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When $\frac{2}{9}$ of the votes on a certain resolution have been counted, $\frac{3}{4}$ of those counted 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 to 1 against the resolution?

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\frac{11}{14}

\frac{13}{18}

\frac{4}{7}

\frac{3}{14}

answer is A.

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Detailed Solution

According to question,

Votes in favor (from counted) = \frac{3}{4} \times \frac{2}{9} = \frac{1}{6}

Votes against (from counted) = \frac{2}{9} - \frac{1}{6} = \frac{3}{9 \times 6} = \frac{1}{18}

Remaining votes = 1 - \frac{2}{9} = \frac{7}{9}

Let assume x be fraction of votes against the resolution from uncounted votes

Therefore,

Votes against (from uncounted) = \frac{7}{9} x

Votes in favor (from uncounted) = \frac{7}{9} - \frac{7}{9} x = \frac{7}{9} (1 - x)

Total Votes in favor = \frac{1}{6} + \frac{7}{9} (1 - x) = \frac{17 - 14 x}{18}

Total Votes against = \frac{1}{18} + \frac{7}{9} x = \frac{1 + 14 x}{18}

Now, \frac{\frac{1 + 14 x}{18}}{\frac{17 - 14 x}{18}} = \frac{2}{1}

1 + 14 x = 2 (17 - 14 x)

42 x = 33

x = \frac{33}{42} = \frac{11}{14}

Fraction of remaining votes against resolution = \frac{11}{14}

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