# Counting Problem: In a school 315 girls play at least one sports

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• Avro1
In summary, there are 75 girls who play exactly two sports, with a total of 315 girls playing at least one sport. Of those, 100 play a fall sport, 150 play a winter sport, and 200 play a spring sport. Using the formula provided, we can find that $|A\cap B\cap C| = 40$, meaning that 40 girls play all three sports.
Avro1
In a school 315 girls play at least one sport. 100 play a fall sport, 150 play a winter sport, and 200 play a spring sport. If 75 girls play exactly 2 sports, how many play three?

Hello, and welcome to MHB! (Wave)

I would begin by constructing a Venn diagram:

View attachment 9104

We've got 7 variables...can you construct equations involving these variables from the given information?

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Hi Avro.

You can also use this formula for any sets $A$, $B$, $C$:
$$|A\cup B\cup C|\ =\ |A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|.$$
So, in this problem, $A$ might be the set of girls playing fall sports, $B$ the set of those playing winter sports, and $C$ the set of those playing spring sports; then you want to find $|A\cap B\cap C|$. Also, note that while you are not given $|A\cap B|$, $|B\cap C|$ or $|C\cap A|$ separately, you are given $|A\cap B|+|B\cap C|+|C\cap A|$, which you can use in the formula above

## 1. How many girls play sports in the school?

There are 315 girls who play at least one sport in the school.

## 2. What percentage of girls in the school play sports?

This cannot be determined without knowing the total number of girls in the school.

## 3. How many sports do the girls play in the school?

This cannot be determined without knowing how many sports are offered in the school.

## 4. Are there any girls who do not play sports in the school?

It is possible that there are girls who do not play sports in the school, but this information cannot be determined from the given information.

## 5. How does the number of girls who play sports compare to the number of boys?

This cannot be determined without knowing the number of boys who play sports in the school.

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