In an election between two candidates, 70% of the enrolled voters cast their votes.\nOut of the votes cast, 2% were declared invalid.\nOne candidate received 7203 votes, which was 60% of the total valid votes.\nWhat was the total number of voters enrolled in that election?

Introduction / Context:
This question combines several percentage concepts in the context of an election: turnout, invalid votes, and a candidate's share of valid votes. You are given the number of votes obtained by one candidate and the percentages at each stage, and you must work backwards to find the total number of voters enrolled. This tests multi step percentage reasoning and careful algebra.

Given Data / Assumptions:

• Let the total number of enrolled voters be T.
• 70% of T cast their votes, so total votes cast = 70% of T.
• 2% of the votes cast are invalid.
• Therefore 98% of the votes cast are valid.
• One candidate receives 7203 votes, which is 60% of the valid votes.
• We must find T.

Concept / Approach:
Work backwards from the candidate's votes. If 7203 is 60% of the valid votes, then valid votes = 7203 / 0.60. Once we know the number of valid votes, we can find total votes cast by recognizing that valid votes are 98% of votes cast. Then, knowing that votes cast are 70% of total enrolled voters, we can find T. This stepwise reverse calculation is typical in layered percentage problems.

Step-by-Step Solution:
Step 1: Let V be total valid votes. Step 2: Candidate's votes = 60% of V = 0.60V = 7203. Step 3: Solve for V: V = 7203 / 0.60. Step 4: 0.60 = 3/5, so V = 7203 * (5 / 3). Step 5: Compute 7203 / 3 = 2401, then multiply by 5: V = 2401 * 5 = 12,005 valid votes. Step 6: Let C be total votes cast. Valid votes are 98% of C, so 0.98C = 12,005. Step 7: Solve for C: C = 12,005 / 0.98. Step 8: 0.98 = 98 / 100, so C = 12,005 * 100 / 98 = 12,005 * (50 / 49). Step 9: Compute 12,005 / 49 = 245, then multiply by 50: C = 245 * 50 = 12,250 votes cast. Step 10: Votes cast are 70% of total voters T, so 0.70T = 12,250. Step 11: Solve for T: T = 12,250 / 0.70 = 12,250 * (10 / 7) = 17,500.

Verification / Alternative check:
Check the entire chain with T = 17,500. Votes cast = 70% of 17,500 = 0.70 * 17,500 = 12,250. Invalid votes = 2% of 12,250 = 0.02 * 12,250 = 245. Valid votes = 12,250 − 245 = 12,005. Candidate's votes = 60% of 12,005 = 0.60 * 12,005 = 7203. All figures match the problem statement, confirming T = 17,500.

Why Other Options Are Wrong:
18,050; 17,000; 7,203; 20,000: Using any of these values as T leads to a different number of votes cast, valid votes, and candidate votes, which will not match 7203 under the given percentages. For example, if T were 17,000, votes cast would be 11,900, leading to different valid votes and a different 60% value.

Common Pitfalls:
Some students mistakenly apply 2% invalid to total voters rather than to votes cast, while others use 60% of total votes instead of valid votes. Another frequent error is trying to jump directly from 7203 to T in one step without carefully tracking each percentage stage. It is safer to proceed step by step: candidate votes → valid votes → votes cast → total voters.

Final Answer:
The total number of voters enrolled in the election was 17,500.