Question

In an election between two candidates, $75\%$ of the voters cast their vote, out of which $2\%$ were declared invalid. A candidate gets 9261 votes, which are $75\%$ of the total valid votes. The total number of voters enrolled in that election was
(a) 16000
(b) 16400
(c) 16800
(d) 18000

Answer 1

Hint: Assume that ‘x’ voters were enrolled for elections. Use the fact that the value of $a\%$ of ‘b’ is $\dfrac{ab}{100}$. Write equations based on the data given in the question. Simplify the equation to calculate the value of variable ‘x’.

Complete step-by-step solution -
We have data regarding the number of voters in an election. We have to calculate the number of voters enrolled in the election.
Let’s assume that ‘x’ voters were enrolled for elections. We know that $75\%$ of these voters cast their votes.
We know that the value of $a\%$ of ‘b’ is $\dfrac{ab}{100}$.
Thus, the value of $75\%$ of ‘x’ is $=\dfrac{75x}{100}=\dfrac{3x}{4}$. We know that $2\%$ of these votes were invalid.
So, the percentage of valid votes $=100\%-2\%=98\%$.
The number of valid votes is equal to $98\%$ of $\dfrac{3x}{4}$.
Thus, the number of valid votes $=\dfrac{98}{100}\left( \dfrac{3x}{4} \right)=\dfrac{49}{50}\left( \dfrac{3x}{4} \right)=\dfrac{147x}{200}$.
We know that a candidate gets 9261 votes, which is $75\%$ of the valid votes. We know the number of valid votes is $\dfrac{147}{200}$.
So, the number of votes a candidate gets $=\dfrac{75}{100}\left( \dfrac{147x}{200} \right)=\dfrac{3}{4}\left( \dfrac{147x}{200} \right)=\dfrac{441x}{800}$. We know this value is equal to 9261.
Thus, we have $\dfrac{441x}{800}=9261$.
Rearranging the terms of the above equation, we have $x=\dfrac{9261\times 800}{441}=16800$.
Hence, the number of voters enrolled for the election is 16800, which is option (c).

Note: It’s necessary to use the fact that the value of $a\%$ of ‘b’ is $\dfrac{ab}{100}$. We should also keep in mind that $2\%$ of the votes are invalid. So, we have to calculate the percentage of valid votes. If we solve the question by taking $2\%$ as the percentage of valid votes, we will get an incorrect answer.