Group Questions
Group Questions
Occasional Math IIC questions will ask about groups with overlapping members. For example:
In a particular school, the school band has 42 members, and the school orchestra has 35 members. Seven students play in both the band and the orchestra, and 231 students play in neither the band nor the orchestra. How many students are in this particular school?
To answer this question, carefully count the students. 231 + 42 + 35 = 308 is a tempting answer, but in this solution, we are counting the students who play in both the band and orchestra twice. We must subtract 7 from this total to get the right answer: 308 – 7 = 301.
This question illustrates the formula for answering such questions. If two subgroups of a population share members, the equation that governs the total number of people in the population is:
The last term of this formula subtracts the elements that were double-counted earlier.
Try another example:
A room contains 80 people. Thirty have curly hair, 24 have blond hair, and 40 have hair that is neither curly nor blond. How many people in the room have curly, blond hair?
Use the formula: 80 = 30 + 24 + 40 – x. Thus, x = 14 (14 people in the room have curly and blond hair). This formula works for all group problems in which there are only two groups involved.
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