Also, If the Yoga club has 42 members and the Sports club has 25 members and there are 18 students that belong to both clubs, how many people total will be at a joint meeting of both clubs, if all members attended? Steps please?
let x=those only in yoga club, y=those in both, z in only sports. Then y=18, x+y=42, y+z=25. So then x=24 and z=7. That means x+y+z=49
@avneety
Number of members in both clubs = 42 + 25 = 67
However this counts the students in both clubs twice, so we have to subtract 18.
==> Number of members in either or both clubs = 42 + 25 - 18 = 49, so the answer is 49.
This is known as the principle of inclusion-exclusion which is a very useful topic to know for the SAT. To count the number of people in both clubs, we can add up the number of people in either club, then subtract their intersection because we overcounted it. We cannot simply do 42+25.
Formally, if A and B are two finite sets, then principle of inclusion-exclusion says that
|A ∪ B| = |A| + |B| - |A ∩ B|
(optional) PIE generalizes to more sets. Here is PIE with 3 sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |C ∩ A| + |A ∩ B ∩ C|
wait then wouldn’t we only subtract 9? so we only count those people once? Because if its 18 doesn’t just take those people out completely
The 18 people are counted twice: in the 42 as well as the 25, therefore you subtract 18 to count them only once.
@avneety no, you subtract 18.
PIE is pretty easy to visualize using a Venn diagram.
I think it’s clearer to think of Venn/double counting problems like this:
Total = (everyone in group 1 - overlap) + (everyone in group 2 - overlap) + overlap + everyone not in group 1 or 2
everyone in group 1 - overlap --> 42 yoga - 18 people in both yoga and sports club = 24 people who are ONLY in yoga
everyone in group 2 - overlap --> 25 sports club - 18 people in both yoga and sports club = 7 people who are ONLY in sportsclub
overlap --> 18 people who are in both yoga and sports club
everyone not in group 1 or 2 --> 0 in this problem
So now we’re clearly only counting everyone once:
24 people who are ONLY in yoga + 7 people who are ONLY in sportsclub + 18 people who are in both yoga and sportsclub = 49 people
Many students find this graphic interpretation easier to grasp:
Picture two overlapping angles: <AOB=42deg and <COD=25deg with an overlap <COB=18deg.
The measure of the encompassing <AOD is
<AOC+<COB+<BOD =
(42-18) + 18 + (25-18) = 42 + 25 -18.
That’s exactly what is done when solving the original question:
[quote=avneety]
If the Yoga club has 42 members and the Sports club has 25 members and there are 18 students that belong to both clubs, how many people total will be at a joint meeting of both clubs, if all members attended?
[quote]
Draw a Venn Diagram with 2 overlapping circles.
The first number that goes in is the overlap: the 18.
Now subtract to find the numbers in the non-overlap
Yoga is 42 - 18 or 24
Sports is 25 - 18 or 7.
The total number of people in your diagram is 24 + 18 + 7, or 49.