how many play both?

AndrewTom

Out of $220$ children, $163$ play rugby, $175$ play football and $24$ play neither. How many play both?

**Posted:** Sun Nov 01, 2009 5:11 am

AIME15USAMO

$220 - 24 = 196$ people play either rugby or football. $163 + 175 = 338$ is the number of people that play football+the number of people who play rugby. Since we over-counted, we do $338 - 196 = \boxed {142}$ play both.

**Posted:** Sun Nov 01, 2009 5:48 am

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Dojo

We can use a venn diagram to display this data:

draw(circle((0,0),20),linewidth(1)+red);
draw(circle((19,0),20),linewidth(1)+blue);

label("Rugby",(0,20),N);
label...

These four values add to 220 total kids. Solve for $x$ , and you have your answer.

**Posted:** Mon Nov 02, 2009 6:03 pm

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limac

(Here's a generalization)

Basically, for two sets: if you have two sets let's say $A$ and $B,$ you can count the items in their union by using PIE:

$\#(A \cup B) = \#(A) + \#(B) + \#(A \cap B)$

Here $\#(X)$ denotes the number of elements in set $X.$

Here you are basically just adding up the number of elements in set $A$ , and set $B$ , but you have to subtract off the elements that are common to both, because those elements are counted twice, once for set $A$ and once for set $B$ . Therefore, you need to subtract those elements once.

Using this method, you can find any of the four things, given (or hinted) the other three.

**Posted:** Mon Nov 02, 2009 8:02 pm

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steve123456

You simply need to do 163+175+24-220 to find the answer. That equals 142. (Because 163 play rugby, 175 play football, and 24 play neither. If x is the number that play both, then 163-x+175-x+x+24=220)

**Posted:** Wed Nov 04, 2009 5:56 pm