In an examination there are 30 questions. 1 mark is given for each correct answer and 0.25 is deducted for every incorrect answer. Ankur attempted all the questions and scored 13.75. How many incorrect answers did he have?

### Concept & Strategy This requires calculating the penalty cost per mistake. When calculating how many questions were wrong based on total marks, the "cost" of a mistake is the positive mark you didn't get plus the negative mark you received. $$ \text{Total Cost of a Mistake} = \text{Positive Mark} + \text{Negative Penalty} $$ ### Step-by-Step Solution * **Given:** Total questions = 30. Correct mark = +1. Penalty = -0.25. Actual score = 13.75. * **Assume all correct:** If Ankur got all 30 right, his score would be $30 \times 1 = 30$ marks. * **Calculate total marks lost:** Maximum score minus actual score. * Marks lost = $30 - 13.75 = 16.25$. * **Determine the cost per mistake:** Getting a question wrong costs the 1 mark he could have earned, plus a 0.25 penalty. * Cost per incorrect answer = $1 + 0.25 = 1.25$. * **Calculate number of incorrect answers:** Divide the total marks lost by the cost per mistake. * Incorrect answers = $\frac{16.25}{1.25}$. * Multiply numerator and denominator by 100 to remove decimals: $\frac{1625}{125}$. * Divide by 25: $\frac{65}{5} = 13$. * The number of incorrect answers is 13. ### Exam Strategy & Shortcut Avoid decimals by scaling the equation. Let $w$ be wrong answers. Equation: $(30 - w) - 0.25w = 13.75$. Multiply the entire equation by 4 to eliminate the 0.25 decimal. $4(30 - w) - w = 55$ $120 - 4w - w = 55$ $5w = 65 \rightarrow w = 13$. Since 13 is not in options (a) through (d), the answer is "None of these". ### Common Pitfall Blindly guessing when the calculated answer (13) isn't in the main options. Many students will re-calculate and try to force an arithmetic error to arrive at 12 or 15. Trust your math—this is exactly why "None of these" is an option. ### Final Answer **Therefore, the correct answer is None of these.**