71 men live in a town. 25 of them belong to the Elks Club, 28 belong to the Kiwanis Club, and 15 belong to the Optimist Club. 7 belong to Elks and Kiwanis, 4 belong to Elks and Optimist, and 4 belong to Kiwanis and Optimist. One of the men belongs to all three clubs. Put this information into a Venn diagram.
How many men belong to exactly two of the clubs?
How many belong to none of the clubs?
How many belong to Elks and Kiwanis but not the Optimist Club?
2007-01-19 02:33:33 · 4 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Kind of hard to draw a Venn diagram on here, but you'll want one that looks like this:
http://tieguy.org/events/2004/gnomesummit/marketing-notes/venn-diagram.png
The "universe" is 71...the total of all men must equal 71.
Start at the middle and work your way out.
1 guy belongs to all 3 clubs.
That means that 7 - 1 = 6 men belong to Elks and Kiwanis (not Optimist), 4 - 1 = 3 men belong to Elks and Optimist (not Kiwanis), and 4 - 1 = 3 men belong to Kiwanis and Optimist (not Elks).
Now's when it gets a little more complicated.
There are 25 men in the Elks Club. 25 men - 6 men in Elks & Kiwanis - 3 men in Elks and Optimist - 1 man in all 3 = 15 men in Elks only.
28 in Kiwanis - 6 in Elks & Kiwanis - 3 in Kiwanis and Optimist - 1 in all 3 = 18 men in Kiwanis only.
15 in Optimist - 3 in Elks & Optimist - 3 in Kiwanis and Optimist - 1 in all 3 = 8 men in Optimist only.
So we have:
8 in Optimist only
18 in Kiwanis only
15 in Elks only
6 in Elks and Kiwanis
3 in Elks and Optimist
3 in Kiwanis and Optimist
1 in Elks, Kiwanis, and Optimist
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54 men in at least one club.
That means there are 71 - 54 = 17 men who are "outside" all 3 circles and who are in NO clubs.
This gives you a complete picture and makes the questions easy.
How many men belong to exactly 2 clubs? 6 + 3 + 3 = 12, as the others said.
How many men belong to no clubs? 17, as I said...the other answerers double-counted. (SaintCady did too, but she fixed it.)
How many in Elks and Kiwanis but not Optimist? Calculated above...it's 6.
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Just so you know, for future reference, there's a neat little formula for solving your middle question. It's called the Inclusion/Exclusion Principle. In this case it would look like:
|(A∪B∪C)'| = |U| - (|A| + |B| + |C|) + (|A∩B| + |B∩C| + |A∩C|) - |A∩B∩C|
What this means is: "To find the number of people who are not in any of the three clubs, from the total number of men, subtract the total in each club, add back in the totals who are in both clubs, and subtract back out the total who are in all three clubs."
71 - (25 + 28 + 15) + (7 + 4 + 4) - 1
71 - 68 + 15 - 1 = 17
2007-01-19 02:45:38 · answer #1 · answered by Jim Burnell 6 · 0⤊ 0⤋
How many men belong to exactly two of the clubs?
12
How many belong to none of the clubs?
17
How many belong to Elks and Kiwanis but not the Optimist Club?
6
This can all be done with addition/subtraction, but if you need to draw the venn diagram and are have trouble, for this problem it would help to work from the center out. Draw the three overlapping circles (or whatever shape). You know a 1 goes in the very center where they all overlap. Then, you know there's a total of 4 in the Elks-Optimist overlap, but you've already counted one (the one in all three), so there are 3 in the Elks-Optimist-only over lap. And continue...
2007-01-19 10:42:38 · answer #2 · answered by saintcady 2 · 0⤊ 0⤋
Two of the clubs: one is in all three, so the ones in two but not all three are 6, 3 and 3. 12
None of the clubs. 71 - 25 - 28 - 15 = 3
Elks and Kiwanis but not Optimist = 7 - the one in all three = 6.
2007-01-19 10:38:14 · answer #3 · answered by bequalming 5 · 0⤊ 1⤋
32 elk =25+7=32, elk and Kiwanis=32 elks + 35 kiw. =67, none of the clubs = 68 in all clubs-71 men in town. = 3 nonmembers.
2007-01-19 10:50:31 · answer #4 · answered by ruth4526 7 · 0⤊ 1⤋