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In an election contested between two candidates, winner beat the loser by 5040 votes. If out of total votes cast, 16% were declared invalid and winner received 75% of the total valid votes, then find the number of invalid votes cast.

Question

In an election contested between two candidates, winner beat the loser by 5040 votes. If out of total votes cast, 16% were declared invalid and winner received 75% of the total valid votes, then find the number of invalid votes cast.

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Solution

Step 1: Let's denote the total number of votes cast as "T". We know that 16% of these votes were declared invalid. Therefore, the number of invalid votes is 0.16T.

Step 2: The valid votes are the total votes minus the invalid votes, which is T - 0.16T = 0.84T.

Step 3: The winner received 75% of the total valid votes, which is 0.75 * 0.84T = 0.63T.

Step 4: The loser received the rest of the valid votes, which is 0.84T - 0.63T = 0.21T.

Step 5: We know that the winner beat the loser by 5040 votes. Therefore, 0.63T - 0.21T = 5040 votes.

Step 6: Solving the equation from step 5, we get T = 5040 / 0.42 = 12000 votes.

Step 7: Substituting T = 12000 into the equation from step 1, we find that the number of invalid votes is 0.16 * 12000 = 1920 votes.

This problem has been solved

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