Each book in a certain library is classified in one of three categories: fiction, nonfiction, or biography. The ratio of...

1. Translate the problem requirements: We have three categories of books with given ratios between them, and we need the smallest total number of books that's at least 10,000.
2. Express all categories in terms of a common unit: Use the given ratios to find how many books of each type exist for every basic unit.
3. Find the total books per unit pattern: Add up all three categories to see how many total books exist for each basic unit.
4. Scale up to meet the minimum requirement: Determine how many units we need to have at least 10,000 books total.
5. Calculate the exact minimum: Multiply the pattern by the required number of units to get the smallest possible total.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have:

• Three types of books: fiction, nonfiction, and biography
• Fiction to nonfiction ratio is \(4:3\) (for every 4 fiction books, there are 3 nonfiction books)
• Nonfiction to biography ratio is \(3:2\) (for every 3 nonfiction books, there are 2 biography books)
• We need at least 10,000 total books
• We want the smallest possible number that satisfies all these conditions

Process Skill: TRANSLATE - Converting the ratio relationships into a mathematical understanding we can work with

2. Express all categories in terms of a common unit

Here's the key insight: we need to find a common way to express all three types of books. Notice that nonfiction appears in both ratios, so we can use it as our connecting link.

From the ratios:

• Fiction : Nonfiction = \(4 : 3\)
• Nonfiction : Biography = \(3 : 2\)

Let's say we have 3 units of nonfiction books. Then:

• Fiction books = 4 units (since \(4:3\) ratio means 4 fiction for every 3 nonfiction)
• Nonfiction books = 3 units
• Biography books = 2 units (since \(3:2\) ratio means 2 biography for every 3 nonfiction)

So our basic pattern is: Fiction : Nonfiction : Biography = \(4 : 3 : 2\)

3. Find the total books per unit pattern

Now let's add up our basic pattern:

Total books in one complete pattern = \(4 + 3 + 2 = 9\) books

This means that no matter how many books we have, the total must be a multiple of 9 to maintain our ratios perfectly.

Process Skill: INFER - Recognizing that the total must be a multiple of 9 is a crucial non-obvious conclusion

4. Scale up to meet the minimum requirement

We need at least 10,000 books, and our total must be a multiple of 9.

Let's find the smallest multiple of 9 that is at least 10,000:

\(10,000 \div 9 = 1,111.11...\)

Since we need a whole number of patterns, we need at least 1,112 complete patterns.

Let's verify: \(1,112 \times 9 = 10,008\)

Let's also check: \(1,111 \times 9 = 9,999\) (this is less than 10,000, so it doesn't work)

5. Calculate the exact minimum

The smallest number of books that satisfies all our conditions is:

\(1,112 \text{ patterns} \times 9 \text{ books per pattern} = 10,008 \text{ books}\)

Let's verify this works:

• Fiction books: \(1,112 \times 4 = 4,448\)
• Nonfiction books: \(1,112 \times 3 = 3,336\)
• Biography books: \(1,112 \times 2 = 2,224\)
• Total: \(4,448 + 3,336 + 2,224 = 10,008\) ✓

Checking our ratios:

• Fiction to Nonfiction: \(4,448 : 3,336 = 4 : 3\) ✓
• Nonfiction to Biography: \(3,336 : 2,224 = 3 : 2\) ✓

Final Answer

The least number of books that could be in the library is 10,008.

This corresponds to answer choice D. 10,008.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting the ratio relationships

Students often struggle to correctly link the two given ratios. They might treat "Fiction:Nonfiction = \(4:3\)" and "Nonfiction:Biography = \(3:2\)" as separate, unconnected ratios instead of recognizing that nonfiction is the common element that allows them to create a unified ratio of Fiction:Nonfiction:Biography = \(4:3:2\).

Faltering Point 2: Missing the constraint that total books must be a whole number multiple

Students may understand the ratios but fail to recognize that since we need whole numbers of books in each category, the total number of books must be a multiple of the sum of the ratio parts \((4+3+2=9)\). This leads them to simply round 10,000 up to the nearest convenient number rather than finding the smallest multiple of 9 that exceeds 10,000.

Faltering Point 3: Incorrectly handling the "at least 10,000" constraint

Some students might interpret "at least 10,000 books" to mean exactly 10,000 books, missing that we need to find the smallest number ≥ 10,000 that satisfies all ratio constraints. This leads them to immediately select answer choice A (10,000) without checking if this number actually works with the given ratios.

Errors while executing the approach

Faltering Point 1: Arithmetic errors when finding the multiple of 9

Students might make calculation errors when dividing 10,000 by 9 (getting the wrong quotient) or when multiplying back to verify their answer. For example, they might calculate \(10,000 \div 9 = 1,111\) (forgetting about the remainder) and then compute \(1,111 \times 9 = 9,999\), not realizing they need the next multiple.

Faltering Point 2: Confusion about rounding up vs. rounding down

When students get \(10,000 \div 9 = 1,111.11...\), they might round down to 1,111 patterns instead of understanding that they need to round up to 1,112 patterns to satisfy the "at least 10,000" requirement. This leads them to calculate \(1,111 \times 9 = 9,999\), which doesn't meet the minimum requirement.

Errors while selecting the answer

Faltering Point 1: Selecting a close but incorrect answer choice

After correctly calculating that they need 10,008 books, students might second-guess themselves and select a nearby answer choice like 10,005 (choice C) or 10,009 (choice E), thinking their calculation might be slightly off rather than trusting their systematic approach and verification.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose smart numbers based on the ratio relationships

Given ratios:

• Fiction : Nonfiction = \(4 : 3\)
• Nonfiction : Biography = \(3 : 2\)

Since nonfiction appears in both ratios with the value 3, we can use 3 as our smart number for nonfiction books. This allows us to directly connect both ratios.

Let's set: Nonfiction = 3 books

Step 2: Calculate other categories using the smart number

From Fiction : Nonfiction = \(4 : 3\), and we chose Nonfiction = 3:

Fiction = 4 books

From Nonfiction : Biography = \(3 : 2\), and we chose Nonfiction = 3:

Biography = 2 books

Step 3: Find the basic unit pattern

Total books in our basic unit = Fiction + Nonfiction + Biography = \(4 + 3 + 2 = 9\) books

This means the library must have books in multiples of 9 to maintain the given ratios.

Step 4: Scale up to meet the minimum requirement

We need at least 10,000 books, so we need to find the smallest multiple of 9 that is ≥ 10,000.

\(10,000 \div 9 = 1,111.11...\)

Since we need a whole number of units, we need at least 1,112 units.

Step 5: Calculate the minimum number of books

Minimum books = \(1,112 \times 9 = 10,008\)

Therefore, the least number of books that could be in the library is 10,008.