A college admissions officer predicts that 20 percent of the students who are accepted will not attend the college. According...

1. Translate the problem requirements: We need to find how many students to accept when \(20\%\) won't attend, so that we end up with exactly \(\mathrm{x}\) students enrolled
2. Establish the attendance relationship: If \(20\%\) don't attend, then \(80\%\) do attend, so we need to work with the \(80\%\) who will actually enroll
3. Set up the enrollment equation: The number we accept × \(80\%\) = our target enrollment of \(\mathrm{x}\) students
4. Solve for acceptances needed: Divide our target by the attendance rate to find how many we should accept

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in plain English. The college wants exactly \(\mathrm{x}\) students to actually show up and enroll. However, they know from experience that not everyone who gets accepted will actually attend - specifically, \(20\%\) of accepted students won't come.

So the question is asking: if we want \(\mathrm{x}\) students to actually enroll, and we know \(20\%\) of our accepted students won't show up, how many students should we accept in the first place?

Think of it like planning a dinner party. If you want \(10\) people to actually come, but you know \(2\) out of every \(10\) invited guests usually don't show up, you'd need to invite more than \(10\) people to end up with exactly \(10\) attendees.

Process Skill: TRANSLATE - Converting the problem scenario into a clear mathematical relationship

2. Establish the attendance relationship

Now let's think about what "\(20\%\) won't attend" really means for our planning.

If \(20\%\) of accepted students don't attend, that means \(80\%\) DO attend. In other words:

• Out of every \(100\) students we accept, only \(80\) will actually enroll
• The attendance rate is \(80\%\) or \(0.8\)

This is the key insight: we need to work with the \(80\%\) who will actually show up, not the \(20\%\) who won't.

3. Set up the enrollment equation

Let's call the number of students we should accept "\(\mathrm{A}\)" (for Accepted).

In plain English: The number we accept × the percentage who attend = our target enrollment

So: \(\mathrm{A} \times 80\% = \mathrm{x}\) students
Or: \(\mathrm{A} \times 0.8 = \mathrm{x}\)

This equation captures exactly what we need: if we accept \(\mathrm{A}\) students, and \(80\%\) of them attend, we'll end up with exactly \(\mathrm{x}\) enrolled students.

4. Solve for acceptances needed

Now we solve for \(\mathrm{A}\) (the number to accept):

Starting with: \(\mathrm{A} \times 0.8 = \mathrm{x}\)

To find \(\mathrm{A}\), we divide both sides by \(0.8\):
\(\mathrm{A} = \mathrm{x} \div 0.8\)

Let's simplify this division:
\(\mathrm{A} = \mathrm{x} \div 0.8 = \mathrm{x} \div \frac{8}{10} = \mathrm{x} \times \frac{10}{8} = \mathrm{x} \times \frac{5}{4} = 1.25\mathrm{x}\)

So we need to accept \(1.25\mathrm{x}\) students.

Final Answer

We should accept \(1.25\mathrm{x}\) students to achieve a planned enrollment of \(\mathrm{x}\) students.

Let's verify this makes sense: If we accept \(1.25\mathrm{x}\) students and \(80\%\) attend, then:
\(1.25\mathrm{x} \times 0.8 = \mathrm{x}\) students will enroll ✓

The answer is D. \(1.25\mathrm{x}\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the percentage relationship

Students often focus on the "\(20\%\) won't attend" and try to work directly with this percentage instead of recognizing they need to work with the \(80\%\) who WILL attend. They might think: "If \(20\%\) don't come, I need to accept \(\mathrm{x} + 20\%\) more" leading to incorrect setups like \(\mathrm{A} = \mathrm{x} + 0.2\mathrm{x} = 1.2\mathrm{x}\).

2. Setting up the equation backwards

Students may confuse what they're solving for and set up the relationship as: "\(\mathrm{x}\) students accepted × attendance rate = number who attend" instead of "\(\mathrm{A}\) students accepted × attendance rate = \(\mathrm{x}\) target enrollment." This leads them to solve for the wrong variable.

3. Misunderstanding the constraint

Students might not clearly identify that \(\mathrm{x}\) represents the final enrollment target (the constraint they must meet exactly), instead treating it as just a reference number. This confusion makes them unsure whether they're solving for total acceptances or something else.

Errors while executing the approach

1. Arithmetic errors when converting percentages

Students often make mistakes when converting \(20\%\) to \(80\%\), or when working with \(0.8\). Common errors include using \(0.2\) instead of \(0.8\) in their calculations, or incorrectly converting between decimal and fraction forms.

2. Division errors with decimals

When dividing by \(0.8\), students frequently make computational mistakes. The step \(\mathrm{A} = \mathrm{x} \div 0.8 = \mathrm{x} \times \frac{5}{4} = 1.25\mathrm{x}\) involves multiple conversions that can trip up students, especially the fraction manipulation \(\mathrm{x} \div \frac{8}{10} = \mathrm{x} \times \frac{10}{8}\).

Errors while selecting the answer

No likely faltering points - once students correctly calculate \(1.25\mathrm{x}\), the answer choice D clearly matches their result.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient target enrollment
Let's say we want \(\mathrm{x} = 100\) students to actually enroll at the college. This gives us a concrete number to work with.

Step 2: Understand the attendance pattern
We know that \(20\%\) of accepted students will not attend, which means \(80\%\) of accepted students will attend.

Step 3: Set up the relationship
If we accept some number of students, and \(80\%\) of them attend, we need that \(80\%\) to equal our target of \(100\) students.
Let \(\mathrm{A}\) = number of students we should accept
Then: \(80\%\) of \(\mathrm{A} = 100\)
\(0.8\mathrm{A} = 100\)

Step 4: Solve for acceptances needed
\(\mathrm{A} = 100 \div 0.8 = 125\)

Step 5: Express as a multiple of target enrollment
We need to accept \(125\) students to get \(100\) enrolled students.
\(125 = 1.25 \times 100\)
So we need to accept \(1.25\mathrm{x}\) students to achieve enrollment of \(\mathrm{x}\) students.

Step 6: Verify our answer
If we accept \(125\) students and \(20\%\) don't attend:
Students who don't attend: \(20\% \times 125 = 25\)
Students who do attend: \(125 - 25 = 100\) ✓

The answer is D. \(1.25\mathrm{x}\)