To solve the problem of how many students offer Biology but not Mathematics, we can use the principle of set theory. Let's break down the information given and find the answer step by step. ### Step 1: Understand the Information Given - Total number of students (B ∪ M) = 220 - Students offering Biology (B) = 125 - Students offering Mathematics (M) = 110 ### Step 2: Use the Formula for Union of Two Sets In set theory, the number of students who are in either Biology or Mathematics or both can be expressed using the formula for the union of two sets: \[ |B \cup M| = |B| + |M| - |B \cap M| \] Where: - \(|B \cup M|\) is the total number of students offering either Biology or Mathematics or both. - \(|B|\) is the number of students offering Biology. - \(|M|\) is the number of students offering Mathematics. - \(|B \cap M|\) is the number of students offering both subjects. ### Step 3: Substitute the Known Values We know: - \(|B \cup M| = 220\) - \(|B| = 125\) - \(|M| = 110\) Substituting these values into the formula gives us: \[ 220 = 125 + 110 - |B \cap M| \] ### Step 4: Solve for \(|B \cap M|\) Now, we can rearrange the equation to find \(|B \cap M|\): \[ 220 = 235 - |B \cap M| \] Subtract 235 from both sides: \[ 220 - 235 = -|B \cap M| \] \[ -15 = -|B \cap M| \] Thus, we find: \[ |B \cap M| = 15 \] This means that 15 students are taking both Biology and Mathematics. ### Step 5: Find Students Offering Only Biology To find the number of students who offer only Biology (not Mathematics), we can use the following formula: \[ |B \text{ only}| = |B| - |B \cap M| \] Substituting the known values: \[ |B \text{ only}| = 125 - 15 = 110 \] ### Conclusion The number of students who offer Biology but not Mathematics is **110**. ### Explanation of Other Options - **Option A (95)**: This is incorrect because it does not account for the total number of students offering Biology correctly. - **Option B (80)**: This is also incorrect as it underestimates the number of students offering only Biology. - **Option C (125)**: This option suggests that all Biology students do not take Mathematics, which contradicts the information that 15 students take both subjects. - **Option D (110)**: This is the correct answer, as calculated above. ### Revision Summary - Use the union formula for sets to find the total number of students in overlapping categories. - Rearranging equations is key to isolating variables in set theory problems. - Always double-check the interpretation of the problem to ensure you account for students in both categories correctly. - Remember to differentiate between students taking only one subject versus those taking both.