Question

In an election between 3 candidates A, B, and C, 10% registered voters did not cast their votes and 5% votes were declared invalid. A obtained 834 votes more than B and 1254 votes more than C. If 178,800 voters were registered for the election, then find the number of votes obtained by the winner.

• A. 51,580
• B. 51,624
• C. 51,644
• D. 51,654

Hint:

In election problems, carefully calculate the number of valid votes after accounting for invalid votes, and use the given relationships between candidates' votes to solve for the unknowns.

Answer 1

The Correct Option is D

Let the total number of voters be 178,800. Step 1: Find the total number of voters who cast their votes. 10% of registered voters did not cast their votes, so 90% cast their votes: \[ \text{Voters who cast their votes} = 0.90 \times 178,800 = 160,920. \] Step 2: Find the number of valid votes. 5% of the votes were declared invalid, so 95% of the votes are valid: \[ \text{Valid votes} = 0.95 \times 160,920 = 152,874. \] Step 3: Define the votes obtained by A, B, and C. Let the number of votes obtained by B be \( x \). Then: - A obtained 834 votes more than B, so A's votes are \( x + 834 \). - A obtained 1254 votes more than C, so C's votes are \( x + 834 - 1254 = x - 420 \). Thus, the total number of valid votes is: \[ x + (x + 834) + (x - 420) = 152,874. \] Step 4: Solve for \( x \). Simplifying the equation: \[ 3x + 414 = 152,874, \] \[ 3x = 152,874 - 414 = 152,460, \] \[ x = \frac{152,460}{3} = 50,820. \] So, B obtained 50,820 votes. Step 5: Find the number of votes obtained by A (the winner). A's votes are \( x + 834 = 50,820 + 834 = 51,654 \). Thus, the number of votes obtained by the winner (A) is 51,654.