When 2/9 of the votes on a certain resolution have been counted, 3/4 of those counted are in favor of...

1. Translate the problem requirements: We need to understand that \(\frac{2}{9}\) of votes are counted (\(\frac{3}{4}\) favor, \(\frac{1}{4}\) against), \(\frac{7}{9}\) remain uncounted, and we want the final result to be \(2:1\) against (meaning \(\frac{1}{3}\) favor, \(\frac{2}{3}\) against overall)
2. Set up the current vote situation: Calculate the actual fractions of total votes that are currently for and against the resolution
3. Determine the final target distribution: Figure out what fractions of total votes need to be for and against to achieve a \(2:1\) ratio against
4. Calculate the gap to fill: Find how many additional 'against' votes are needed from the remaining uncounted votes
5. Express as fraction of remaining votes: Convert the required additional 'against' votes into a fraction of the \(\frac{7}{9}\) uncounted votes

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in plain English:

• We've counted \(\frac{2}{9}\) of all the votes so far
• Of those counted votes, \(\frac{3}{4}\) are in favor and \(\frac{1}{4}\) are against
• We still have \(\frac{7}{9}\) of all votes left to count
• We want the final result to show a "\(2\) to \(1\) against" ratio

What does "\(2\) to \(1\) against" mean? This means for every \(1\) vote in favor, there are \(2\) votes against. So out of every \(3\) votes total, \(1\) is in favor and \(2\) are against. This translates to:

• Final target: \(\frac{1}{3}\) of all votes should be in favor
• Final target: \(\frac{2}{3}\) of all votes should be against

Process Skill: TRANSLATE - Converting the ratio language into fraction targets

2. Set up the current vote situation

Now let's figure out what fraction of ALL votes are currently counted as "for" and "against":

We've counted \(\frac{2}{9}\) of all votes. Of these counted votes:

• Votes currently FOR = \(\frac{3}{4} \times \frac{2}{9} = \frac{6}{36} = \frac{1}{6}\) of all votes
• Votes currently AGAINST = \(\frac{1}{4} \times \frac{2}{9} = \frac{2}{36} = \frac{1}{18}\) of all votes

Let's verify: \(\frac{1}{6} + \frac{1}{18} = \frac{3}{18} + \frac{1}{18} = \frac{4}{18} = \frac{2}{9}\) ✓

3. Determine the final target distribution

We established that for a "\(2\) to \(1\) against" final result:

• Total FOR votes needed = \(\frac{1}{3}\) of all votes
• Total AGAINST votes needed = \(\frac{2}{3}\) of all votes

We currently have \(\frac{1}{6}\) of all votes FOR, and we need \(\frac{1}{3}\) total FOR.

Since \(\frac{1}{3} = \frac{2}{6}\), and we already have \(\frac{1}{6}\) FOR, we don't need any more FOR votes from the remaining votes.

4. Calculate the gap to fill

For the AGAINST votes:

• We currently have: \(\frac{1}{18}\) of all votes against
• We need total: \(\frac{2}{3}\) of all votes against
• Additional AGAINST votes needed = \(\frac{2}{3} - \frac{1}{18}\)

Converting to common denominator: \(\frac{2}{3} = \frac{12}{18}\)

Additional AGAINST votes needed = \(\frac{12}{18} - \frac{1}{18} = \frac{11}{18}\) of all votes

5. Express as fraction of remaining votes

We need \(\frac{11}{18}\) of all votes to be against from the remaining uncounted votes.

The remaining uncounted votes represent \(\frac{7}{9}\) of all votes.

So we need to find: What fraction of the remaining \(\frac{7}{9}\) votes must be against?

This is: \(\frac{11}{18} \div \frac{7}{9}\)

Converting division to multiplication: \(\frac{11}{18} \times \frac{9}{7} = \frac{11 \times 9}{18 \times 7} = \frac{99}{126}\)

Simplifying: \(\frac{99}{126} = \frac{11}{14}\)

Process Skill: MANIPULATE - Converting the division of fractions correctly

4. Final Answer

We need \(\frac{11}{14}\) of the remaining votes to be against the resolution.

Let's verify: If \(\frac{11}{14}\) of the remaining \(\frac{7}{9}\) votes are against:

• Additional against votes = \(\frac{11}{14} \times \frac{7}{9} = \frac{77}{126} = \frac{11}{18}\) ✓
• Total against = \(\frac{1}{18} + \frac{11}{18} = \frac{12}{18} = \frac{2}{3}\) ✓
• Total for = \(\frac{1}{6} = \frac{3}{18} = \frac{1}{3}\) ✓
• Ratio against to for = \(\frac{2}{3}:\frac{1}{3} = 2:1\) ✓

Answer: A. \(\frac{11}{14}\)

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting the "\(2\) to \(1\) against" ratio

Students often confuse what "\(2\) to \(1\) against" means. They might think it means \(\frac{2}{3}\) are FOR and \(\frac{1}{3}\) are AGAINST, when it actually means the opposite: \(\frac{2}{3}\) are AGAINST and \(\frac{1}{3}\) are FOR. This fundamental misunderstanding will lead to completely incorrect target fractions and a wrong final answer.

Faltering Point 2: Confusion about what fraction to find

The question asks for "what fraction of the REMAINING votes must be against", but students might set up their calculation to find what fraction of ALL votes must be against. This misreads the constraint and leads to solving for the wrong unknown, even if their arithmetic is correct.

Faltering Point 3: Misunderstanding the current vote breakdown

Students may incorrectly assume that \(\frac{3}{4}\) and \(\frac{1}{4}\) represent fractions of ALL votes, rather than fractions of the \(\frac{2}{9}\) that have been counted so far. This leads to wrong starting values when calculating how many additional votes are needed.

Errors while executing the approach

Faltering Point 1: Arithmetic errors with fraction operations

Students frequently make mistakes when adding, subtracting, or finding common denominators with fractions like \(\frac{2}{3} - \frac{1}{18}\) or when converting \(\frac{2}{3}\) to eighteenths (\(\frac{12}{18}\)). These computational errors compound throughout the solution.

Faltering Point 2: Incorrect division of fractions

When calculating \(\frac{11}{18} \div \frac{7}{9}\), students often forget to flip the second fraction and multiply, or make errors in the multiplication step. This is the critical final calculation that determines the answer.

Faltering Point 3: Fraction simplification errors

Students may correctly get \(\frac{99}{126}\) but then incorrectly simplify it, perhaps not recognizing that both numerator and denominator are divisible by \(9\), or making errors when reducing to \(\frac{11}{14}\).

Errors while selecting the answer

Faltering Point 1: Selecting a partial result instead of the final answer

Students might calculate intermediate values like \(\frac{11}{18}\) (the fraction of ALL votes that need to be against) and mistakenly select this if it appears among the choices, rather than completing the final step to find what fraction of the REMAINING votes this represents.

Alternate Solutions

Smart Numbers Approach

We can solve this problem by choosing a convenient total number of votes that makes the fractional calculations clean.

Step 1: Choose a smart total number of votes
Since we're dealing with fractions like \(\frac{2}{9}\), let's choose a total that's divisible by \(9\). We also need to consider the final ratio of \(2:1\) against, which means the total should be divisible by \(3\).
Let's use \(63\) total votes (\(63 = 9 \times 7 = 3 \times 21\), so it works well with our fractions).

Step 2: Calculate the current vote situation
Votes counted so far: \(\frac{2}{9} \times 63 = 14\) votes
Of these \(14\) counted votes:
• In favor: \(\frac{3}{4} \times 14 = 10.5\) votes
• Against: \(\frac{1}{4} \times 14 = 3.5\) votes
Remaining uncounted votes: \(63 - 14 = 49\) votes

Step 3: Determine the target final distribution
For a \(2:1\) ratio against the resolution:
• Total against votes needed: \(\frac{2}{3} \times 63 = 42\) votes
• Total in favor votes needed: \(\frac{1}{3} \times 63 = 21\) votes

Step 4: Calculate how many additional 'against' votes are needed
We currently have \(3.5\) against votes and need \(42\) total against votes.
Additional against votes needed: \(42 - 3.5 = 38.5\) votes
These must come from the \(49\) remaining votes.

Step 5: Express as a fraction of remaining votes
Fraction of remaining votes that must be against: \(\frac{38.5}{49} = \frac{77}{98} = \frac{11}{14}\)

Verification:
• Remaining against votes: \(\frac{11}{14} \times 49 = 38.5\)
• Total against: \(3.5 + 38.5 = 42\)
• Total in favor: \(10.5 + 10.5 = 21\)
• Ratio: \(42:21 = 2:1\) ✓

The answer is A. \(\frac{11}{14}\)