Candidate McFe received 12,000 votes, which was 1/4 of the total number of votes. If x additional people had voted...

1. Translate the problem requirements: McFee got 12,000 votes which was \(\frac{1}{4}\) of total votes. If x more people voted for McFee, then McFee would have \(\frac{1}{3}\) of the new total. We need to find x.
2. Find the original total votes: Use the fact that McFee's 12,000 votes represented \(\frac{1}{4}\) of all votes to determine the original total.
3. Set up the new scenario equation: After x additional votes, McFee has (12,000 + x) votes out of a new total, and this equals \(\frac{1}{3}\) of the new total votes.
4. Solve for x using the constraint: Use the equation that McFee's new vote count equals \(\frac{1}{3}\) of the new total to find the value of x.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in this voting scenario in plain English. McFee received 12,000 votes, and we're told this represents exactly \(\frac{1}{4}\) of all the votes cast. Think of it like a pie chart - McFee's slice is \(\frac{1}{4}\) of the whole pie.

Then we have a hypothetical situation: if x additional people had voted and all of them voted for McFee, then McFee's share would have been \(\frac{1}{3}\) of the total votes (including those x new votes).

We need to find the value of x - how many additional votes would create this scenario.

Process Skill: TRANSLATE - Converting the voting scenario into mathematical relationships

2. Find the original total votes

Since McFee got 12,000 votes and this was \(\frac{1}{4}\) of the total, we can think about this simply: if 12,000 is one-fourth, then the whole amount is four times as much.

Original total votes = \(4 × 12,000 = 48,000\) votes

Let's verify this makes sense: \(12,000 ÷ 48,000 = \frac{1}{4}\) ✓

3. Set up the new scenario equation

Now let's think through the hypothetical scenario. If x additional people vote (all for McFee), then:

- McFee's new vote count = 12,000 + x
- Total new vote count = 48,000 + x (original votes plus the x new votes)

We're told that in this scenario, McFee would have \(\frac{1}{3}\) of the total votes. So McFee's votes would equal one-third of the new total.

In mathematical terms: \((12,000 + x) = \frac{1}{3} × (48,000 + x)\)

4. Solve for x using the constraint

Let's solve this step by step, keeping the arithmetic simple:

\((12,000 + x) = \frac{1}{3} × (48,000 + x)\)

Multiply both sides by 3 to eliminate the fraction:
\(3(12,000 + x) = 48,000 + x\)
\(36,000 + 3x = 48,000 + x\)

Subtract x from both sides:
\(36,000 + 2x = 48,000\)

Subtract 36,000 from both sides:
\(2x = 12,000\)

Divide by 2:
\(x = 6,000\)

Let's verify: If 6,000 additional people vote for McFee:

- McFee's new total: 12,000 + 6,000 = 18,000 votes
- New total votes: 48,000 + 6,000 = 54,000 votes
- McFee's fraction: \(\frac{18,000}{54,000} = \frac{1}{3}\) ✓

4. Final Answer

The value of x is 6,000.

This matches answer choice B.

Common Faltering Points

Errors while devising the approach

Students often misunderstand what happens in the hypothetical scenario. They might think that McFee's original 12,000 votes stay the same and x additional people vote for other candidates, rather than understanding that all x additional people vote specifically for McFee. This leads to setting up the wrong equation where McFee still has only 12,000 votes instead of (12,000 + x) votes.

Students may set up the equation using the original total (48,000) instead of the new total (48,000 + x). They might write: \((12,000 + x) = \frac{1}{3} × 48,000\), forgetting that when x additional people vote, the total number of votes also increases by x.

Some students might set up a proportion incorrectly, thinking that if McFee goes from \(\frac{1}{4}\) to \(\frac{1}{3}\) of votes, they can simply use the difference between these fractions \((\frac{1}{3} - \frac{1}{4})\) to solve for x, rather than setting up the proper equation with the new vote totals.

Errors while executing the approach

When solving \((12,000 + x) = \frac{1}{3} × (48,000 + x)\), students often make mistakes when multiplying both sides by 3. They might incorrectly get \(3(12,000) + x = 48,000 + x\) instead of \(3(12,000) + 3x = 48,000 + x\), forgetting to distribute the 3 to both terms on the left side.

Students may make simple computational errors, such as calculating \(3 × 12,000 = 32,000\) instead of \(36,000\), or making errors when performing the subtraction steps (36,000 - 48,000 instead of 48,000 - 36,000).

Errors while selecting the answer

No likely faltering points - The question asks for the value of x directly, and once students solve the equation correctly, the answer choice selection is straightforward with clear verification possible.