All 15 members of a foreign language club speak one or more of three languages-Spanish, French, and German. If 1/3...

1. Translate the problem requirements: Convert fractions to actual numbers of members speaking each language, and clarify that we need to find members who speak exactly two languages (not including those who speak all three)
2. Calculate individual language speakers: Determine how many members speak Spanish, French, and German individually using the given fractions
3. Apply inclusion-exclusion principle: Use the fact that all members speak at least one language to find the total number of overlapping memberships
4. Determine exactly two language speakers: Subtract those who speak all three languages from the total overlapping memberships to find those who speak exactly two

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know in simple terms. We have 15 club members, and each person speaks at least one of three languages: Spanish, French, or German.

First, let's convert the fractions to actual numbers:

• Spanish speakers: \(\frac{1}{3} \times 15 = 5\) members
• French speakers: \(\frac{2}{5} \times 15 = 6\) members
• German speakers: \(\frac{2}{3} \times 15 = 10\) members
• All three languages: 1 member

The key question asks for members who speak "exactly two" languages. This means people who speak two languages but NOT all three.

Process Skill: TRANSLATE - Converting fractions to concrete numbers makes the problem much more manageable

2. Calculate individual language speakers

Now we have our concrete numbers:

• 5 people speak Spanish
• 6 people speak French
• 10 people speak German
• 1 person speaks all three languages

If we simply add up all the language speakers (\(5 + 6 + 10 = 21\)), we get more than 15 people. This makes sense because some people are being counted multiple times - once for each language they speak.

3. Apply inclusion-exclusion principle

Here's the key insight: Since every member speaks at least one language, the total count of "language instances" minus the overlaps should equal 15.

Let's think about this step by step:

• Total language instances: \(5 + 6 + 10 = 21\)
• But we only have 15 people
• This means we've overcounted by \(21 - 15 = 6\)

This overcounting of 6 represents people who speak multiple languages. Specifically:

• People who speak exactly 2 languages get counted twice (so they contribute 1 extra count each)
• People who speak all 3 languages get counted three times (so they contribute 2 extra counts each)

Since 1 person speaks all three languages, that person contributes 2 to our overcount.
So the remaining overcount from people speaking exactly two languages is: \(6 - 2 = 4\)

Process Skill: INFER - The key insight is recognizing that overcounting directly tells us about overlaps

4. Determine exactly two language speakers

Since each person who speaks exactly two languages contributes 1 to the overcount, and we have an overcount of 4 from people speaking exactly two languages, there must be 4 people who speak exactly two languages.

Let's verify this makes sense:

• 1 person speaks all 3 languages
• 4 people speak exactly 2 languages
• This leaves \(15 - 1 - 4 = 10\) people who speak exactly 1 language

Check: \((10 \times 1) + (4 \times 2) + (1 \times 3) = 10 + 8 + 3 = 21\) total language instances ✓

5. Final Answer

The number of members who speak exactly two languages is 4.

The answer is E.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "exactly two languages"

Students often confuse "exactly two languages" with "at least two languages." The question asks specifically for members who speak two languages but NOT all three. This distinction is crucial because the person who speaks all three languages should not be included in the count of those speaking exactly two.

2. Attempting to use Venn diagram without proper setup

Many students jump straight into drawing a three-circle Venn diagram but struggle because they don't establish the relationship between overlaps systematically. Without recognizing that the inclusion-exclusion principle can be applied through overcounting analysis, students get lost trying to assign values to individual regions of the Venn diagram.

3. Not recognizing the overcounting insight

Students often miss the key insight that when total language instances (21) exceeds total people (15), this difference (6) directly represents the overcounting due to people speaking multiple languages. Instead, they may attempt more complex algebraic approaches that are harder to execute correctly.

Errors while executing the approach

1. Incorrect fraction-to-number conversion

Students may make arithmetic errors when converting fractions to actual numbers of people: 1/3 of 15 = 5, 2/5 of 15 = 6, and 2/3 of 15 = 10. Simple multiplication errors here will throw off the entire solution since all subsequent calculations depend on these values.

2. Misallocating the overcounting contribution

When students understand that people speaking all three languages contribute 2 to the overcount (since they're counted 3 times but should be counted once), they may incorrectly subtract this from the wrong part of their calculation, or forget to account for it entirely when determining how much overcount comes from people speaking exactly two languages.

3. Verification errors in the final check

Students may make arithmetic mistakes when verifying their answer by checking that \((10 \times 1) + (4 \times 2) + (1 \times 3) = 21\) total language instances. These verification errors can cause them to doubt their correct answer and change it unnecessarily.

Errors while selecting the answer

No likely faltering points - the question asks directly for the number of members speaking exactly two languages, and once students complete their calculations correctly, the final answer selection is straightforward.