Question

In a competitive examination, 1 mark is awarded for each correct answer while \(\frac{1}{2}\) mark is deducted for each wrong answer. If a student answered 120 questions and got 90 marks, then the number of questions that the student answered correctly is

• A. 90
• B. 100
• C. 110
• D. None of these

Answer 1

The Correct Option is B

Let:

\(x\) be the number of correct answers.

\(y\) be the number of wrong answers. 

Step 1: Forming equations

Given that the total number of questions answered is:

\(x + y = 120\)  ...(1)

Marks are awarded as follows:

- Each correct answer gives \(+1\) mark.

- Each wrong answer results in a deduction of \(\frac{1}{2}\) mark.

Total marks obtained:

\(x - \frac{1}{2}y = 90\)  ...(2)

Step 2: Solve for \(x\) and \(y\)

From equation (1):

\(y = 120 - x\)

Substituting in equation (2):

\(x - \frac{1}{2}(120 - x) = 90\)

\(x - 60 + \frac{x}{2} = 90\)

\(\frac{3x}{2} = 150\)

\(3x = 300\)

\(x = 100\)

Final Answer: 100

Answer 2

To solve the problem, let's define variables to represent the number of correct and incorrect answers. 
Let \( x \) be the number of correct answers and \( y \) be the number of incorrect answers. 
According to the problem, the student answered a total of 120 questions: 
\( x + y = 120 \) 
Additionally, the student received 90 marks. 
Since each correct answer gives 1 mark and each incorrect answer results in a deduction of \(\frac{1}{2}\) mark, the equation for the total marks is:
\( x - \frac{1}{2}y = 90 \) 
Now, we have a system of linear equations:
\( \begin{cases} x + y = 120 \\ x - \frac{1}{2}y = 90 \end{cases} \) 
We can solve these equations by substituting \( y \) from the first equation into the second equation. From \( x + y = 120 \), we get:
\( y = 120 - x \) 
Substitute this value of \( y \) into the second equation:
\( x - \frac{1}{2}(120-x) = 90 \) 
Simplify the equation:
\( x - 60 + \frac{1}{2}x = 90 \) 
\( \frac{3}{2}x - 60 = 90 \) 
Add 60 to both sides:
\( \frac{3}{2}x = 150 \) 
Multiply both sides by \(\frac{2}{3}\):
\( x = 100 \) 
Therefore, the number of questions that the student answered correctly is 100.