A small School has three foreign language classes, one in French, one in Spanish, and one in German. How many...

Understanding the Question

The question asks us to find the exact number of students enrolled in both Spanish and French classes.

Given Information

• Small school with three language classes: French, Spanish, and German
• 34 students enrolled in Spanish class
• We need a specific number for students taking both Spanish AND French

What We Need to Determine

We need to find how many of the 34 Spanish students are also taking French. Since we need an exact value, a statement will be sufficient only if it allows us to calculate this specific number.

Analyzing Statement 1

Statement 1 tells us: 27 students are enrolled in French, and 49 students are enrolled in either French OR Spanish (or both).

Let's think about what this means. We have:

• French students: 27
• Spanish students: 34 (from the question)
• Students taking French OR Spanish: 49

Here's the key insight: If we simply added French + Spanish, we'd get \(27 + 34 = 61\) students. But we're told only 49 students take at least one of these languages.

Why the difference? When we add \(27 + 34\), we're counting some students twice—those taking BOTH languages. The difference (\(61 - 49 = 12\)) represents exactly these double-counted students.

Using the set relationship: \(|\mathrm{F} \cup \mathrm{S}| = |\mathrm{F}| + |\mathrm{S}| - |\mathrm{F} \cap \mathrm{S}|\)

• \(49 = 27 + 34 - |\mathrm{F} \cap \mathrm{S}|\)
• \(49 = 61 - |\mathrm{F} \cap \mathrm{S}|\)
• \(|\mathrm{F} \cap \mathrm{S}| = 12\)

Therefore, exactly 12 students are enrolled in both Spanish and French.

[STOP - Sufficient!]

Statement 1 is sufficient.

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: Half of the Spanish students are enrolled in more than one foreign language class.

Let's break this down:

• 34 students take Spanish (from the question)
• Therefore, 17 students (half of 34) take Spanish plus at least one other language

But here's the problem: "more than one language" could mean several things:

• Spanish + French only
• Spanish + German only
• Spanish + French + German

We have no way to determine how many of these 17 multilingual students specifically take French. Consider these possibilities:

• All 17 could take Spanish + French (making Spanish ∩ French = 17)
• All 17 could take Spanish + German (making Spanish ∩ French = 0)
• Any combination in between

Since we cannot determine the exact number taking both Spanish and French, Statement 2 is NOT sufficient.

This eliminates choices B and D.

The Answer: A

Only Statement 1 provides enough information to determine the exact number of students taking both Spanish and French (12 students).

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."