According to a survey, 7% of teenagers have not used a computer to play games, 11% have not used a...

1. Translate the problem requirements: We need to understand what each percentage represents - \(7\%\) haven't played games (so \(93\%\) have), \(11\%\) haven't written reports (so \(89\%\) have), and \(95\%\) have done at least one activity. We're looking for the percentage who have done both activities.
2. Convert negative statements to positive: Transform the "have not used" percentages into "have used" percentages to work with what people actually do rather than what they don't do.
3. Apply the inclusion-exclusion principle logically: Use the relationship that when we add two overlapping groups, we double-count the overlap, so we must subtract it to get the total who did at least one activity.
4. Solve for the overlap: Set up the equation where (games users) + (report writers) - (both users) = (at least one user) and solve for the "both users" percentage.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what each piece of information tells us about the teenagers in the survey.

We're told that:

• \(7\%\) have NOT used a computer to play games
• \(11\%\) have NOT used a computer to write reports
• \(95\%\) have used a computer for AT LEAST ONE of these purposes

We need to find what percent have used a computer for BOTH purposes.

Think of this like asking about students in a class: some play computer games, some write reports on computers, some do both, and a few do neither. We want to find the overlap - those who do both activities.

Process Skill: TRANSLATE - Converting the problem's language about what people haven't done into a clear picture of what we're looking for

2. Convert negative statements to positive

It's much easier to work with what people DO rather than what they DON'T do. Let's flip these percentages:

• If \(7\%\) have NOT played games, then \(100\% - 7\% = 93\%\) HAVE played games
• If \(11\%\) have NOT written reports, then \(100\% - 11\% = 89\%\) HAVE written reports
• \(95\%\) have done at least one activity (this stays the same)

Now we're working with:

• \(93\%\) play games
• \(89\%\) write reports
• \(95\%\) do at least one of these activities
• We want to find: percent who do BOTH

3. Apply the inclusion-exclusion principle logically

Here's the key insight: imagine we're counting teenagers twice - once for gaming and once for report writing.

If we simply add \(93\% + 89\% = 182\%\), we get more than \(100\%\)! This happens because we're double-counting the students who do BOTH activities.

To get the correct total of students who do "at least one" activity, we need to subtract out this double-counting:

(Students who game) + (Students who write reports) - (Students who do BOTH) = Students who do at least one

This makes intuitive sense: we add the two groups, then subtract the overlap to avoid counting anyone twice.

Process Skill: VISUALIZE - Understanding how overlapping groups work prevents us from making counting errors

4. Solve for the overlap

Now we can set up our equation using the numbers we know:

\(93\% + 89\% - \mathrm{(Both)} = 95\%\)

Solving for "Both":

\(182\% - \mathrm{(Both)} = 95\%\)

\(\mathrm{(Both)} = 182\% - 95\%\)

\(\mathrm{(Both)} = 87\%\)

Let's verify this makes sense: if \(87\%\) do both activities, then:

• All \(87\%\) are included in the \(93\%\) who play games ✓
• All \(87\%\) are included in the \(89\%\) who write reports ✓
• When we count \(93\% + 89\% - 87\% = 95\%\) who do at least one ✓

4. Final Answer

\(87\%\) of the teenagers have used a computer both to play games and to write reports.

This corresponds to answer choice D. \(87\%\).

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting "at least one" as "exactly one"
Students often confuse "\(95\%\) have used a computer for at least one of the above purposes" to mean "exactly one purpose." This leads them to think that people doing both activities are separate from this \(95\%\), when in reality, those doing both are included within the \(95\%\). This fundamental misunderstanding makes it impossible to set up the inclusion-exclusion principle correctly.

Faltering Point 2: Working directly with negative percentages instead of converting
Many students attempt to work directly with "\(7\%\) have NOT used" and "\(11\%\) have NOT used" without converting these to positive statements. This approach becomes confusing quickly and often leads to incorrect equation setup. The key insight is to convert these to "\(93\%\) HAVE used for games" and "\(89\%\) HAVE used for reports" to make the problem manageable.

Faltering Point 3: Not recognizing this as a sets/Venn diagram problem
Students may not immediately recognize that this is a classic overlapping sets problem requiring the inclusion-exclusion principle. Instead, they might try to solve it using other approaches like basic percentage calculations or ratios, missing the fundamental structure of the problem entirely.

Errors while executing the approach

Faltering Point 1: Setting up the inclusion-exclusion formula incorrectly
Even when students recognize they need inclusion-exclusion, they often write the formula incorrectly. Common mistakes include: \(|A \cup B| = |A| + |B| + |A \cap B|\) (adding instead of subtracting the intersection) or confusing which values represent which sets in the formula.

Faltering Point 2: Arithmetic errors in percentage calculations
Students may make simple calculation mistakes such as: \(100\% - 7\% = 92\%\) instead of \(93\%\), or \(93\% + 89\% = 181\%\) instead of \(182\%\). These small errors compound and lead to incorrect final answers even when the approach is right.

Errors while selecting the answer

Faltering Point 1: Selecting the complement of the correct answer
Once students calculate that \(87\%\) do both activities, they might mistakenly select the percentage who do NOT do both (\(100\% - 87\% = 13\%\)) and choose answer A instead of D. This happens when they lose track of what the question is actually asking for in their final step.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient total number of teenagers

Let's say there are \(100\) teenagers in the survey (this makes percentage calculations straightforward since each person represents \(1\%\)).

Step 2: Convert the given percentages to actual numbers

• \(7\%\) have not used a computer to play games → \(7\) teenagers haven't played games
• Therefore, \(100 - 7 = 93\) teenagers have played games
• \(11\%\) have not used a computer to write reports → \(11\) teenagers haven't written reports
• Therefore, \(100 - 11 = 89\) teenagers have written reports
• \(95\%\) have used a computer for at least one purpose → \(95\) teenagers have done at least one activity

Step 3: Apply the inclusion-exclusion principle with concrete numbers

Using the formula: (Games players) + (Report writers) - (Both) = (At least one)

Substituting our numbers: \(93 + 89 - \mathrm{(Both)} = 95\)

Step 4: Solve for the number who did both activities

\(182 - \mathrm{(Both)} = 95\)

\(\mathrm{(Both)} = 182 - 95 = 87\)

Step 5: Convert back to percentage

Since we started with \(100\) teenagers, \(87\) out of \(100\) means \(87\%\) have used a computer both to play games and to write reports.

Verification: This smart numbers approach works perfectly because we can logically choose \(100\) as our total (making percentages easy to work with), and all the given constraints result in whole numbers, confirming our choice was sound.