Question

A test has 50 questions. A student scores 1 mark for a correct answer, \(-1/3\) for a wrong answer, and \(-1/6\) for not attempting a question. If the net score of a student is 32, the number of questions answered wrongly by that student cannot be less than

• A. 6
• B. 12
• C. 3
• D. 9

Hint:

Write one equation for the number of questions and one for the net score, then remove the unattempted count. Only a couple of whole number cases will survive.

Answer 1

The Correct Option is C

Step 1: Name the unknowns.
Let \(c\) be the number of correct answers, \(w\) the number of wrong answers and \(u\) the number of questions left blank. All three are whole numbers that are zero or more.
There are 50 questions, so \(c + w + u = 50\).

Step 2: Write the score equation.
Correct gives \(+1\), wrong gives \(-\frac{1}{3}\) and unattempted gives \(-\frac{1}{6}\). The net score is 32:
\[ c - \frac{w}{3} - \frac{u}{6} = 32 \]
Multiplying by 6 clears the fractions:
\[ 6c - 2w - u = 192 \]

Step 3: Eliminate \(u\).
From the first equation \(u = 50 - c - w\). Put it in:
\[ 6c - 2w - (50 - c - w) = 192 \]
\[ 7c - w = 242 \Rightarrow w = 7c - 242 \]

Step 4: Use the limits on the variables.
A wrong count cannot be negative, so \(7c - 242 \geq 0\), which gives \(c \geq 34.57\). Since \(c\) is a whole number, \(c \geq 35\).
Also \(c + w \leq 50\) because \(u \geq 0\). That gives \(c + 7c - 242 \leq 50\), so \(8c \leq 292\) and \(c \leq 36.5\). So \(c \leq 36\).
So \(c\) can only be 35 or 36.

Step 5: List the two possible cases.
If \(c = 35\): \(w = 7(35) - 242 = 3\) and \(u = 50 - 35 - 3 = 12\). Score: \(35 - 1 - 2 = 32\). This works.
If \(c = 36\): \(w = 7(36) - 242 = 10\) and \(u = 4\). Score: \(36 - \frac{10}{3} - \frac{4}{6} = 36 - 4 = 32\). This also works.
So the student answered either 3 or 10 questions wrongly.

Step 6: Pick the least and compare with the options.
The smallest possible number of wrong answers is 3, so the number of wrong answers cannot be less than 3.
Options 6, 12 and 9 are not the lower bound because 3 is actually possible, which is smaller than all of them.

Final Answer:
The least possible number of wrongly answered questions is 3, which is option 3. \[ \boxed{3} \]