Solving Test Question Problem

The problem describes a test with 20 questions and a specific scoring system. We are given the total score obtained by a candidate who had a few wrong answers and need to find the number of questions they attempted.

Let's define the variables:

• C = Number of correct answers
• W = Number of wrong answers
• U = Number of unattempted questions

The total number of questions is 20. So, the sum of correct, wrong, and unattempted questions must be 20:

\begin{equation*} C + W + U = 20 \end{equation*}

The scoring system is:

• Correct answer: +4 marks
• Wrong answer: -1 mark
• Unattempted question: 0 marks

The candidate secured a total score of 44 marks. The total score is calculated as:

\begin{equation*} (4 \times C) + (-1 \times W) + (0 \times U) = 44 \end{equation*}

This simplifies to:

\begin{equation*} 4C - W = 44 \end{equation*}

We are also told that the candidate had 'a few wrong answers', which means the number of wrong answers (W) must be greater than 0:

\begin{equation*} W > 0 \end{equation*}

From the score equation ($4C - W = 44$), we can express W in terms of C:

\begin{equation*} W = 4C - 44 \end{equation*}

Since $W > 0$, we must have:

\begin{equation*} 4C - 44 > 0 \end{equation*}

\begin{equation*} 4C > 44 \end{equation*}

\begin{equation*} C > 11 \end{equation*}

So, the number of correct answers must be an integer greater than 11.

The number of attempted questions is the sum of correct and wrong answers: $C + W$. Since the total questions are 20, the number of attempted questions cannot exceed 20.

Let's test integer values for C starting from 12 (since $C > 11$):

Case 1: If $C = 12$

Using the equation $W = 4C - 44$, we find W:

\begin{equation*} W = 4(12) - 44 = 48 - 44 = 4 \end{equation*}

In this case, $W=4$, which is an integer and $W > 0$. This is a valid number of wrong answers.

The number of attempted questions would be $C + W = 12 + 4 = 16$.

The number of unattempted questions would be $U = 20 - (C + W) = 20 - 16 = 4$.

Let's verify the total questions and total score with these values:

• Total Questions: $C + W + U = 12 + 4 + 4 = 20$ (Correct)
• Total Score: $4C - W = 4(12) - 4 = 48 - 4 = 44$ (Correct)

This solution fits all the conditions of the problem.

Case 2: If $C = 13$

Using the equation $W = 4C - 44$, we find W:

\begin{equation*} W = 4(13) - 44 = 52 - 44 = 8 \end{equation*}

In this case, $W=8$, which is an integer and $W > 0$. This is valid so far.

The number of attempted questions would be $C + W = 13 + 8 = 21$.

However, the total number of questions in the test is only 20. It is impossible to attempt 21 questions out of 20.

Therefore, $C=13$ is not a valid solution.

Any value of C greater than 13 will result in a number of attempted questions ($C+W$) that is even larger than 21, and thus also impossible given the total of 20 questions.

The only valid case is when $C=12$, which leads to $W=4$.

The number of attempted questions is $C + W = 12 + 4 = 16$.

Thus, the candidate attempted 16 questions.