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.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are not sufficient

Hint:

When working with word problems involving sets, use the given conditions to form equations and solve for the unknown.

Answer 1

The Correct Option is C

Step 1: Analyze statement (1).
Statement (1) tells us that of the fiction books, there are 6 more that are not written in Spanish than are written in Spanish. Let \( x \) represent the number of fiction books written in Spanish. Then, the number of fiction books not written in Spanish is \( x + 6 \). Since there are 24 fiction books total, we can write the equation: \[ x + (x + 6) = 24 \quad \implies \quad 2x + 6 = 24 \quad \implies \quad 2x = 18 \quad \implies \quad x = 9 \] So, there are 9 fiction books written in Spanish.
Step 2: Analyze statement (2).
Statement (2) tells us that of the books written in Spanish, there are 5 more nonfiction books than fiction books. Let \( y \) represent the number of fiction books written in Spanish. Then, the number of nonfiction books written in Spanish is \( y + 5 \). Since there are 23 books written in Spanish, we can write the equation: \[ y + (y + 5) = 23 \quad \implies \quad 2y + 5 = 23 \quad \implies \quad 2y = 18 \quad \implies \quad y = 9 \] So, there are 9 fiction books written in Spanish.
Step 3: Combine both statements.
Combining both statements, we find that there are 9 fiction books written in Spanish.
\[ \boxed{9} \]