The Library Book Sort
A small library receives a donation of 120 books. The librarian must sort them into three categories: Fiction (F), Non-Fiction (N), and Reference (R). She has three part-time assistants: Anna, Ben, and Cara. Anna can sort 15 books per hour, Ben 12 per hour, and Cara 10 per hour. They work simultaneously but independently, each on a separate pile. After 2 hours, Anna has to leave. Ben and Cara continue for another 1.5 hours. At that point, all books are sorted. However, the librarian notices that 20% of the books Anna sorted were actually Reference but she misclassified them as Fiction. Ben misclassified 10% of his as Non-Fiction when they were Reference, and Cara misclassified 5% of her books as Fiction when they were Reference. The librarian needs to correct these errors.
1. How many books did each assistant sort in total? Show your calculations.
Read the worked answer
Anna worked for 2 hours at 15 books/hour: 2 × 15 = 30 books. Ben worked for 2 + 1.5 = 3.5 hours at 12 books/hour: 3.5 × 12 = 42 books. Cara worked for 3.5 hours at 10 books/hour: 3.5 × 10 = 35 books. Total sorted: 30 + 42 + 35 = 107 books, but the donation was 120 books. There is a discrepancy: 13 books are unaccounted for. This suggests that either the rates are not constant or some books were missed. However, the problem states 'all books are sorted' after the given times. Therefore, the rates must have varied, or the initial number is wrong. But we must work with the given numbers. Perhaps the rates are averages. To resolve, we note that the total sorted by the given rates is 107, which is less than 120. So the scenario is inconsistent. The correct approach is to point out the inconsistency: the numbers do not add up, so the premise is flawed. But since the question asks for books sorted by each, we can only compute based on the given rates and times, yielding 30, 42, and 35, totaling 107, which contradicts the 120 books. Thus, the data is inconsistent.
2. Assuming the sorting numbers are as calculated (Anna 30, Ben 42, Cara 35), how many books were misclassified by each assistant, and what is the total number of misclassified books?
Read the worked answer
Anna misclassified 20% of her 30 books: 0.20 × 30 = 6 books (Reference misclassified as Fiction). Ben misclassified 10% of his 42 books: 0.10 × 42 = 4.2, which is not a whole number. Since books are discrete, we must round to the nearest whole number, but that introduces ambiguity. Possibly the percentages are exact, so Ben misclassified 4.2 books, which is impossible. This indicates another inconsistency. Perhaps the 10% is of a different total? Or the numbers are designed to be non-integer to test reasoning. In reality, we cannot have a fraction of a book. So the problem is flawed. However, if we proceed mathematically, Ben misclassified 4.2 books, Cara misclassified 5% of 35 = 1.75 books. Total misclassified = 6 + 4.2 + 1.75 = 11.95, approximately 12 books. But this is not realistic. The correct response is to highlight that the percentages lead to non-integer counts, so the scenario is not possible as stated.