Question 17 A group of 538 students were surveyed about the courses they were taking at their college with the following results: 336 students said they were taking Math. 268 students said they were taking English. 325 students said they were taking History. 174 students said they were taking Math and English. 178 students said they were taking Math and History. 136 students said they were taking English and History. 82 students said they were taking all three courses. Remember to create a Venn diagram What percentage of students said that they were taking Math? Question Help: Video Submit Question (rounding to the nearest tenth of a percent) How many students took English and History, but not Math? Stats

To solve the problem, we can use the principle of inclusion-exclusion and a Venn diagram to organize the information. ### Given Data: - Total students surveyed: \( N = 538 \) - Students taking Math: \( |M| = 336 \) - Students taking English: \( |E| = 268 \) - Students taking History: \( |H| = 325 \) - Students taking Math and English: \( |M \cap E| = 174 \) - Students taking Math and History: \( |M \cap H| = 178 \) - Students taking English and History: \( |E \cap H| = 136 \) - Students taking all three courses: \( |M \cap E \cap H| = 82 \) ### Step 1: Calculate the number of students taking only two subjects 1. **Math and English only**: \[ |M \cap E| - |M \cap E \cap H| = 174 - 82 = 92 \] 2. **Math and History only**: \[ |M \cap H| - |M \cap E \cap H| = 178 - 82 = 96 \] 3. **English and History only**: \[ |E \cap H| - |M \cap E \cap H| = 136 - 82 = 54 \] ### Step 2: Calculate the number of students taking only one subject 1. **Only Math**: \[ |M| - (|M \cap E| + |M \cap H| - |M \cap E \cap H|) = 336 - (174 + 178 - 82) = 336 - 270 = 66 \] 2. **Only English**: \[ |E| - (|M \cap E| + |E \cap H| - |M \cap E \cap H|) = 268 - (174 + 136 - 82) = 268 - 228 = 40 \] 3. **Only History**: \[ |H| - (|M \cap H| + |E \cap H| - |M \cap E \cap H|) = 325 - (178 + 136 - 82) = 325 - 232 = 93 \] ### Step 3: Summary of counts - Only Math: 66 - Only English: 40 - Only History: 93 - Math and English only: 92 - Math and History only: 96 - English and History only: 54 - All three: 82 ### Step 4: Calculate the percentage of students taking Math To find the percentage of students taking Math: \[ \text{Percentage of students taking Math} = \left( \frac{|M|}{N} \right) \times 100 = \left( \frac{336}{538} \right) \times 100 \approx 62.3\% \] ### Step 5: Calculate the number of students taking English and History, but not Math The number of students taking English and History but not Math is: \[ |E \cap H| - |M \cap E \cap H| = 136 - 82 = 54 \] ### Final Answers: 1. **Percentage of students taking Math**: **62.3%** 2. **Number of students taking English and History, but not Math**: **54**