Of the students at a certain school, 40 took French, 30 took Latin, and 20 took Spanish. How many students...

Understanding the Question

We need to find how many students took exactly two of the three languages (French, Latin, Spanish).

Given Information

• \(40\) students took French
• \(30\) students took Latin
• \(20\) students took Spanish
• Total language enrollments: \(40 + 30 + 20 = 90\)

Key Insight: The Double-Counting Principle

Here's the crucial insight: when we add up the individual language totals (\(90\)), we're counting some students multiple times:

• Students taking \(1\) language → counted once
• Students taking \(2\) languages → counted twice (once in each language total)
• Students taking \(3\) languages → counted three times (once in each language total)

So the total of \(90\) represents not just students, but "student-language enrollments." This double and triple counting is the key to solving our problem.

What "Sufficient" Means Here

We need enough information to determine a unique value for the number of students taking exactly two languages. If we can narrow it down to one specific number, we have sufficiency.

Analyzing Statement 1

Statement 1 tells us: \(5\) students took all three languages.

These \(5\) students contribute \(15\) to our total count (since \(5 \times 3 = 15\)). This leaves us with \(90 - 15 = 75\) enrollments from students taking either one or two languages.

But here's the problem: we still don't know how to split those \(75\) enrollments between:

• Students taking just \(1\) language (each counted once)
• Students taking exactly \(2\) languages (each counted twice)

Let's test some scenarios to see why this matters:

• Scenario A: If \(47\) students took one language and \(14\) took two:
  Count = \(47(1) + 14(2) = 47 + 28 = 75\) ✓
• Scenario B: If \(19\) students took one language and \(28\) took two:
  Count = \(19(1) + 28(2) = 19 + 56 = 75\) ✓

Statement 1 is NOT sufficient. [STOP - Not Sufficient!]

This eliminates choices A and D.

Analyzing Statement 2

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

Statement 2 tells us: \(52\) students total took at least one language.

We know:

• Total unique students = \(52\)
• Total enrollments = \(90\)

But without knowing how many students took all three languages, we can't determine the breakdown. The \(90\) enrollments could be distributed many different ways among the \(52\) students.

For instance:

• If \(0\) students took all three: various combinations of one-language and two-language students could total \(52\) students and \(90\) enrollments
• If \(10\) students took all three: we'd need a completely different distribution to maintain both \(52\) total students and \(90\) enrollments

Without knowing the number taking all three languages, we can't determine how many took exactly two.

Statement 2 is NOT sufficient. [STOP - Not Sufficient!]

This eliminates choice B.

Combining Both Statements

Now we have two crucial pieces of information:

• \(5\) students took all three languages (Statement 1)
• \(52\) students total took at least one language (Statement 2)
• Total enrollments = \(90\) (given)

Let me set up our equations. If:

• \(\mathrm{x}\) = students taking exactly \(1\) language
• \(\mathrm{y}\) = students taking exactly \(2\) languages
• \(5\) = students taking all \(3\) languages (from Statement 1)

Then:

• Total students equation: \(\mathrm{x} + \mathrm{y} + 5 = 52\)
  Simplifying: \(\mathrm{x} + \mathrm{y} = 47\)
• Total enrollments equation: \(\mathrm{x}(1) + \mathrm{y}(2) + 5(3) = 90\)
  This means: \(\mathrm{x} + 2\mathrm{y} + 15 = 90\)
  Simplifying: \(\mathrm{x} + 2\mathrm{y} = 75\)

We now have a system of two equations with two unknowns:

1. \(\mathrm{x} + \mathrm{y} = 47\)
2. \(\mathrm{x} + 2\mathrm{y} = 75\)

Solving by subtracting equation 1 from equation 2:

• \((\mathrm{x} + 2\mathrm{y}) - (\mathrm{x} + \mathrm{y}) = 75 - 47\)
• \(\mathrm{y} = 28\)

Therefore, exactly \(28\) students took two languages. [STOP - Sufficient!]

Both statements together are sufficient.

This eliminates choice E.

The Answer: C

With both statements combined, we can determine that exactly \(28\) students took two of the three languages. Neither statement alone provides enough information.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."