Solving the Election Votes Problem

This problem involves calculating the number of votes received by three candidates based on the given relationships and differences. We are given information about the percentage difference in votes between pairs of candidates and the exact difference between two candidates. We need to find half of the difference in votes between Arjun and Bhaskar.

Understanding the Election Scenario and Relationships

Let's denote the number of votes received by Arjun, Bhaskar, and Saral as A, B, and S respectively. Based on the problem statement, we have the following relationships:

• Arjun got 50% votes more than Saral. This means Arjun's votes are 150% of Saral's votes.
  $\text{A} = \text{S} + 50\% \text{ of S} = \text{S} + 0.50\text{S} = 1.5\text{S}$
• Saral got 2% votes less than Bhaskar. This means Saral's votes are 98% of Bhaskar's votes.
  $\text{S} = \text{B} - 2\% \text{ of B} = \text{B} - 0.02\text{B} = 0.98\text{B}$
• The difference between the votes of Bhaskar and Saral is 1296. Since Saral got less votes than Bhaskar, the difference is Bhaskar's votes minus Saral's votes.
  $\text{B} - \text{S} = 1296$

Setting Up Equations for Votes

We can write these relationships as a system of equations:

1. $\text{A} = 1.5\text{S}$
2. $\text{S} = 0.98\text{B}$
3. $\text{B} - \text{S} = 1296$

Solving for Bhaskar's and Saral's Votes

We can use equations (2) and (3) to find the values of B and S. Substitute the expression for S from equation (2) into equation (3):

$\text{B} - (0.98\text{B}) = 1296$

This simplifies to:

$(1 - 0.98)\text{B} = 1296$

$0.02\text{B} = 1296$

Now, solve for B:

$\text{B} = \frac{1296}{0.02}$

To make the division easier, we can multiply the numerator and denominator by 100:

$\text{B} = \frac{1296 \times 100}{0.02 \times 100} = \frac{129600}{2}$

$\text{B} = 64800$

So, Bhaskar received 64800 votes.

Now we can find Saral's votes using equation (2):

$\text{S} = 0.98\text{B} = 0.98 \times 64800$

$\text{S} = 63504$

Saral received 63504 votes.

Let's quickly check if the difference between B and S is 1296:

$64800 - 63504 = 1296$. This matches the given information.

Calculating Arjun's Votes

Now we use equation (1) to find Arjun's votes:

$\text{A} = 1.5\text{S} = 1.5 \times 63504$

$\text{A} = \frac{3}{2} \times 63504$

$\text{A} = 3 \times \frac{63504}{2}$

$\text{A} = 3 \times 31752$

$\text{A} = 95256$

Arjun received 95256 votes.

Finding the Difference Between Arjun and Bhaskar

The difference between the votes of Arjun and Bhaskar is:

Difference = $\text{A} - \text{B}$

Difference = $95256 - 64800$

Difference = $30456$

Determining Half the Difference

The question asks for half of the difference between the votes of Arjun and Bhaskar. So, we need to calculate:

Half Difference = $\frac{\text{Difference}}{2}$

Half Difference = $\frac{30456}{2}$

Half Difference = $15228$

The half of the difference between the votes of Arjun and Bhaskar is 15228.

Final Answer

The half of the difference between the votes of Arjun and Bhaskar is 15228.

Revision Table: Election Vote Calculation

Additional Information: Percentage Calculations in Problems

Problems involving percentages are common in quantitative aptitude sections of exams. Understanding how to represent percentage increase or decrease as a multiplication factor is crucial.

• Percentage Increase: If a quantity increases by x%, the new quantity is the original quantity plus x% of the original quantity. This can be written as Original Quantity $\times (1 + \frac{x}{100})$. In our problem, Arjun got 50% more than Saral, so Arjun's votes are Saral's votes $\times (1 + \frac{50}{100}) = \text{S} \times 1.5$.
• Percentage Decrease: If a quantity decreases by y%, the new quantity is the original quantity minus y% of the original quantity. This can be written as Original Quantity $\times (1 - \frac{y}{100})$. In our problem, Saral got 2% less than Bhaskar, so Saral's votes are Bhaskar's votes $\times (1 - \frac{2}{100}) = \text{B} \times 0.98$.
• Setting up Equations: For problems with multiple variables, define variables clearly and translate each sentence of the problem into a mathematical equation.
• Solving System of Equations: Use substitution or elimination methods to solve the system of equations. In this case, substitution was effective, substituting the expression for S from one equation into another.