Understanding the Exam Question and Marks Calculation

This problem involves calculating the number of correct answers in an exam based on the total questions, marking scheme, attempted questions, and final score. We are given that the exam has 80 questions. The marking scheme is +1 for a correct answer, -1 for a wrong answer, and 0 for an unattempted question. The student attempted only 80% of the questions and scored 32 marks. We need to determine how many questions were answered correctly.

Step-by-Step Problem Solving

Let's break down the problem into smaller, manageable steps to find the number of correctly answered questions.

Calculating the Number of Attempted Questions

The total number of questions in the exam is 80. The student attempted only 80% of the total questions.

Number of attempted questions = 80% of 80

In mathematical terms, this is: $$ \text{Attempted questions} = \frac{80}{100} \times 80 $$ $$ \text{Attempted questions} = 0.80 \times 80 $$ $$ \text{Attempted questions} = 64 $$

So, the student attempted a total of 64 questions. The remaining questions (80 - 64 = 16) were not attempted, and thus received 0 marks.

Setting Up Equations for Correct and Wrong Answers

Among the 64 attempted questions, some were answered correctly, and others were answered wrongly. Let's use variables to represent these:

• Let $C$ be the number of questions answered correctly.
• Let $W$ be the number of questions answered wrongly.

The total number of attempted questions is the sum of correct and wrong answers:

$$ C + W = 64 \quad (\text{Equation 1}) $$

Now, let's consider the marks obtained. The student received 1 mark for each correct answer and lost 1 mark for each wrong answer. The total score is 32 marks.

• Marks from correct answers = $C \times 1 = C$
• Marks from wrong answers = $W \times (-1) = -W$

The total marks obtained is the sum of marks from correct and wrong answers:

$$ C + (-W) = 32 $$ $$ C - W = 32 \quad (\text{Equation 2}) $$

Solving the System of Equations

We now have a system of two linear equations with two variables ($C$ and $W$):

1. $C + W = 64$
2. $C - W = 32$

We can solve this system using the elimination method. Adding Equation 1 and Equation 2 will eliminate $W$:

$$ (C + W) + (C - W) = 64 + 32 $$ $$ C + W + C - W = 96 $$ $$ 2C = 96 $$

Now, solve for $C$:

$$ C = \frac{96}{2} $$ $$ C = 48 $$

So, the number of questions answered correctly is 48.

We can also find the number of wrong answers ($W$) by substituting the value of $C$ into Equation 1:

$$ 48 + W = 64 $$ $$ W = 64 - 48 $$ $$ W = 16 $$

The student answered 48 questions correctly and 16 questions wrongly.

Verification

Let's check if these numbers yield the correct total marks:

• Marks from correct answers = $48 \times 1 = 48$
• Marks from wrong answers = $16 \times (-1) = -16$

Total marks = $48 + (-16) = 48 - 16 = 32$.

The calculated total marks match the given information, confirming our solution is correct.

The student answered 48 questions correctly.

Summary of Results

Conclusion on Correct Answers

Based on our calculations, the student answered 48 questions correctly out of the 64 attempted questions in the exam.

Revision Table: Exam Marking Scheme

Additional Information: Solving Equations

The method used to solve for the number of correct and wrong answers involved setting up and solving a system of two linear equations. This is a common technique in quantitative problems. The two equations represented:

• The total count of attempted questions ($C + W = \text{Total attempted}$).
• The relationship between correct/wrong answers and the final score ($C \times (\text{marks per correct}) + W \times (\text{marks per wrong}) = \text{Total score}$).

In this specific problem, the marks per correct answer were +1 and per wrong answer were -1, simplifying the second equation to $C - W = \text{Total score}$. Solving such systems can be done through substitution or elimination methods, both leading to the same unique solution for $C$ and $W$.