Of the seniors, some study French (F), some study Spanish (S), and some both. If 45 study at least one of the two languages, 20 study Spanish, and 12 study both, how many study French?

Rationale

37 seniors study French.

To determine how many seniors study French, we can use the principle of inclusion-exclusion. We know that 45 seniors study at least one language, 20 study Spanish, and 12 study both languages. By calculating the number of students studying only Spanish and then finding the remaining students who study French, we arrive at the answer.

A (25) YOUR ANSWER

If 25 seniors studied French, it would imply that 13 seniors study only French. However, this calculation does not account for the overlap correctly, as the total students studying at least one language would then exceed 45. Thus, this choice does not satisfy the given conditions.

B (33) YOUR ANSWER

Selecting 33 seniors studying French would imply that 21 study only French. When combined with the 20 studying Spanish and the 12 studying both, the total would again exceed 45, suggesting that not all conditions are met. Therefore, this choice is incorrect.

C (37) CORRECT

This choice accurately indicates that 37 seniors study French, allowing for 25 students who study only French (37 - 12) and 12 who study both languages. When combined with the 20 studying Spanish, the total is 45 (25 only French + 8 only Spanish + 12 both), confirming that all conditions are satisfied.

D (41) YOUR ANSWER

If 41 seniors were studying French, it would imply that at least 29 study only French. When added to the 20 studying Spanish and 12 studying both, the total would exceed 45, contradicting the stated numbers. Thus, this option cannot be correct.

Conclusion

In summary, the calculations reveal that 37 seniors study French, aligning perfectly with the problem's constraints. By applying the principle of inclusion-exclusion, we ensure that the total number of students studying at least one language remains consistent with the given data, confirming that 37 is the only viable solution.

Of the seniors, some study French (F), some study Spanish (S), and some both. If 45 study at least one of the two languages, 20 study Spanish, and 12 study both, how many study French?

37 seniors study French.

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