In an election between two candidates, 20% of the total votes were invalid and the remaining votes were valid. One candidate received 55% of the total valid votes. If the total number of votes (valid plus invalid) was 7,500, how many valid votes did the other candidate receive?

Introduction / Context:
This election-based percentage problem tests the ability to work with percentages of totals and to reason carefully about valid and invalid votes. Students must distinguish between percentages calculated on total votes and those calculated on valid votes only. The question provides the total number of votes, the percentage of invalid votes and the percentage share of valid votes obtained by one candidate. We are asked to determine the number of valid votes received by the other candidate.

Given Data / Assumptions:
- Total votes cast (including valid and invalid) = 7,500.- 20% of the total votes were invalid.- Therefore, 80% of the total votes were valid.- Of the valid votes, one candidate got 55%.- There are only two candidates.- We must find how many valid votes the other candidate received.

Concept / Approach:
The solution involves two levels of percentage calculations. First, we find the number of valid votes by applying the given invalid vote percentage to the total votes. Then, within the valid votes, we calculate the vote share for each candidate. Since there are only two candidates, the second candidate receives the remaining valid votes after subtracting the first candidate's share from the total valid votes.

Step-by-Step Solution:
Step 1: Total votes = 7,500.Step 2: Invalid votes = 20% of 7,500.Step 3: 20% of 7,500 = (20 / 100) * 7,500 = 0.20 * 7,500 = 1,500.Step 4: Valid votes = total votes - invalid votes = 7,500 - 1,500 = 6,000.Step 5: Of the valid votes, one candidate gets 55%.Step 6: Votes for the first candidate = 55% of 6,000.Step 7: 55% of 6,000 = (55 / 100) * 6,000 = 0.55 * 6,000 = 3,300.Step 8: Since there are only two candidates, the other candidate gets the remaining valid votes.Step 9: Votes for the other candidate = valid votes - first candidate's votes = 6,000 - 3,300 = 2,700.

Verification / Alternative check:
We can verify quickly by checking total valid votes.Total valid votes = 2,700 + 3,300 = 6,000, which matches the earlier calculation of valid votes.Among 6,000 valid votes, one candidate has 3,300 votes, which is 3,300 / 6,000 = 0.55 or 55%. The other candidate has 2,700 votes, which is 2,700 / 6,000 = 0.45 or 45% of valid votes. All figures are consistent.

Why Other Options Are Wrong:
- 2500: This would imply total valid votes of 3,300 + 2,500 = 5,800, contradicting the 6,000 valid votes derived from the total and invalid percentages.- 2900: Gives a total of 3,300 + 2,900 = 6,200 valid votes, exceeding the correct valid count.- 3100: Again, 3,300 + 3,100 = 6,400 valid votes, not matching the correct figure of 6,000.

Common Pitfalls:
- Computing 55% of the total votes instead of valid votes only.- Forgetting to subtract invalid votes from total votes before allocating votes between candidates.- Misapplying percentages, for example, subtracting 55% from 100% incorrectly or miscalculating 20% of 7,500.- Confusing valid vote percentages with proportions of the total voter list.

Final Answer:
The other candidate received 2,700 valid votes.