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of the 911 girls at East Statistics H.S. last year, 98 played fall sports, 91 played winter sports, and 84 played spring sports. 22 girls played sports all 3 seasons while 30 played only in the fall, 20 played only in the winter, and 10 played only in the spring. how many girls did not play sports in any of the 3 seasons?

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Jan 29, 2010
Sports Participation
by: Staff

Note ERROR - Data given for the problem contains an error

There is a problem with the head count for the number of girls participating in sports which is given in the problem.

The annual head count of the number of girls participating in sports is FALL+WINTER+SPRING = 98+91+84 = 273.

273 is an odd number.

However, an odd number makes the other data given invalid. Or . . . the other data given makes the number 273 invalid. One or the other is invalid.



That said, here is how the problem can be solved. Note that the solution is not realistic because it involves ½ a person.



The solution



1. Use the total headcount to understand the problem:

Headcount of girls participating 1 season + Headcount of girls participating 2 seasons + Headcount of girls participating 3 seasons = Total Headcount for the year (FALL+WINTER+SPRING)


2. Enter known values:

Headcount for girls participating 1 season = 30 + 20 + 10
The total of the three seasons is counted only once in the yearly headcount.

Headcount for girls participating 2 seasons = unknown
The number of girls who participate for only 2 seasons is counted twice in the yearly headcount. The unknown value can be represented as 2x (where “x” represents the number of individual girls who participate for only 2 seasons).

Headcount for girls participating 3 seasons = (3)(22)
The 22 girls must be counted 3 times in the yearly headcount since they participate 3 times.

The total headcount for the year = 98 + 91 + 84
FALL+WINTER+SPRING = 98+91+84


The final equation would be:

Headcount of girls participating 1 season + Headcount of girls participating 2 seasons + Headcount of girls participating 3 seasons = Total Headcount for the year (FALL+WINTER+SPRING)


(30 + 20 + 10) + (2x) + (3)(22) = (98 + 91 + 84)


2. Solve the equation for the number of girls participating 2 seasons

(30 + 20 + 10) + (2x) + (3)(22) = (98 + 91 + 84)

60 + 2x + 66 = 273

2x + 66 + 60 = 273

2x + 126 = 273

2x + 126 - 126 = 273 - 126

2x + 0 = 147

2x = 147

(2x)/2 = 147/2

x = 73.5 individual girls participate in 2 seasons
(since a decimal of .5 girls is not possible in real life, then one of the headcount values given in the statement of the problem is incorrect)




3. Calculate the total number of individual girls participating in sports for the year


Count each individual girl only once.

Total number of individual girls participating 1 season only + Total number of individual girls participating 2 seasons only + Total number of individual girls participating in all 3 seasons = Total number of individual girls who participated in sports during the year.


(30 + 20 + 10) + (73.5) + (22) = 155.5


4. Calculate the total number of unique girls who did NOT participate in any sports for the year


911 – 155.5 = 755.5

The final answer is: 755.5 girls did not participate in any sports in any of the 3 seasons.
(½ a girl is impossible in real life)



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