4}$ of which were brown. The millionaire sold $\frac{2}{3}$ of the hats including $\frac{4}{5}$ of the brown hats. What fraction of the unsold hats were brown?

### Concept & Strategy The goal is to determine the ratio of remaining brown hats to the total remaining unsold hats. This can be modeled algebraically by tracking the fractions of populations, or more efficiently by assuming a total headcount that eliminates fraction denominators. ### Step-by-Step Solution * **Given:** * Initial brown hats = $\frac{1}{4}$ of total * Initial non-brown hats = $\frac{3}{4}$ of total * Total hats sold = $\frac{2}{3}$ of total $\implies$ Unsold hats = $1 - \frac{2}{3} = \frac{1}{3}$ of total * Brown hats sold = $\frac{4}{5}$ of initial brown hats * **Calculation:** * Let the total number of hats be $H$. * Total initial brown hats = $\frac{1}{4}H$ * Unsold brown hats = $\text{Total initial brown hats} \times \left(1 - \frac{4}{5}\right)$ $$ \text{Unsold brown hats} = \frac{1}{4}H \times \frac{1}{5} = \frac{1}{20}H $$ * Total unsold hats = $\frac{1}{3}H$ * Required fraction of unsold hats that are brown: $$ \text{Fraction} = \frac{\text{Unsold brown hats}}{\text{Total unsold hats}} = \frac{\frac{1}{20}H}{\frac{1}{3}H} = \frac{3}{20} $$ ### Exam Strategy & Shortcut **Plug-in Number Method:** Choose a total quantity of hats that is a common multiple of the denominators (4, 3, 5). Let the total number of hats be $60$. * Initial brown hats = $\frac{1}{4} \times 60 = 15$ * Total hats sold = $\frac{2}{3} \times 60 = 40 \implies$ Total unsold hats = $60 - 40 = 20$ * Brown hats sold = $\frac{4}{5} \times 15 = 12$ * Remaining unsold brown hats = $15 - 12 = 3$ * Fraction of unsold hats that are brown = $\frac{3}{20}$ ### Common Pitfall Students often calculate the fraction of unsold brown hats relative to the *total* initial hats ($\frac{1}{20}$) instead of the *unsold* hats ($\frac{3}{20}$). Always verify the final reference set specified by the text query. ### Final Answer **Therefore, the correct answer is 3/20.**