In a class of 30 students, 18 students have used electronic mail to submit their homework and 9 students have...

1. Translate the problem requirements: We have 30 total students divided into groups based on homework submission methods. We need to find how many students used BOTH electronic mail AND faxes, given the individual usage numbers and those who used neither method.
2. Identify the given information systematically: Extract all the numerical data about student groups - total students, email users, fax users, and students using neither method.
3. Apply the inclusion-exclusion principle logically: Use the fundamental relationship that \(\mathrm{Total} = \mathrm{Email\ users} + \mathrm{Fax\ users} - \mathrm{Both} + \mathrm{Neither\ users}\), recognizing that we subtract 'Both' to avoid double-counting students who appear in both groups.
4. Solve for the overlap: Rearrange the equation to isolate 'Both' and calculate the final answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this problem is asking us to find. We have a class of students who submitted homework using different methods - some used email, some used fax, some used both methods, and some used neither method.

Think of it like this: imagine you're looking at a classroom and asking "Who used what method to submit homework?" We need to find how many students are in the overlap - those who used BOTH email AND fax.

What we're looking for: The number of students who used both electronic mail and faxes to submit their homework.

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical question about overlapping groups

2. Identify the given information systematically

Let's carefully extract all the numbers from the problem:

• Total students in class = 30
• Students who used electronic mail = 18
• Students who used faxes = 9
• Students who used neither method = 8
• Students who used both methods = ? (this is what we need to find)

Notice that some students might appear in multiple categories. For example, a student who used both email and fax would be counted in both the "email users" group AND the "fax users" group.

3. Apply the inclusion-exclusion principle logically

Here's the key insight: when we add up email users (18) and fax users (9), we get 27. But this counts students who used both methods twice - once in each group.

Let's think about it step by step:
• If we have 30 total students
• And 8 students used neither method
• Then \(30 - 8 = 22\) students used at least one method (email OR fax OR both)

Now, when we count email users (18) plus fax users (9), we get \(18 + 9 = 27\). But we know only 22 students actually used at least one method. This means we've double-counted some students.

The number of students we double-counted = \(27 - 22 = 5\)

These 5 students are exactly the ones who used BOTH methods (they got counted once in the email group and once in the fax group).

Mathematically: \(\mathrm{Total} = \mathrm{Email} + \mathrm{Fax} - \mathrm{Both} + \mathrm{Neither}\)
\(30 = 18 + 9 - \mathrm{Both} + 8\)

4. Solve for the overlap

Now let's solve our equation:
\(30 = 18 + 9 - \mathrm{Both} + 8\)
\(30 = 27 - \mathrm{Both} + 8\)
\(30 = 35 - \mathrm{Both}\)
\(\mathrm{Both} = 35 - 30\)
\(\mathrm{Both} = 5\)

Let's verify this makes sense:
• Students using only email: \(18 - 5 = 13\)
• Students using only fax: \(9 - 5 = 4\)
• Students using both: 5
• Students using neither: 8
• Total: \(13 + 4 + 5 + 8 = 30\) ✓

5. Final Answer

The number of students who used both electronic mail and faxes to submit their homework is 5.

This matches answer choice C.

The answer is C.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding overlapping categories

Students often fail to recognize that this is a Venn diagram problem involving overlapping sets. They may think that students who used email (18) and students who used fax (9) are completely separate groups, not realizing that some students could have used both methods. This leads them to incorrectly assume that \(18 + 9 = 27\) different students used these methods.

2. Confusing "both" with "either"

Students may misinterpret what the question is asking for. The question asks for students who used "both electronic mail AND faxes" but students might think it's asking for students who used "either electronic mail OR faxes." This fundamental misunderstanding leads to setting up the wrong equation entirely.

3. Forgetting to account for the "neither" group

Students may focus only on the email and fax numbers (18 and 9) and forget that 8 students used neither method. They might try to solve the problem without considering that these 8 students must be subtracted from the total to find how many actually used at least one method.

Errors while executing the approach

1. Setting up the inclusion-exclusion formula incorrectly

Even if students recognize this as a Venn diagram problem, they may write the inclusion-exclusion formula incorrectly. Common mistakes include writing \(30 = 18 + 9 + \mathrm{Both} + 8\) (adding "Both" instead of subtracting it) or forgetting to include the "neither" group in the equation altogether.

2. Arithmetic errors in solving the equation

Students may set up the correct equation (\(30 = 18 + 9 - \mathrm{Both} + 8\)) but make calculation errors. For example, they might incorrectly add \(18 + 9 + 8 = 34\) instead of 35, or make sign errors when isolating the "Both" variable, leading to answers like 3 or 7 instead of 5.

Errors while selecting the answer

No likely faltering points - once students correctly execute the calculation and get Both = 5, this directly corresponds to answer choice C with no additional interpretation needed.