In a two-candidate election, the candidate who received 35% of all valid votes lost by 1,80,000 votes. How many valid votes were cast in total?

The correct answer is 6,00,000.

Step-by-Step Explanation:

Step 1: Understand the percentage share of votes for each candidate
In a two-candidate election, the total valid votes cast represent 100%.
The candidate who lost the election received 35% of all valid votes.

Since there are only two candidates, the winning candidate must have received the remaining percentage of the total valid votes:

$\text{Winning candidate's vote percentage}=100\%-35\%=65\%$

Step 2: Determine the vote difference in terms of percentage
The loss margin (the difference in percentage between the winner and the loser) is calculated by subtracting the loser's percentage from the winner's percentage:

$\text{Percentage difference}=65\%-35\%=30\%$

Step 3: Formulate an equation using the given number of votes
We are given that the candidate lost by 1,80,000 votes. This means that 30% of the total valid votes is equal to 1,80,000 votes.

$30\%\text{of total valid votes}=\mathrm{1,80,000}$

Step 4: Calculate the total number of valid votes (100%)
First, find 1% of the total valid votes by dividing the margin of votes by 30:

$1\%\text{of total valid votes}=\frac{\mathrm{1,80,000}}{30}=\mathrm{6,000}$

Now, multiply this value by 100 to find 100% of the valid votes (the total valid votes cast):

$\text{Total valid votes}=\mathrm{6,000}\times 100=\mathrm{6,00,000}$

Therefore, the total number of valid votes cast in the election was 6,00,000.