A test consists of 20 questions and the students are awarded 4 marks for a correct answer. -1 mark for a wrong answer and 0 mark for an unattempted question. Which of the following could be a possible number of questions attempted by a student who secured 27 marks?
18
Understanding the Test Scoring
The question describes a test with a specific scoring system. There are a total of 20 questions. Marks are awarded based on whether a question is answered correctly, wrongly, or not attempted.
- Correct Answer: +4 marks
- Wrong Answer: -1 mark
- Unattempted Question: 0 marks
A student secured a total score of 27 marks. We need to find which of the given options could be the possible number of questions attempted by the student.
Setting up the Equations
Let's define the variables:
- \(Q_{total}\) = Total number of questions = 20
- \(Q_{attempted}\) = Number of questions attempted = \(A\)
- \(Q_{unattempted}\) = Number of questions unattempted = \(U\)
- \(Q_{correct}\) = Number of correct answers = \(C\)
- \(Q_{wrong}\) = Number of wrong answers = \(W\)
From the definitions, we know that the total number of questions is the sum of attempted and unattempted questions:
\(Q_{total} = Q_{attempted} + Q_{unattempted}\)
\(20 = A + U\)
The number of attempted questions is the sum of correct and wrong answers:
\(Q_{attempted} = Q_{correct} + Q_{wrong}\)
\(A = C + W\)
The total score is calculated from the number of correct, wrong, and unattempted questions:
Total Score = \((Q_{correct} \times 4) + (Q_{wrong} \times -1) + (Q_{unattempted} \times 0)\)
We are given that the total score is 27. So,
\(27 = (C \times 4) + (W \times -1) + (U \times 0)\)
\(27 = 4C - W + 0\)
\(4C - W = 27\)
From the equation \(A = C + W\), we can express \(W\) as \(W = A - C\). Substitute this into the score equation:
\(4C - (A - C) = 27\)
\(4C - A + C = 27\)
\(5C - A = 27\)
This equation \(5C - A = 27\) relates the number of correct answers (\(C\)) to the number of attempted questions (\(A\)) for a total score of 27.
Remember that \(C\) and \(W\) must be non-negative integers (\(C \ge 0\), \(W \ge 0\)). Since \(A = C + W\), this also means \(C \le A\).
Checking the Possible Number of Questions Attempted (A)
We are given several options for the number of attempted questions (\(A\)). We will test each option using the equation \(5C - A = 27\) to see if it yields a valid number of correct answers (\(C\)), which must be a non-negative integer and less than or equal to \(A\).
| Option (A) | Equation: \(5C - A = 27\) | Solving for \(C\) | Is \(C\) a non-negative integer? | Is \(C \le A\)? | Possible? |
|---|---|---|---|---|---|
| 15 | \(5C - 15 = 27\) | \(5C = 42 \implies C = 42/5 = 8.4\) | No | N/A | No |
| 16 | \(5C - 16 = 27\) | \(5C = 43 \implies C = 43/5 = 8.6\) | No | N/A | No |
| 17 | \(5C - 17 = 27\) | \(5C = 44 \implies C = 44/5 = 8.8\) | No | N/A | No |
| 18 | \(5C - 18 = 27\) | \(5C = 45 \implies C = 45/5 = 9\) | Yes | Yes (\(9 \le 18\)) | Yes |
Analyzing the Results
From the table, only when the number of attempted questions (\(A\)) is 18, do we get a valid integer value for the number of correct answers (\(C\)), which is 9.
Let's check the details for \(A=18\):
- Attempted Questions (\(A\)) = 18
- Correct Answers (\(C\)) = 9
- Wrong Answers (\(W\)) = \(A - C = 18 - 9 = 9\)
- Unattempted Questions (\(U\)) = \(Q_{total} - A = 20 - 18 = 2\)
Let's calculate the score with these numbers:
Score = \((C \times 4) + (W \times -1) + (U \times 0)\)
Score = \((9 \times 4) + (9 \times -1) + (2 \times 0)\)
Score = \(36 - 9 + 0\)
Score = \(27\)
This matches the given total score. Therefore, 18 is a possible number of questions attempted by the student.
Conclusion
Based on the analysis of the scoring system and the options provided, only attempting 18 questions allows for a combination of correct and wrong answers that results in a total score of 27 while adhering to the constraints of the problem.
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