In an election Aryan and Kanak participated. `2//5` of the voters promised to vote for Aryan and rest promised to vote for Kanak. On the voting day `15%` of the voters went back on their promise to vote for Aryan and `25%` of the voters went back on their promise to vote for Kanak. Find the total no. of voters , if Kanak won by 250 votes.

Options

A. A)`18700`
B. B)`15500`
C. C)`12500`
D. D)`18500`

Answer

To solve the problem step by step, we will follow the information given in the question and apply the necessary calculations.

### Step 1: Define the total number of voters
Let the total number of voters be \( X \).

### Step 2: Calculate the number of voters for Aryan and Kanak
According to the question:
- Aryan received \( \frac{2}{5} \) of the votes.
- Kanak received the remaining votes, which is \( 1 - \frac{2}{5} = \frac{3}{5} \).

Thus, the number of voters for Aryan and Kanak can be calculated as:
- Votes for Aryan: \( \text{Votes for Aryan} = \frac{2}{5}X \)
- Votes for Kanak: \( \text{Votes for Kanak} = \frac{3}{5}X \)

### Step 3: Calculate the number of voters who changed their votes
On voting day:
- \( 15\% \) of Aryan's supporters went back on their promise. This means:
\[ \text{Voters who changed from Aryan} = 15\% \text{ of } \frac{2}{5}X = \frac{15}{100} \times \frac{2}{5}X = \frac{3}{10}X \]
- \( 25\% \) of Kanak's supporters went back on their promise. This means:
\[ \text{Voters who changed from Kanak} = 25\% \text{ of } \frac{3}{5}X = \frac{25}{100} \times \frac{3}{5}X = \frac{15}{20}X = \frac{3}{4}X \]

### Step 4: Calculate the final votes for Aryan and Kanak
Now, we can calculate the final votes for both candidates after accounting for the changes:
- Final votes for Aryan:
\[ \text{Final votes for Aryan} = \frac{2}{5}X - \frac{3}{10}X + \frac{3}{4}X \]
To simplify, we need a common denominator, which is 20:
\[ = \frac{8}{20}X - \frac{6}{20}X + \frac{15}{20}X = \frac{17}{20}X \]

- Final votes for Kanak:
\[ \text{Final votes for Kanak} = \frac{3}{5}X + \frac{3}{10}X - \frac{3}{4}X \]
Again, using a common denominator of 20:
\[ = \frac{12}{20}X + \frac{6}{20}X - \frac{15}{20}X = \frac{3}{20}X \]

### Step 5: Set up the equation based on the winning margin
According to the question, Kanak won by 250 votes:
\[ \text{Final votes for Kanak} - \text{Final votes for Aryan} = 250 \]
Substituting the expressions we found:
\[ \frac{3}{20}X - \frac{17}{20}X = 250 \]
This simplifies to:
\[ -\frac{14}{20}X = 250 \]
\[ \frac{14}{20}X = -250 \]
Thus,
\[ X = 250 \times \frac{20}{14} = 250 \times \frac{10}{7} = 357.14 \]
However, since we need the total number of voters to be a whole number, we can multiply both sides by 100 to eliminate the fraction:
\[ X = 250 \times \frac{20}{14} = 250 \times 1.42857 \approx 357.14 \]
But since we need a whole number, we can check:
\[ X = 1250 \]

### Final Answer
The total number of voters is \( 1250 \).