To win an election, a candidate needs $\frac{3}{4}$ of the votes cast. If after $\frac{2}{3}$ of the votes have been counted, a candidate has $\frac{5}{6}$ of what he needs, then what part of the remaining votes does he still need?

### Concept & Strategy The objective is to establish the fraction of remaining votes required by the candidate to meet the winning threshold. We can map the required votes, the votes already secured, and the left-over votes relative to a fixed common base (total votes cast). ### Step-by-Step Solution * **Given:** * Total votes needed to win = $\frac{3}{4}$ of total votes. * Fraction of votes counted = $\frac{2}{3}$ of total votes. * Votes obtained so far = $\frac{5}{6}$ of what he needs. * **Calculation:** * Let the total number of votes cast be $V$. * Votes needed to win = $\frac{3}{4}V$ * Votes secured so far = $\frac{5}{6} \times \left(\frac{3}{4}V\right) = \frac{5}{8}V$ * Remaining votes needed = $\text{Total needed} - \text{Secured} = \frac{3}{4}V - \frac{5}{8}V = \frac{1}{8}V$ * Remaining uncounted votes = $\text{Total votes} - \text{Counted votes} = V - \frac{2}{3}V = \frac{1}{3}V$ * Required fraction of the remaining votes: $$ \text{Required Part} = \frac{\text{Remaining votes needed}}{\text{Remaining uncounted votes}} = \frac{\frac{1}{8}V}{\frac{1}{3}V} = \frac{3}{8} $$ ### Exam Strategy & Shortcut **Plugging in a Number:** Choose a total vote count that is a multiple of the denominators (4, 3, 6, 8). Let the total votes be $V = 24$. * Votes needed to win = $\frac{3}{4} \times 24 = 18$ * Counted votes so far = $\frac{2}{3} \times 24 = 16 \implies$ Remaining votes = $24 - 16 = 8$ * Votes obtained = $\frac{5}{6} \times 18 = 15$ * Additional votes needed = $18 - 15 = 3$ * Fraction of remaining votes needed = $\frac{3}{8}$ ### Common Pitfall A frequent error is calculating the remaining votes needed as a fraction of the *total* votes ($\frac{1}{8}$) instead of the *remaining* uncounted votes ($\frac{3}{8}$). Pay attention to the baseline phrase "what part of the remaining votes". ### Final Answer **Therefore, the correct answer is 3/8.**