In an election `20%` of population do not caste the vote and `120` votes are invalid winning candidate gets 200 votes more than his competitor and get `41%` votes of toal vote. Then how much percentage of casted votes looser candidate got ?

Options

A. `35`
B. `45`
C. `65.5`
D. `85`

Answer

To solve the problem step by step, let's break down the information given and calculate the required percentage of votes that the loser candidate received.

### Step 1: Determine the total population
Let’s assume the total population is \( P \).

### Step 2: Calculate the number of people who cast their votes
Since \( 20\% \) of the population did not cast their vote, the percentage of the population that did cast their vote is \( 80\% \). Therefore, the number of people who cast their votes is:
\[ \text{Votes cast} = 0.8P \]

### Step 3: Calculate the total valid votes
From the total votes cast, we need to subtract the invalid votes. We know there are \( 120 \) invalid votes. Thus, the total valid votes are:
\[ \text{Valid votes} = \text{Votes cast} - \text{Invalid votes} = 0.8P - 120 \]

### Step 4: Determine the votes received by the winning candidate
The winning candidate received \( 41\% \) of the total valid votes. Therefore, the votes received by the winning candidate can be expressed as:
\[ \text{Votes for winner} = 0.41 \times (\text{Valid votes}) = 0.41 \times (0.8P - 120) \]

### Step 5: Determine the votes received by the losing candidate
We know that the winning candidate received \( 200 \) votes more than the losing candidate. Let’s denote the votes received by the losing candidate as \( L \). Therefore, we can write:
\[ \text{Votes for winner} = L + 200 \]

### Step 6: Set up the equation
From the previous steps, we can set up the equation:
\[ 0.41 \times (0.8P - 120) = L + 200 \]

### Step 7: Express \( L \) in terms of \( P \)
Substituting \( L \) from the equation gives:
\[ L = 0.41 \times (0.8P - 120) - 200 \]

### Step 8: Calculate the total valid votes
Now, we need to express \( L \) in terms of \( P \) and solve for \( P \). We also know that the total valid votes can be expressed as:
\[ L + (L + 200) = 0.8P - 120 \]
This simplifies to:
\[ 2L + 200 = 0.8P - 120 \]

### Step 9: Substitute \( L \) from the previous equation
Substituting \( L \) into this equation gives:
\[ 2\left(0.41 \times (0.8P - 120) - 200\right) + 200 = 0.8P - 120 \]

### Step 10: Solve for \( P \)
Now, we can solve for \( P \) to find the total population. After solving, we can find the number of votes for the loser candidate \( L \).

### Step 11: Calculate the percentage of votes received by the loser candidate
Finally, to find the percentage of cast votes that the loser candidate received, we can use:
\[ \text{Percentage of votes for loser} = \left(\frac{L}{\text{Valid votes}}\right) \times 100 \]