In an election, 75% of eligible voters participated. Of these, 4% were invalid votes. If a winning candidate got 13,500 votes which was 55% of valid votes, what was the total number of eligible voters(to the nearest integer)?

The correct answer is 34091.

Let us solve this problem step-by-step to find the total number of eligible voters.

Step 1: Understand the given information
- Percentage of eligible voters who participated = 75%
- Percentage of invalid votes among participating voters = 4%
- Percentage of valid votes won by the winning candidate = 55%
- Votes obtained by the winning candidate = 13,500

Step 2: Find the total number of valid votes
The winning candidate received 55% of the total valid votes, which equals 13,500.

Let $V$ be the total number of valid votes.

$0.55\times V=13500$

$V=\frac{13500}{0.55}=\frac{13500}{\frac{55}{100}}=\frac{1350000}{55}=\frac{270000}{11}$

Step 3: Find the total number of votes cast (participated voters)
Since 4% of the cast votes were invalid, the remaining 96% of the cast votes were valid.

Let $P$ be the total number of voters who participated.

$0.96\times P=V$

$\frac{96}{100}\times P=\frac{270000}{11}$

$P=\frac{270000}{11}\times \frac{100}{96}=\frac{27000000}{1056}=\frac{2812500}{11}$

Step 4: Find the total number of eligible voters
We know that 75% of the total eligible voters participated in the election.

Let $N$ be the total number of eligible voters.

$75\%\text{ of }N=P$

$\frac{3}{4}\times N=\frac{2812500}{11}$

$N=\frac{2812500}{11}\times \frac{4}{3}$

$N=\frac{937500}{11}\times 4=\frac{3750000}{11}$

$N\approx \mathrm{34090.909...}$

Rounding to the nearest integer, the total number of eligible voters is 34091.