In a college election between two candidates, one candidate got $55\%$ of the total valid votes. $15\%$ of the votes were invalid. If the total votes were $15200$, what is the number of valid votes the other candidate got?

### Concept & Strategy This is a successive percentage problem. First, you must isolate the valid votes from the total votes. Then, you distribute the valid votes based on the percentages given for the two candidates. $$ \text{Valid Votes} = \text{Total Votes} \times (100\% - \text{Invalid \%}) $$ ### Step-by-Step Solution * **Given:** * Total votes = $15200$ * Invalid votes = $15\%$ * Winning candidate gets $55\%$ of *valid* votes. * **Calculation:** * **Step 1: Find the number of valid votes.** Since $15\%$ are invalid, $85\%$ are valid. Valid votes = $85\%$ of $15200$ $= 0.85 \times 15200 = 12920$ * **Step 2: Find the other candidate's share.** The first candidate received $55\%$ of the valid votes. Therefore, the other candidate received $100\% - 55\% = 45\%$ of the valid votes. * **Step 3: Calculate the exact number of votes for the other candidate.** Other candidate's votes = $45\%$ of $12920$ $= 0.45 \times 12920 = 5814$ ### Exam Strategy & Shortcut Chain the multipliers to do it in one single step and look for digital roots or unit digits if options permit. Other Candidate's Votes = $15200 \times \frac{85}{100} \times \frac{45}{100}$ $= 152 \times 85 \times 0.45$ To multiply $152 \times 85 \times 0.45$ quickly: $152 \times 85 = 152 \times (100 - 15) = 15200 - 2280 = 12920$ $12920 \times 45\%$: $50\%$ is $6460$, minus $5\%$ ($646$) = $5814$. ### Common Pitfall A major mistake is calculating $55\%$ of the *total* $15200$ votes rather than the *valid* votes. Always pay strict attention to the base of the percentage. Another error is solving for the winning candidate ($55\%$) when the question specifically asks for the *other* candidate ($45\%$). ### Final Answer **Therefore, the correct answer is 5814.**