Loading...
Among the \(\mathrm{N}\) books in Estelle's library, 40 are fiction and the rest are nonfiction. Fifty of the \(\mathrm{N}\) books are hardcovers and the rest are paperbacks. If 30 of the \(\mathrm{N}\) books are hardcover fiction books, what is the least possible value of \(\mathrm{N}\)?
Let's start by understanding what we have in plain English. Estelle has N books total in her library. Think of this like organizing books on shelves where each book can be categorized in two different ways:
Here's the key piece of information: 30 books are both hardcover and fiction. This is like finding books that belong to both categories at the same time.
We need to find the smallest possible value of N that makes all of this work logically.
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationshipsNow let's think about what constraints (rules) must be true for any valid library:
Constraint 1: Since 30 books are hardcover fiction, and we only have 40 fiction books total, this means at most 40 books can be hardcover fiction. Since \(30 \leq 40\), this constraint is satisfied.
Constraint 2: Since 30 books are hardcover fiction, and we only have 50 hardcover books total, this means at most 50 books can be hardcover fiction. Since \(30 \leq 50\), this constraint is also satisfied.
Constraint 3: Here's the key insight - we need enough total books to accommodate all categories. Since we have 40 fiction books and \((\mathrm{N}-40)\) nonfiction books, plus 50 hardcover books and \((\mathrm{N}-50)\) paperback books, with 30 books being hardcover fiction, we need to ensure there are enough books for the remaining categories.
The remaining hardcover books that are nonfiction: \(50 - 30 = 20\) books
The remaining fiction books that are paperback: \(40 - 30 = 10\) books
Process Skill: INFER - Drawing non-obvious conclusions about the relationships between categoriesTo minimize N, we want to minimize the number of books that are paperback nonfiction. Let's think about this step by step:
We know we must have:
Let's call the number of paperback nonfiction books "x". For the total to work out:
Total nonfiction books = hardcover nonfiction + paperback nonfiction
\((\mathrm{N} - 40) = 20 + \mathrm{x}\)
Also, total paperback books = paperback fiction + paperback nonfiction
\((\mathrm{N} - 50) = 10 + \mathrm{x}\)
For N to be minimized, x should be minimized. Since x represents a number of books, the smallest value x can take is 0.
If x = 0 (meaning no paperback nonfiction books), then:
From \((\mathrm{N} - 50) = 10 + \mathrm{x}\):
\(\mathrm{N} - 50 = 10 + 0 = 10\)
Therefore: \(\mathrm{N} = 60\)
Let's verify this works with our other equation:
From \((\mathrm{N} - 40) = 20 + \mathrm{x}\):
\(60 - 40 = 20 + 0\)
\(20 = 20\) ✓
Let's double-check our solution by organizing all 60 books:
Total: \(30 + 20 + 10 + 0 = 60\) books ✓
Fiction check: \(30 + 10 = 40\) fiction books ✓
Hardcover check: \(30 + 20 = 50\) hardcover books ✓
Process Skill: APPLY CONSTRAINTS - Using all given conditions to verify our solutionThe least possible value of N is 60.
Looking at our answer choices:
Our answer matches choice B.
Students often struggle to visualize that books can belong to TWO categories simultaneously (both hardcover AND fiction). They might think of the 30 hardcover fiction books as separate from the 40 fiction books or the 50 hardcover books, rather than understanding that these 30 books are counted within both the fiction total and the hardcover total. This leads to incorrect setup where they might add 30 + 40 + 50, double-counting books.
Many students fail to recognize that with two binary classifications (fiction/nonfiction and hardcover/paperback), there must be exactly four distinct categories: hardcover fiction, hardcover nonfiction, paperback fiction, and paperback nonfiction. Instead, they might try to work with just the given numbers (40, 50, 30) without systematically organizing all possible book types.
Students often focus on just finding ANY valid value of N that satisfies the constraints, rather than understanding that the question specifically asks for the LEAST possible value. They might stop after finding one solution without exploring whether N could be smaller.
When setting up equations like \((\mathrm{N}-40) = 20 + \mathrm{x}\) and \((\mathrm{N}-50) = 10 + \mathrm{x}\), students frequently make basic arithmetic mistakes. For example, they might incorrectly calculate that if there are 50 hardcovers total and 30 are fiction, then 30 (instead of 20) must be nonfiction hardcovers, leading to wrong values throughout their solution.
Students might arrive at negative values for one of the four categories (especially paperback nonfiction) and not recognize this as impossible. They may continue with their calculations even when getting results like "paperback nonfiction = -10 books" instead of understanding this indicates their approach or arithmetic is wrong.
Students may calculate \(\mathrm{N} = 60\) but fail to double-check that this value actually satisfies all three original conditions: 40 fiction books, 50 hardcover books, and 30 hardcover fiction books. Without this verification step, they might select an incorrect answer choice even if their arithmetic was initially correct.