Question

Three candidates \(A, B\) and \(C\) contested an election. Out of the total votes on a voter list \(25 \%\) did not vote and \(6.66 \%\) votes polled were invalid. C got 2450 valid votes, which were \(40 \%\) more than that of \(B\). If \(A\) got only \(40 \%\) of the total votes, then who is the winner? (a) \(\mathrm{A}\) (b) B (c) \(\mathrm{C}\) (d) can't be determined

Short answer

Given the information regarding a voting scenario and the percentages of the total votes, we need to determine the winner of the election. To do this, we first find the total votes and valid votes for each candidate. We then compare the valid votes for each candidate to determine the winner. Based on our calculations, we find: A got \((0.4x)(0.6834)\) valid votes B got 1750 valid votes C got 2450 valid votes Since 2450 is greater than both 1750 and \((0.4x)(0.6834)\), candidate C received the most valid votes and is the winner.

Step-by-step solution

First step: Find the total votes

We are not given the total number of votes directly, but we know that \(40\%\) of them were cast for candidate A. Let's call the total number of votes \(x\). Then: A got \(40\%\) of the total votes \((0.4x)\) Since it doesn't matter which candidate won the most votes, \(x\) remains free: we're simply comparing who got the most votes.

Second step: Calculate the percentage of valid votes

Out of the total votes, we know that: - \(25\%\) did not vote - \(6.66\%\) votes were invalid Thus, the percentage of valid votes is: \(100\% - 25\% - 6.66\% = 68.34\% \text{ of the total votes}\) We will use this percentage to find the valid votes cast for each candidate.

Third step: Calculate valid votes cast for candidate C

We are given that candidate C received 2450 valid votes. Since we know the percentage of valid votes from the total votes, we can write an equation to find the valid votes for C as a percentage of the total votes: C got \(2450 = (0.6834 * x)\) valid votes

Fourth step: Calculate valid votes cast for candidate B

We are given that candidate C received \(40\%\) more valid votes than candidate B. Using the valid votes for C, we can calculate the valid votes for B: C got \(40\%\) more votes than B, so \((1 + 40\%)B = 2450\) Rearrange to find B: \(B = \frac{2450}{1.4} \approx 1750\) B got 1750 valid votes.

Fifth step: Calculate valid votes cast for candidate A

We know that candidate A received \(40\%\) of the total votes. Since we have found the valid votes for candidates B and C, we can now find the valid votes for candidate A: A got \(40\%\) of the total votes, which is: A got \(0.4x\) votes And since \(68.34\%\) of the total votes were valid: A got \((0.4x)(0.6834)\) valid votes

Sixth step: Compare valid votes of all candidates

Now we will plug in our values of valid votes for candidates A, B, and C: A got \((0.4x)(0.6834)\) valid votes B got 1750 valid votes C got 2450 valid votes Since 2450 > 1750 and 2450 > \((0.4x)(0.6834)\), candidate C received the most valid votes and is the winner: Winner: C (c)

Key concepts

Percentage Calculations

To solve election mathematics problems, a solid understanding of percentage calculations is essential. Percentages allow us to express numbers as parts of a hundred, which makes it easier to compare values. In the context of the given exercise, percentages help in determining the proportions of votes related to different scenarios:

• The percentage of the total votes not cast (25% did not vote).
• The percentage of total votes that were invalid (6.66%).
• The percentage of votes received by candidate A, which was 40% of total votes cast.

Understanding how to calculate these percentages of a total allows us to determine the valid vote counts, and helps us compare candidates effectively. For instance, calculating the valid votes involves subtracting non-voting and invalid percentages from 100%. Thus, the valid percentage of votes becomes 68.34%. This reflects the real competition each candidate had during the election.

Vote Analysis

Vote analysis lets us break down the competitive dynamics among candidates. In our problem, it's imperative to analyze vote distributions. Here's how:

• Determine Total Votes: In the absence of an exact total vote count, using placeholders (like 'x') simplifies comparisons.
• Valid Votes: With 68.34% of total votes being valid, compute the valid votes which help us pinpoint each candidate's performance.

Candidate C received 2450 valid votes. Given that this was 40% more than candidate B's valid votes, we calculate B's votes by factoring in this additional percentage. By doing so, candidate B ends up with 1750 votes. Such detailed breakdowns help us see the whole picture. It illustrates who did better, despite lacking some raw data. In essence, vote analysis provides a clear portrait of which candidate led, based on specified factors.

Problem Solving Steps

Following logical problem-solving steps ensures precise answers in election mathematics. Here’s a brief journey through solving the given election problem:

• Identifying Known Values: Start by identifying any given data; percent of turnout, invalid votes, and known valid votes for each candidate.
• Determining Valid Votes: Calculate percentage of valid votes from the total registered ones; this leverages results effectively.
• Interlinked Candidates' Votes: Use relationships between candidates' votes. Candidate C having 40% more votes than B can guide us to calculate B's votes effectively.

Each step interconnects to build a full understanding of the scenario. Concluding with comparisons of every candidate's valid votes allows us to identify the winner. Such structured approaches equip students to tackle similar, sometimes intricate, election scenarios. It’s not just about math; it’s about reading between the lines of the numbers presented.