Question

The number of students who take both the subjects mathematics and chemistry is 30. This represents 10% of the enrolment in mathematics and 12% of the enrolment in chemistry. How many students take at least one of these two subjects?

• A. \( 520 \)
• B. \( 490 \)
• C. \( 560 \)
• D. \( 480 \)
• E. \( 540 \)

Hint:

"At least one" is a keyword for the Union of sets. Always subtract the intersection (those taking both) to avoid counting them twice when you add the individual totals.

Answer 1

The Correct Option is A

Concept: In set theory, the number of elements in the union of two sets (students taking at least one subject) is given by the Principle of Inclusion-Exclusion: \[ n(M \cup C) = n(M) + n(C) - n(M \cap C) \] We first calculate the total enrolment for each subject individually.

Step 1: Calculate total enrolment for Mathematics and Chemistry.
Given $n(M \cap C) = 30$. - For Mathematics: $10% \text{ of } n(M) = 30 \Rightarrow 0.10 \times n(M) = 30 \Rightarrow n(M) = \frac{30}{0.10} = 300$. - For Chemistry: $12% \text{ of } n(C) = 30 \Rightarrow 0.12 \times n(C) = 30 \Rightarrow n(C) = \frac{30}{0.12} = 250$.

Step 2: Apply the Inclusion-Exclusion Principle.
To find students taking at least one subject ($n(M \cup C)$): \[ n(M \cup C) = n(M) + n(C) - n(M \cap C) \] \[ n(M \cup C) = 300 + 250 - 30 \] \[ n(M \cup C) = 550 - 30 = 520 \]