31. The 25 students in a sixth-grade class each reported whether or not they have at least one dog, at least one cat, or at least one parakeet. The results are shown in the Venn diagram above, with two of the numbers represented by x and y. If all of the students have at least one of these pets, what is the value of x + y?
Answer: C
x + y = 8
The value of x + y is determined by analyzing the Venn diagram data, which indicates the total number of students with pets. Given that all 25 students have at least one of the pets, the sum of the numbers represented by x and y in the diagram must equal 8 to satisfy the conditions of the problem.
A) 6
Option A is incorrect because if x + y equals 6, that would imply that there are 19 students accounted for by other pet combinations, which contradicts the total of 25 students having pets. Thus, this option does not satisfy the problem's requirements.
B) 7
Option B is also incorrect as it suggests that x + y equals 7. This would leave 18 students accounted for by other combinations, which again does not fulfill the total of 25 students reported in the class. Therefore, this option cannot be valid.
C) 8
Option C is correct because x + y equaling 8 ensures that the remaining 17 students have combinations of pets that meet the total of 25. This calculation aligns perfectly with the requirement that all students have at least one pet, making this option consistent with the information provided.
D) 10
Option D is incorrect since x + y equaling 10 would imply that 15 students have other pet combinations. This would also not satisfy the total of 25 students with pets, making this choice untenable in the context of the problem.
E) 14
Option E is incorrect as well, because if x + y were 14, then only 11 students would be accounted for by other combinations. This would also contradict the condition that all students must have at least one pet, further invalidating this option.
Conclusion
The correct answer, C (x + y = 8), accurately reflects the total number of students conforming to the conditions of the problem, while all other options fail to meet the necessary total of 25 students. Thus, only option C aligns with the requirements established by the Venn diagram analysis.