In a group of learners a total of 54 take history, 48 take geography and 12 do not take history or geography. If 16 learners take both history and geography what is the total number of learners in the group?
Learners in a group
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0i got an answer of 98 – 2012-03-21
1 Answers
Hint: First find the number of learners who take only history and the number of learners that take only geography. Then note the total number of learners can be split amongst 4 mutually exclusive classes: those taking only history, those taking only geography, those taking both subjects, and those taking neither subject.
A Venn diagram approach to this problem would be useful. Here you would have 2 sets, say $\color{maroon}H$ for history and $\color{darkgreen}G$ for geography (both inside your "universal set" of all learners).
There are four regions in the diagram corresponding to four classes mentioned above. Write the number of learners in the $H\cap G$ region first. This is region $\color{gray}I$ below; these are the learners who take both subjects.
Then write the number of learners in the $H\setminus G$ region. This is region $\color{maroon}{II}$ below, the number of learners who take only history. Note the number here is not 54, since you have already accounted for 16 of them when you considered region $\color{gray}I$...
Fill in the sizes for the other two regions. Then you can find the total number of learners by adding the sizes of the four regions.
An alternate approach would be to just add things up; but here, we need to be careful:
Just adding $54+48+12+16$ would not give the correct answer. In this sum, we added the 16 students who take both subjects three times (16 of the 48 students taking history also take geography, same deal for the geography students). But we can correct this by subtracting 32 to account for the over counting. The total number of students is $ 54+48+12+16 -32= 54+48 +12-16 = 98. $
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0@Daniel $54-16=38$ take history only. $48-16=32$ take geography only. 12 take neither and 16 take both. So the total number of learners is $38+32+12+16=98$. – 2012-03-21