Calculating Percentage of Valid Votes in an Election

Let's break down this election problem step-by-step to find the percentage of total valid votes secured by Dharam.

Step 1: Calculate the Total Number of Votes Cast

The total number of people on the electoral roll is 3,00,000. 85% of these people cast their votes.

Total votes cast = 85% of 3,00,000

Using LaTeX for the calculation:

\( \text{Votes Cast} = \frac{85}{100} \times 3,00,000 \)

\( \text{Votes Cast} = 85 \times 3,000 \)

\( \text{Votes Cast} = 2,55,000 \)

So, 2,55,000 votes were cast during the election.

Step 2: Calculate the Number of Invalid Votes

10% of the votes cast were declared invalid. The total votes cast were 2,55,000.

Number of invalid votes = 10% of 2,55,000

Using LaTeX for the calculation:

\( \text{Invalid Votes} = \frac{10}{100} \times 2,55,000 \)

\( \text{Invalid Votes} = \frac{1}{10} \times 2,55,000 \)

\( \text{Invalid Votes} = 25,500 \)

So, 25,500 votes were invalid.

Step 3: Calculate the Total Number of Valid Votes

The total valid votes are the votes cast minus the invalid votes.

Total Valid Votes = Total votes cast - Invalid votes

Using LaTeX for the calculation:

\( \text{Total Valid Votes} = 2,55,000 - 25,500 \)

\( \text{Total Valid Votes} = 2,29,500 \)

So, there were 2,29,500 valid votes in the election.

Step 4: Calculate Dharam's Percentage of Total Valid Votes

Dharam secured 1,37,700 valid votes. The total number of valid votes is 2,29,500.

Percentage secured by Dharam = \(\left( \frac{\text{Dharam's Valid Votes}}{\text{Total Valid Votes}} \right) \times 100\%\)

Using LaTeX for the calculation:

\( \text{Percentage for Dharam} = \left( \frac{1,37,700}{2,29,500} \right) \times 100\% \)

We can simplify the fraction:

\( \frac{137700}{229500} = \frac{1377}{2295} \)

Let's perform the division:

\( \frac{1377}{2295} \)

Divide both numerator and denominator by a common factor. Both are divisible by 5, but let's look for larger factors. The sum of digits for 1377 is 18 (divisible by 9), and for 2295 is 18 (divisible by 9). So, both are divisible by 9.

\( 1377 \div 9 = 153 \)

\( 2295 \div 9 = 255 \)

Now we have \(\frac{153}{255}\). Both are divisible by 3 (sum of digits 9 and 12).

\( 153 \div 3 = 51 \)

\( 255 \div 3 = 85 \)

Now we have \(\frac{51}{85}\). Both are divisible by 17.

\( 51 \div 17 = 3 \)

\( 85 \div 17 = 5 \)

So the fraction simplifies to \(\frac{3}{5}\).

Now substitute back into the percentage calculation:

\( \text{Percentage for Dharam} = \left( \frac{3}{5} \right) \times 100\% \)

\( \text{Percentage for Dharam} = 0.6 \times 100\% \)

\( \text{Percentage for Dharam} = 60\% \)

Dharam secured 60% of the total number of valid votes.

Summary of Calculations

Revision Table: Election Vote Calculation

Understanding the different stages of vote counting is crucial for solving such problems.

• Electoral Roll: The total number of registered voters.
• Votes Cast: The number of people from the electoral roll who actually voted.
• Invalid Votes: Votes that are not counted due to errors or other reasons.
• Valid Votes: Votes cast minus invalid votes. These are the votes that determine the winner.

Additional Information: Working with Percentages in Elections

Percentages are frequently used in election analysis to understand participation rates and results relative to the total numbers.

• When calculating a percentage of a number, convert the percentage to a decimal (divide by 100) and multiply by the number. For example, 85% of 3,00,000 is \(\frac{85}{100} \times 3,00,000\).
• To find what percentage one number is of another, divide the first number by the second number and multiply by 100. For example, to find what percentage 1,37,700 is of 2,29,500, calculate \(\left( \frac{1,37,700}{2,29,500} \right) \times 100\).
• It's important to carefully read the question to identify which total the percentage is based on (e.g., percentage of electoral roll, percentage of votes cast, percentage of valid votes).