CAT 2022Slot 2QAQuestion & Solution
Question
In an election, there were four candidates and 80% of the registered voters casted their votes. One of the candidates received 30% of the casted votes while the other three candidates received the remaining casted votes in the proportion 1 : 2 : 3. If the winner of the election received 2512 votes more than the candidate with the second highest votes, then the number of registered voters was
Options
50240
40192
60288
62800
Solution
1. Concept Used
- Topic: Ratio and Proportion combined with Percentages
- Formula: $$\text{Votes for candidate} = \text{Fraction of total casted votes} \times \text{Total casted votes}$$
2. Calculation
Let the total number of registered voters be ( 100x ).
Since 80% of registered voters cast their votes, the total casted votes ( = 80x ).
Candidate A received 30% of casted votes: $$\text{Votes}_A = \frac{30}{100} \times 80x = 24x$$
The remaining votes distributed among the other three candidates (B, C, D) in ratio ( 1:2:3 ): $$\text{Remaining votes} = 80x - 24x = 56x$$
The sum of ratio parts ( = 1 + 2 + 3 = 6 ), so: $$\text{Votes}_B = \frac{1}{6} \times 56x = \frac{56x}{6}$$ $$\text{Votes}_C = \frac{2}{6} \times 56x = \frac{112x}{6}$$ $$\text{Votes}_D = \frac{3}{6} \times 56x = \frac{168x}{6} = 28x$$
Now let's list all four candidates' votes:
- Candidate A: ( 24x )
- Candidate B: ( \approx 9.33x )
- Candidate C: ( \approx 18.67x )
- Candidate D: ( 28x )
Winner = Candidate D with ( 28x ) votes. Second highest = Candidate A with ( 24x ) votes.
Given that the winner received 2512 more votes than the second-highest candidate: $$28x - 24x = 2512$$ $$4x = 2512$$ $$x = 628$$
Therefore, the total number of registered voters: $$= 100x = 100 \times 628 = 62800$$
3. Solution
Answer = Option D ✅
The total number of registered voters is 62,800.