Occasionally, the Math IC will pose questions about groups
with overlapping members. For example:
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:
Total Population = Group A Population + Group
B Population + Neither Group A nor B population – Group A and B
The last term of this formula subtracts the elements that
were double-counted earlier.
Try another example:
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 will work for all group problems,
as long as there are only two groups involved.