In a university election of two candidates, `80%` of the students cast their votes out of which 120 votes are invalid. The winner gets `37.5%` of the total students votes and won by only 30 votes. Then find the number of students who don’t cast their votes?

Options

A. 360
B. 320
C. 340
D. 300

Answer

To solve the problem step by step, we need to analyze the information given in the question and use it to find the number of students who did not cast their votes.

### Step 1: Define the total number of students
Let the total number of students be \( x \).

### Step 2: Calculate the number of students who cast their votes
According to the question, \( 80\% \) of the students cast their votes. Therefore, the number of students who cast their votes is:
\[ 0.8x \]

### Step 3: Determine the number of valid votes
Out of the votes cast, \( 120 \) votes are invalid. Thus, the number of valid votes is:
\[ 0.8x - 120 \]

### Step 4: Calculate the votes received by the winner
The winner receives \( 37.5\% \) of the total votes. Therefore, the number of votes received by the winner is:
\[ 0.375 \times (0.8x) = 0.3x \]

### Step 5: Calculate the votes received by the loser
Since there are only two candidates, the votes received by the loser can be calculated as:
\[ \text{Votes by loser} = \text{Total valid votes} - \text{Votes by winner} \]
\[ = (0.8x - 120) - 0.3x = 0.5x - 120 \]

### Step 6: Set up the equation based on the winning margin
The winner won by \( 30 \) votes. Therefore, we can set up the equation:
\[ \text{Votes by winner} - \text{Votes by loser} = 30 \]
Substituting the values we have:
\[ 0.3x - (0.5x - 120) = 30 \]

### Step 7: Simplify the equation
Now, simplify the equation:
\[ 0.3x - 0.5x + 120 = 30 \]
\[ -0.2x + 120 = 30 \]

### Step 8: Solve for \( x \)
Rearranging gives:
\[ -0.2x = 30 - 120 \]
\[ -0.2x = -90 \]
\[ x = \frac{-90}{-0.2} = 450 \]

### Step 9: Calculate the number of students who did not cast their votes
The number of students who did not cast their votes is \( 20\% \) of the total students:
\[ 0.2x = 0.2 \times 450 = 90 \]

### Final Answer
The number of students who did not cast their votes is **90**.