SOLUTION: A group of students contains the following: 18 who like math, 32 who like English, 25 like language, 3 who like all 3 subjects, 8 who like both math and English, 16 who like both E

Question 188338: A group of students contains the following: 18 who like math, 32 who like English, 25 like language, 3 who like all 3 subjects, 8 who like both math and English, 16 who like both English and language, and 7 who like both math and language. Everyone likes at least one subject. How many students are there in the overall group?

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Each circle represents a subject, Math, English, and Language. Write the number 3 in the little area in the center, because that is where the three circles all overlap and you have 3 people who like all three subjects.

Then you are told that 8 people like both math and English -- but you aren't told how these 8 people feel about language, so since there are 3 people who like all three subjects, they must be included in the 8 who like both math and English, leaving us with 8 - 3 = 5 people who like math and English but not language. So write a 5 in the segment where the E and M circles overlap but not the L circle.

Your diagram should look like this:

Now, use similar logic to fill in the proper numbers for the other two overlap areas. Once you have done that, the four segments that make up each circle, that is the three overlap areas and the main part of the circle must add up to the total number of students who like that subject. So you total the three overlap areas and subtract from the given totals, 18 for M, 32 for E, and 25 for L.

Once you have all of the regions filled in, the sum of the numbers in all the regions is the total number of students.

John

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