1. **Stating the problem:** You have a question paper originally out of 28 marks, subdivided into three categories: 8, 10, and 10 marks. You want to convert this paper to a total out of 20 marks, with new subdivisions of 10, 5, and 5 marks respectively. 2. **Understanding the conversion:** The original total is 28 marks, and the new total is 20 marks. Each category's marks need to be scaled proportionally to fit the new subdivisions. 3. **Formula used:** To convert marks from the original category to the new category, use the formula: $$\text{New marks} = \frac{\text{Original marks}}{\text{Original category max}} \times \text{New category max}$$ 4. **Applying the formula to each category:** - For the first category (original max 8, new max 10): $$\text{New marks}_1 = \frac{\text{Original marks}_1}{8} \times 10$$ - For the second category (original max 10, new max 5): $$\text{New marks}_2 = \frac{\text{Original marks}_2}{10} \times 5$$ - For the third category (original max 10, new max 5): $$\text{New marks}_3 = \frac{\text{Original marks}_3}{10} \times 5$$ 5. **Explanation:** This method ensures each category's marks are scaled proportionally to the new maximum marks assigned. 6. **Example:** If a student scored 6 in the first category, 7 in the second, and 8 in the third: - New marks in category 1: $$\frac{6}{8} \times 10 = 7.5$$ - New marks in category 2: $$\frac{7}{10} \times 5 = 3.5$$ - New marks in category 3: $$\frac{8}{10} \times 5 = 4$$ Total new marks: $$7.5 + 3.5 + 4 = 15$$ This way, the total is out of 20 with the new subdivisions.