Question

In a certain classroom, there are 80 books, of which 24 are fiction and 23 are written in Spanish. How many of the fiction books are written in Spanish?
(1) Of the fiction books, there are 6 more that are not written in Spanish than are written in Spanish.
(2) Of the books written in Spanish, there are 5 more nonfiction books than fiction books.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

For 2x2 overlapping sets problems, defining a single variable for the quantity you need to find (\(x\)) and then expressing the other quantities in the set in terms of \(x\) and the given totals is a very effective and systematic approach.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is an overlapping sets problem. We can organize the information in a 2x2 table.
Let F be Fiction, NF be Non-Fiction.
Let S be Spanish, NS be Not Spanish.
We are given:

Total Books = 80
Total Fiction (F) = 24
Total Spanish (S) = 23
The question asks for the number of books that are both Fiction and Spanish, which is the intersection of F and S. Let's call this number \(x\).
Step 2: Key Formula or Approach:
We can use variables to represent the quantities. Let \(x\) = number of Fiction books in Spanish. The total number of Fiction books is 24. So, the number of Fiction books not in Spanish is \(24 - x\). The total number of Spanish books is 23. So, the number of Non-Fiction books in Spanish is \(23 - x\).
Step 3: Detailed Explanation:
Analyze Statement (1): Of the fiction books, there are 6 more that are not written in Spanish than are written in Spanish.
This can be written as an equation: (Number of Fiction books not in Spanish) = (Number of Fiction books in Spanish) + 6
Using our variables: \[ 24 - x = x + 6 \] Now, we solve for \(x\): \[ 18 = 2x \] \[ x = 9 \] Since we found a unique value for \(x\), statement (1) is sufficient.
Analyze Statement (2): Of the books written in Spanish, there are 5 more nonfiction books than fiction books.
This can be written as an equation: (Number of Non-Fiction books in Spanish) = (Number of Fiction books in Spanish) + 5
Using our variables: \[ 23 - x = x + 5 \] Now, we solve for \(x\): \[ 18 = 2x \] \[ x = 9 \] Since we found a unique value for \(x\), statement (2) is sufficient.
Step 4: Final Answer:
Each statement alone is sufficient to determine the number of fiction books written in Spanish.