In a two-candidate election, one contestant received 38% of all valid votes and lost by 1,20,000 votes. How many valid votes were cast in total?

Correct Answer: 5,00,000

Step-by-Step Explanation:

Step 1: Understand the total vote percentage
In any election, the total number of valid votes cast represents 100% of the votes.
Percentage of valid votes received by the losing candidate = 38%

Step 2: Calculate the winner's vote percentage
Since there are only two candidates in the election, the winning candidate must have received all the remaining valid votes.

$\text{Winner's Percentage}=100\%-38\%=62\%$

Step 3: Determine the margin of defeat in percentage
The difference in percentage of votes between the winner and the loser is:

$\text{Percentage Difference}=62\%-38\%=24\%$

Step 4: Set up the equation using the vote difference
We are given that the contestant lost by 1,20,000 votes. This means that 24% of the total valid votes is equal to 1,20,000 votes.

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

$24\%\text{of}V=\mathrm{1,20,000}$

$\frac{24}{100}\times V=\mathrm{1,20,000}$

Step 5: Solve for the total valid votes ($V$)

$V=\frac{\mathrm{1,20,000}\times 100}{24}$

Simplifying the division:

$V=\mathrm{5,000}\times 100$

$V=\mathrm{5,00,000}$

Therefore, the total number of valid votes cast in the election was 5,00,000.