Question

10% of the voters did not cast their votes in an election between two candidates, 10% of the votes polled were found invalid. The successful candidate got 54% of the valid votes and won by a margin of 1620 votes. The number of voters enrolled in the voter's list was
(a) 25000
(b) 33000
(c) 35000
(d) 40000

Answer 1

Hint: Assume the number of voters enrolled in the voter’s list as \[x\] because that is what the question is asking us to find and the candidate who lost got 46% of votes. The winning candidate gets 54% of the valid votes which is \[54%\times [90%\times (90%\times x)]\] votes.

Complete step-by-step solution:
Let the total number of voters enrolled in the voter’s list for the election be \[x\].
It is given in the question that 10% of the voters did not cast their votes. So then the total votes polled\[=90%\times \,x\]
10% of the votes polled were found invalid, so the total number of valid votes = 90% of the total votes polled \[=90%\times (90%\times x)\]
Votes achieved by successful candidate \[=54%\times [90%\times (90%\times x)]..........(1)\]
Votes achieved by lost candidate \[=46%\times [90%\times (90%\times x)]........(2)\]
We will get the margin of votes by subtracting equation (2) from equation (1),
\[\Rightarrow 54%\times [90%\times (90%\times x)]-46%\times [90%\times (90%\times x)]........(3)\]
\[\Rightarrow \dfrac{54}{100}\times \left[ \dfrac{90}{100}\times \dfrac{90}{100}\times x \right]-\dfrac{46}{100}\times \left[ \dfrac{90}{100}\times \dfrac{90}{100}\times x \right]..........(4)\]
Taking all the common terms out from equation (4) we get,
\[\Rightarrow \dfrac{90}{100}\times \dfrac{90}{100}\times x\times \dfrac{1}{100}\times \left[ 54-46 \right]\]
\[\Rightarrow \dfrac{90}{100}\times \dfrac{90}{100}\times x\times \dfrac{1}{100}\times 8...........(5)\]
Now equating equation (5) to the margin of votes by which the successful candidate wins the election and that is 1620 votes. We get,
\[\Rightarrow \dfrac{90}{100}\times \dfrac{90}{100}\times x\times \dfrac{1}{100}\times 8=1620\]
Now moving all the numbers to one side and the variable term to another side we get,
\[\Rightarrow x=\dfrac{1620\times 100\times 100\times 100}{90\times 90\times 8}\]
Now we will divide and get the answer.
\[\Rightarrow x=25000\]
Hence 25000 is the number of voters enrolled in the voter’s list. So we will tick option (a).

Note: Reading and grasping the question is important so we need to read it 3 to 4 times. In a hurry instead of doing \[90%\times (90%\times x)\] we might end up doing \[90%\times \,x\] and then we will get the wrong answer. We also need to understand that margin means by how many votes the candidate won from his rival candidate so we need to subtract and this step is important.