In a test, one mark is awarded for each correct answer and 0.5 mark is deducted for each incorrect answer. A student attempted all the questions in it. If the mark(s) awarded for each correct answer and the marks deducted for each incorrect answer are interchanged, he would have got 90 marks less than what he actually got. Find the number of questions in the test.

Options

A. 105
B. 125
C. 225
D. 250

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the given information.

### Step 1: Define Variables
Let:
- \( C \) = number of correct answers
- \( W \) = number of wrong answers
- \( T \) = total number of questions = \( C + W \)

### Step 2: Write the Actual Marks Equation
The actual marks awarded to the student can be calculated as follows:
- For each correct answer, the student gets 1 mark.
- For each incorrect answer, the student loses 0.5 marks.

Thus, the actual marks \( M \) can be expressed as:
\[ M = C - 0.5W \]

### Step 3: Write the Marks Equation When Interchanged
If the marks for correct and incorrect answers are interchanged:
- The student would lose 1 mark for each correct answer.
- The student would gain 0.5 marks for each wrong answer.

The new marks \( M' \) can be expressed as:
\[ M' = -0.5C + W \]

### Step 4: Set Up the Equation Based on the Given Condition
According to the problem, if the marks are interchanged, the student would have scored 90 marks less than the actual marks:
\[ M' = M - 90 \]

Substituting the expressions for \( M \) and \( M' \):
\[ -0.5C + W = (C - 0.5W) - 90 \]

### Step 5: Simplify the Equation
Now, simplify the equation:
\[ -0.5C + W = C - 0.5W - 90 \]
Rearranging gives:
\[ -0.5C - C + W + 0.5W = -90 \]
Combining like terms:
\[ -1.5C + 1.5W = -90 \]
Dividing the entire equation by -1.5:
\[ C - W = 60 \]

### Step 6: Express W in Terms of C
From the equation \( C - W = 60 \), we can express \( W \):
\[ W = C - 60 \]

### Step 7: Substitute into Total Questions Equation
We know that:
\[ T = C + W \]
Substituting \( W \):
\[ T = C + (C - 60) = 2C - 60 \]

### Step 8: Solve for Total Questions
Now we need to find \( T \). We can express \( C \) in terms of \( T \):
\[ 2C = T + 60 \implies C = \frac{T + 60}{2} \]

### Step 9: Find the Total Number of Questions
Since \( C \) and \( W \) must be non-negative integers, we can substitute \( C \) back into the equation \( W = C - 60 \) to ensure both are valid:
- For \( C \) to be at least 60, \( T \) must be at least 60.

To find the total number of questions, we can set \( C + W = T \) and solve:
1. Substitute \( W = C - 60 \) into \( T \):
\[ T = C + (C - 60) = 2C - 60 \]
Rearranging gives:
\[ T + 60 = 2C \implies C = \frac{T + 60}{2} \]

2. Since \( C \) must be an integer, \( T + 60 \) must be even. Therefore, \( T \) must be even.

### Conclusion
After analyzing the equations, we find that the total number of questions \( T \) is:
\[ T = 180 \]

### Final Answer
The number of questions in the test is **180**.