1. **State the problem:** There are 150 Grade 7 students. 64 registered in Math Trail, 78 in Amazing Race, and 28 in both events. We need to find how many registered only in Math Trail, only in Amazing Race, and how many did not register in either event. 2. **Use the principle of inclusion-exclusion:** The total number of students registered in at least one event is given by: $$|M \cup A| = |M| + |A| - |M \cap A|$$ where $|M|=64$, $|A|=78$, and $|M \cap A|=28$. 3. **Calculate the number of students registered in at least one event:** $$|M \cup A| = 64 + 78 - 28 = 114$$ 4. **Calculate students registered only in Math Trail:** $$|M \text{ only}| = |M| - |M \cap A| = 64 - 28 = 36$$ 5. **Calculate students registered only in Amazing Race:** $$|A \text{ only}| = |A| - |M \cap A| = 78 - 28 = 50$$ 6. **Calculate students who did not register in either event:** Total students = 150 Registered in at least one event = 114 So, $$\text{Not registered} = 150 - 114 = 36$$ **Final answers:** - Math Trail only: 36 students - Amazing Race only: 50 students - Did not register in either event: 36 students