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When \(\frac{2}{9}\) of the votes on a certain resolution have been counted, \(\frac{3}{4}\) of those counted are in favor of the resolution. What fraction of the remaining votes must be against the resolution so that the total count will result in a vote of \(2:1\) against the resolution?
Let's break down what we know in plain English:
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:
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:
Let's verify: \(\frac{1}{6} + \frac{1}{18} = \frac{3}{18} + \frac{1}{18} = \frac{4}{18} = \frac{2}{9}\) ✓
We established that for a "\(2\) to \(1\) against" final result:
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.
For the AGAINST votes:
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
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 correctlyWe 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:
Answer: A. \(\frac{11}{14}\)
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.
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 \(\frac{11}{14}\).
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.
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. \(\frac{11}{14}\)