In an election between two candidates K and L, 20% of the total 8,720 votes polled were invalid. The number of valid votes received by K exceeds those received by L by 15% of the total votes polled. How many valid votes did candidate L receive?

Introduction / Context:
This election based percentage problem tests your ability to handle valid and invalid votes, and to convert textual information about vote differences into algebraic equations. Such questions are common in quantitative aptitude and reasoning exams, because they combine percentage calculations with simple algebra and logical interpretation of real life scenarios.

Given Data / Assumptions:

• Total votes polled = 8,720.
• 20% of the total votes are invalid.
• Let the valid votes received by K be V_K and by L be V_L.
• The valid votes of K exceed those of L by 15% of the total votes polled.
• We need to find V_L, the number of valid votes received by L.

Concept / Approach:
First, we compute the number of valid votes by removing the invalid votes from the total. Then we use the statement about K's lead to form two equations. One equation is V_K + V_L = total valid votes. The other comes from the fact that the difference V_K - V_L equals 15% of total votes polled. Solving these two simple linear equations gives values for V_K and V_L. This method is systematic and avoids confusion between percentages of total votes and percentages of valid votes.

Step-by-Step Solution:
Step 1: Total votes polled = 8720. Step 2: Invalid votes = 20% of 8720 = 0.20 * 8720 = 1744. Step 3: Valid votes = 8720 - 1744 = 6976. Step 4: Let V_K be valid votes for K and V_L be valid votes for L. Step 5: We know V_K + V_L = 6976 (all valid votes). Step 6: K exceeds L by 15% of total votes polled, so V_K - V_L = 15% of 8720. Step 7: Compute 15% of 8720 = 0.15 * 8720 = 1308. Step 8: So we have the system: V_K + V_L = 6976 and V_K - V_L = 1308. Step 9: Add both equations: 2V_K = 6976 + 1308 = 8284, so V_K = 4142. Step 10: Substitute into V_K + V_L = 6976 to get V_L = 6976 - 4142 = 2834. Step 11: Therefore, candidate L received 2,834 valid votes.

Verification / Alternative check:
Check the conditions using V_L = 2834 and V_K = 4142. First, V_K + V_L = 4142 + 2834 = 6976, which matches the valid votes count. Second, the difference is 4142 - 2834 = 1308, which equals 15% of 8720, confirming that K leads L by the correct margin. This confirms that our calculations are consistent and that 2834 is correct.

Why Other Options Are Wrong:
Values like 3241, 2457, 2642, and 3000 either do not yield the correct sum of valid votes when combined with K's total, or do not maintain the required difference of 1308. Only V_L = 2834 satisfies both the sum equation and the difference equation simultaneously. Therefore, the other options fail to meet the conditions given in the problem.

Common Pitfalls:
A common error is to treat 15% as a percentage of valid votes instead of total votes polled, which leads to incorrect equations. Another frequent mistake is forgetting to subtract invalid votes before distributing the remaining votes between the candidates. Some learners also make arithmetic errors when solving the pair of linear equations, especially while adding or subtracting them. Careful and systematic calculation avoids these issues.

Final Answer:
Candidate L received 2,834 valid votes.