1. **Problem statement:** In a class of 30 students, 30 like Math, 25 like Science, and 10 like neither subject. We need to find: - Number of students who like both subjects. - Number of students who like only Math. - Number of students who like only one subject. 2. **Understanding the problem:** - Total students = 30 - Students liking Math = 30 - Students liking Science = 25 - Students liking none = 10 3. **Formula used:** Let $M$ = set of students who like Math, $S$ = set of students who like Science. We know from set theory: $$|M \cup S| = |M| + |S| - |M \cap S|$$ Also, total students = $|M \cup S| +$ students who like none. 4. **Calculate $|M \cup S|$:** $$|M \cup S| = 30 - 10 = 20$$ because 10 students like none, so 20 students like at least one subject. 5. **Find number of students who like both subjects $|M \cap S|$:** Using the formula: $$|M \cup S| = |M| + |S| - |M \cap S|$$ Substitute known values: $$20 = 30 + 25 - |M \cap S|$$ Simplify: $$20 = 55 - |M \cap S|$$ Rearranged: $$|M \cap S| = 55 - 20 = 35$$ This is impossible since total students are 30, so we must re-examine the problem statement. **Note:** The problem states "In a class of 30 students, 30 students like Math," which is contradictory because total students are 30 but 30 like Math and 25 like Science, and 10 like none. This suggests the total number of students is more than 30 or the numbers are inconsistent. Assuming the total number of students is 30, and 10 like none, then 20 like at least one subject. But 30 like Math and 25 like Science cannot be true simultaneously. **Assuming the problem meant:** - Total students = 50 - 30 like Math - 25 like Science - 10 like none Then total students liking at least one subject: $$|M \cup S| = 50 - 10 = 40$$ Now calculate $|M \cap S|$: $$40 = 30 + 25 - |M \cap S|$$ $$|M \cap S| = 30 + 25 - 40 = 15$$ 6. **Find number of students who like only Math:** $$|M \text{ only}| = |M| - |M \cap S| = 30 - 15 = 15$$ 7. **Find number of students who like only one subject:** $$|M \text{ only}| + |S \text{ only}|$$ Calculate $|S \text{ only}|$: $$|S \text{ only}| = |S| - |M \cap S| = 25 - 15 = 10$$ Sum: $$15 + 10 = 25$$ **Final answers:** - Number of students who like both subjects = 15 - Number of students who like only Math = 15 - Number of students who like only one subject = 25