A question paper has 20 questions, the answer to each of which is either true or false. A student scores 3 marks for every correct answer and loses 2 marks for every incorrect answer. If a student scores 25 marks, find the number of questions he answered correctly and the questions he answered incorrectly.

Answer

To solve the problem, we will set up two equations based on the information given in the question.

**Step 1: Define Variables**
Let:
- \( x \) = number of correct answers
- \( y \) = number of incorrect answers

**Step 2: Set Up the Equations**
From the problem, we know:
1. The total number of questions is 20. Therefore, we can write the first equation as:
\[ x + y = 20 \quad \text{(Equation 1)} \]

2. The scoring system states that a student earns 3 marks for each correct answer and loses 2 marks for each incorrect answer. If the student scores 25 marks, we can write the second equation as:
\[ 3x - 2y = 25 \quad \text{(Equation 2)} \]

**Step 3: Solve the First Equation for One Variable**
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 20 - x \quad \text{(Substituting into Equation 1)} \]

**Step 4: Substitute into the Second Equation**
Now, substitute \( y \) in Equation 2:
\[ 3x - 2(20 - x) = 25 \]
Expanding this gives:
\[ 3x - 40 + 2x = 25 \]
Combine like terms:
\[ 5x - 40 = 25 \]

**Step 5: Solve for \( x \)**
Add 40 to both sides:
\[ 5x = 65 \]
Now, divide by 5:
\[ x = 13 \]

**Step 6: Find \( y \)**
Now that we have \( x \), we can find \( y \) using the first equation:
\[ y = 20 - x = 20 - 13 = 7 \]

**Final Answer**
The student answered:
- Correct answers (\( x \)): 13
- Incorrect answers (\( y \)): 7

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