In a recent election, James received 0.5% of the 2{,}000 votes cast. To win the election, a candidate needed to...

1. Translate the problem requirements: James got \(0.5\%\) of \(2000\) votes. To win, he needs more than \(50\%\) of total votes. We need to find how many additional votes he needs beyond what he already has.
2. Calculate James's current vote count: Convert the percentage to actual number of votes James received.
3. Determine the winning threshold: Find the minimum number of votes needed to have more than \(50\%\) of the total.
4. Find the additional votes needed: Subtract James's current votes from the winning threshold to get the answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in plain English:

• James received \(0.5\) percent of the \(2000\) votes that were cast
• To win an election, a candidate needs MORE than \(50\) percent of all votes
• We need to find how many additional votes James would need to actually win

The key word here is "additional" - this means we're looking for how many MORE votes beyond what he already has.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical requirements

2. Calculate James's current vote count

Now let's figure out exactly how many votes James actually received.

James got \(0.5\) percent of \(2000\) votes. Let's think about this step by step:

• \(0.5\) percent means \(0.5\) out of every \(100\)
• So out of \(2000\) votes, James got: \(0.5\% \times 2000\)
• \(0.5\% = \frac{0.5}{100} = 0.005\)
• \(0.005 \times 2000 = 10\)

So James received exactly \(10\) votes.

3. Determine the winning threshold

To win, James needs MORE than \(50\) percent of the total votes.

Let's calculate what "more than \(50\%\)" means:

• \(50\%\) of \(2000\) votes \(= 0.50 \times 2000 = 1000\) votes
• But James needs MORE than \(50\%\), not exactly \(50\%\)
• Since we're dealing with whole votes, "more than \(1000\)" means at least \(1001\) votes

Therefore, James needs at least \(1001\) votes to win the election.

Process Skill: INFER - Recognizing that "more than 50%" requires at least 1,001 votes, not just 1,000

4. Find the additional votes needed

Now we can find how many additional votes James needs:

• James currently has: \(10\) votes
• James needs to win: \(1001\) votes
• Additional votes needed \(= 1001 - 10 = 991\) votes

Let's verify this makes sense: If James gets \(991\) more votes, he'll have \(10 + 991 = 1001\) total votes, which is indeed more than \(50\%\) of \(2000\).

Final Answer

James would need \(991\) additional votes to win the election.

Looking at our answer choices, this matches choice D: \(991\).

The answer is D.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "more than 50%" as "at least 50%"
Students often confuse "more than \(50\%\)" with "at least \(50\%\)" or "exactly \(50\%\)". This leads them to think James needs exactly \(1000\) votes instead of MORE than \(1000\) votes (which means at least \(1001\) votes). This conceptual error affects the entire solution.

2. Confusing "additional votes" with "total votes needed"
The question asks for "additional votes" but students might set up their approach to find the total votes James needs to win, forgetting that they need to subtract his current votes from that total.

Errors while executing the approach

1. Percentage calculation errors
Students may incorrectly calculate \(0.5\%\) of \(2000\). Common mistakes include treating \(0.5\%\) as \(0.5\) (instead of \(0.005\)) or confusing it with \(5\%\), leading to incorrect current vote counts like \(1000\) or \(100\) instead of \(10\).

2. Arithmetic mistakes in the final subtraction
Even with correct intermediate values, students may make simple arithmetic errors when calculating \(1001 - 10\), potentially getting \(990\) or \(1000\) instead of \(991\).

Errors while selecting the answer

1. Selecting the total votes needed instead of additional votes
Students who correctly calculate that James needs \(1001\) total votes might mistakenly select answer choice E (\(1001\)) instead of performing the final subtraction to get \(991\) additional votes, forgetting what the question actually asked for.