2009 AMC 10A Problems/Problem 18
At Jefferson Summer Camp, of the children play soccer, of the children swim, and of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
Out of the soccer players, swim. As the soccer players are of the whole, the swimming soccer players are of all children.
The non-swimming soccer players then form of all the children.
Out of all the children, swim. We know that of all the children swim and play soccer, hence of all the children swim and don't play soccer.
Finally, we know that of all the children are non-swimmers. And as of all the children do not swim but play soccer, of all the children do not engage in any activity.
A quick summary of what we found out:
- : swimming yes, soccer yes
- : swimming no, soccer yes
- : swimming yes, soccer no
- : swimming no, soccer no
Now we can compute the answer. Out of all children, are non-swimmers, and again out of all children are non-swimmers that play soccer. Hence the percent of non-swimmers that play soccer is .
Let us set the total number of children as . So children play soccer, swim, and play soccer and swim.
Thus, children only play soccer.
So our numerator is .
Our denominator is simply
And so we get which is roughly
WLOG, let the total number of students be . Draw a venn diagram with 2 circles encompassing these 4 regions:
Non-soccer players, non-swimmers: 34 people
Soccer players, non-swimmers: 36 people
Soccer players, swimmers: 24 people
Non-soccer players, swimmers: 6 people.
Hence the answer is . We know this is a little bit larger than because .
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