A bakery opened with its daily supply of $40$ dozen rolls. Half of the rolls were sold by noon and $60\%$ of the remaining rolls were sold between noon and closing time. How many dozen rolls were left unsold?

### Concept & Strategy This is a successive percentage problem. When a percentage of a quantity is removed, the next percentage calculation must be applied to the *remaining* amount, not the original total. $$ \text{Remaining Amount} = \text{Initial} \times (1 - \text{Fraction Sold}) $$ ### Step-by-Step Solution * **Given:** * Total initial supply = $40$ dozen. * Sold by noon = Half ($50\%$). * Sold after noon = $60\%$ of the *remaining* rolls. * **Calculation:** * First, find the number of rolls remaining at noon: * $40 \text{ dozen} \div 2 = 20 \text{ dozen}$ remaining. * Next, calculate how many were sold in the afternoon: * $60\%$ of $20 = 0.60 \times 20 = 12 \text{ dozen}$ sold. * Finally, subtract the afternoon sales from the noon remainder to find the unsold rolls: * $20 - 12 = 8 \text{ dozen}$ left unsold. ### Exam Strategy & Shortcut Instead of calculating how many were *sold*, calculate how many were *left* at each step to save time: If $50\%$ were sold, $50\%$ are left. If $60\%$ of the remainder were sold, then $40\%$ of the remainder are left. So, Unsold $= 40 \times 0.50 \times 0.40 = 40 \times 0.20 = 8$ dozen. This one-line chain multiplication is much faster and reduces subtraction errors. ### Common Pitfall The most frequent error is calculating the second percentage based on the original total. A student might find $50\%$ of $40$ ($20$) and $60\%$ of $40$ ($24$), then realise they have "sold" more than they had, or misapply the $60\%$. Always watch for the word "remaining". ### Final Answer **Therefore, the correct answer is 8.**