Problem Solving

Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

I can't figure out what parts of the diagram this represents "If 6 students sign up for exactly two clubs" ? Also, the Oe can never be done in 4 minutes, not to mention 2 minutes. Any creativity ?

|P&H| + |H&W| + | P&W| - 3|P&H&W| = 6

59 = |P|+|H|+|W| - |P&H| - |H&W|-|P&W| + |P&H&W|
59 = 22+27+28 -6 - 2|P&H&W|
|P&H&W| = 6

heshamelaziry wrote:Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

I can't figure out what parts of the diagram this represents "If 6 students sign up for exactly two clubs" ? Also, the Oe can never be done in 4 minutes, not to mention 2 minutes. Any creativity ?

If you know the right formula, it's a 30 second question.

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only groups 1/2) - (# in only groups 1/3) - (# in only groups 2/3) - 2(# in all 3 groups) + (# in no groups)

More generally:

True # of items = G1 + G2 + G3 - (# in exactly 2 groups) - 2(# in 3 groups)

(On 3 group questions, the # in no groups is almost always 0).

Applying the formula to this question:

59 = 22 + 27 + 28 - 6 - 2(# in all 3 groups)

59 = 71 - 2(# in all 3 groups)

2(# in all 3 groups) = 12
# in all 3 groups = 6

There is no pattern with overlapping sets problem. I have seen different formulas for different problems. although i understand the relationships between the different parts, i can't relate which method to use .

However, I solved it by simultaneous equations.

heshamelaziry wrote:Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

I can't figure out what parts of the diagram this represents "If 6 students sign up for exactly two clubs" ? Also, the Oe can never be done in 4 minutes, not to mention 2 minutes. Any creativity ?

Another way...

If you draw the venn diagram chart start by filling from inside outside.

Three clubs = x-6 ( which is what we need minus exactly two clubs)
Total exactly two clubs = 6
Poetry = 22
History = 27
Writing = 28

28 + 27 + 22 - x-6 - 6 = 59
77 -12 -x = 59
65- x =59
x = 6

Abdulla wrote:

heshamelaziry wrote:Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

I can't figure out what parts of the diagram this represents "If 6 students sign up for exactly two clubs" ? Also, the Oe can never be done in 4 minutes, not to mention 2 minutes. Any creativity ?

Another way...

If you draw the venn diagram chart start by filling from inside outside.

Three clubs = x-6 ( which is what we need minus exactly two clubs)
Total exactly two clubs = 6
Poetry = 22
History = 27
Writing = 28

28 + 27 + 22 - x-6 - 6 = 59
77 -12 -x = 59
65- x =59
x = 6

I added three equations, one for each club. Then, from the result, I subtracted a+b+c+d+M+k+x = 59, i got a+b+c+2x = 18, then I substituted 6 for a+b+c -----> x = 6

Abdulla wrote:

heshamelaziry wrote:Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

I can't figure out what parts of the diagram this represents "If 6 students sign up for exactly two clubs" ? Also, the Oe can never be done in 4 minutes, not to mention 2 minutes. Any creativity ?

Another way...

If you draw the venn diagram chart start by filling from inside outside.

Three clubs = x-6 ( which is what we need minus exactly two clubs)
Total exactly two clubs = 6
Poetry = 22
History = 27
Writing = 28

28 + 27 + 22 - x-6 - 6 = 59
77 -12 -x = 59
65- x =59
x = 6

Will take my GMAT in two weeks and still struggling with overlapping sets. Thank you for giving a simple and straightforward solution.

A+B+C-AB-BC-AC+ABC???

Is this the correct formula also???

I normally use Venn Diagrams for these, but I decided to be brave with this one.

Since there are six in two groups, I dealt with that as 6*2 against the count of 22+27+28 which left me with 77-12=65. I then subtracted 59 from 65; 65-59=6.

As I said though, I was brave. Hopefully my correct answer wasn't a stroke of dumb luck.