ISEE Upper Level Mathematics Achievement Question of the Day

1. $\frac{1}{3}$ (correct answer)
2. $\frac{10}{30}$
3. $\frac{18}{30}$
4. $\frac{8}{30}$

Explanation: When you encounter problems about students studying multiple subjects, you're dealing with set theory and overlapping groups. The key is organizing the information to avoid double-counting students who belong to both categories. Let's break down what we know: 18 students study French, 16 study Spanish, and 8 study both languages. To find students who study "French but not Spanish," you need to subtract the overlap from the French total: 18 - 8 = 10 students study only French. The probability is therefore $\frac{10}{30} = \frac{1}{3}$, making answer A correct. Now let's examine why the other answers are wrong. Answer B gives $\frac{10}{30}$, which represents the correct number of students (10) but fails to simplify the fraction. While mathematically equivalent to $\frac{1}{3}$, it's not in simplest form. Answer C shows $\frac{18}{30}$, which would be the probability of studying French at all (including those who also study Spanish) – this ignores the "but not Spanish" requirement. Answer D gives $\frac{8}{30}$, which represents students studying both languages, not French exclusively. Study tip: Always draw a Venn diagram for overlap problems. Put the intersection number (8) in the middle first, then work outward to find the exclusive regions. This visual approach prevents the common mistake of forgetting to subtract the overlap when finding "but not" probabilities.