Three Overlapping Sets Formula

[crejoc]

The Three Overlapping Sets Formula

Here it is:
Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither

I got this formual from https://www.gmathacks.com/gmat-math/thre ... -sets.html - you can check this for explanation

But i am still not able to apply it , i am posting two problems, If those problems can be solved using the above formula , kindly explain in detail.

Q1 :
In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?


Q2 :
There are 70 students in Math or English or German. Exactly 40 are in Math, 30 in German, 35 in English and 15 in all three courses. How many students are enrolled in exactly two of the courses? Math, English and German.

The two problems have been already discussed in this thread, but i want to know whether the above formula can be applied to these problems? if so how?
https://www.beatthegmat.com/set-theory-w ... 21635.html
Thanks in advance.

[rah_pandey]

In these type of question, the problem arises from wording.
so lets see

Stmt 150% of those asked liked product 1

Stmt 250% of those asked liked product 1 only

can you distinguish between the two
Stmt 1=> it includes all those people who like 1 i.e they may like 1 only, 1+2 or 1+3 or 1+2+3

where as Stmt2 => people who like 1 and only 1.


by your question let
A= no of people who like 1 only
x= no of people who like 1 and 2
let y= no of people who like 1 and 3
and z= 1+2+3

thus x+y+z+A=50%
=> A=50-x-y-z
similarly let v= people who like 2 and 3
B=no who like 2 only
thus
x+z+v+B=30
let C= no of people who like 3 only
C+y+z+v=20%

also z=5%
and people who do not like anything =15%

thus we know
A+B+C+x+y+z+v=85% from Venn diagram representation
or
(A+x+y+z)+(B+x+v+z)+(-x-z)+(C+y+v+z)+(-v-y-z)
=(A+x+y+z)+(B+x+v+z)+(C+y+v+z)-(x+y+v)-2z=85%---(A)
same as formula given by you

Now your question says

what percentage of the survey participants liked more than one of the three products?

this means we need x+y+v+z
by using (A) we get
(50+30+20)%-85%-5%=x+y+z+v
=10% required answer

I would always suggest to use formula only when you understand why it has been written the way it has been written otherwise start from basics and then later graduate to formula use and in this case Venn diagram is very helpful. It eliminates use of the confusing formula and confusing terminology. The formula as stated by you will not be valid if stmt 2 as mentioned above was given instead of Stmt1

[crejoc]

Thanks for your detailed explanation.. pandey . but am unable to understand your explanation, may be if you please use the same symbols as used in the formula, i may understand sucha group1, group2. Other symbols confusing me, am already exhausted with this problem. It may be lame, but anyway i have to learn it. Thanks in advance.

[Stuart@KaplanGMAT]

Here it is:
Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither


But i am still not able to apply it , i am posting two problems, If those problems can be solved using the above formula , kindly explain in detail.

Q1 :
In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?

Sure.. applying the formula, we get:

100 = 50 + 30 + 20 - (people who like 2 products) - 2(5) + 15

(Since it's a percent question, just let total number = 100.)

100 = 100 - double - 10 + 15
100 = 105 - double
double = 5

So, 5% of people like exactly 2 products; since 5% of the people like all 3 products, a total of 10% like more than 1 product.

Q2 :
There are 70 students in Math or English or German. Exactly 40 are in Math, 30 in German, 35 in English and 15 in all three courses. How many students are enrolled in exactly two of the courses? Math, English and German.

70 = 40 + 30 + 35 - (exactly 2 courses) - 2(15) + 0

(All 70 are in one of the subjects, so "none" = 0.)

70 = 105 - double - 30
70 = 75 - double
double = 5

So, 5 students are enrolled in exactly 2 of the courses.

[rah_pandey]

The Three Overlapping Sets Formula

Here it is:
Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither


By my terminology
Group1= All those people who like 1
Group1=A+x+y+z
A= Like Only 1
x= like 1 and 2
y= like 1 and 3
z= like 1,2 and 3

Group2= B+x+v+z
B= Like Only 2
v= like 2 and 3

Group3=C+y+v+z
C= Like Only 3

Also x+y+v+z= People who like more than one

A+B+C= People who like EXACTLY ONE product

x+y+v= People who like EXACTLY TWO product


According to Question 1 you have to find x+y+v+z

You are given
Group1=50%
Group2=30%
Group3=20%
All three(z)=5%

Also people who do not like any of the three products=(100-85)%
=>Neither=15% in your formula

As you can see for yourself the equation that you have given will yield

sum of 2 group overlap=x+y+v
you already know All three(z)=5%

hence you can find the answer. I HOPE You UNDERSTAND



What i wanted to emphasize was, USE the formula only when you understand what really comprises Group1,Group2,Group3

Q1 :
In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?

[PussInBoots]

<ignore>

[rah_pandey]

This is wrong.
what is your P(A),P(B) etc
if it means only A, only B etc than correct formula is

P(A)+P(B)+P(C)+P(A n B) + P(A n C) +P(B n C) + P(A n B n C)

If P(A)=> people who like A either alone or in combination
P(B)=> people who like B either alone or in combination

than correct formula is
P(A)+P(B)+P(C)-P(A n B) - P(A n C) -P(B n C) - 2P(A n B n C)

I have given proof for both in my earlier post

[PussInBoots]

By definition P(A n B) is intersection of sets A and B. That intersection might include numbers from set C. My formula was correct, just two views on problem https://en.wikipedia.org/wiki/Inclusion- ... _principle

In statistics and math P(A) + P(B) + P(C) – P(A n B) – P(A n C) – P(B n C) + P(A n B n C) is used more.

Here is more info https://gmatclub.com/forum/formulae-for- ... 69014.html

[crejoc]

Q1:

100 = 50 + 30 + 20 - (people who like 2 products) - 2(5) + 15

(Since it's a percent question, just let total number = 100.)



Q2 :

70 = 40 + 30 + 35 - (exactly 2 courses) - 2(15) + 0

(All 70 are in one of the subjects, so "none" = 0.)

Thank you so much , i could understand clearly now, thanks a lot.

[crejoc]

By my terminology
Group1= All those people who like 1
Group1=A+x+y+z
A= Like Only 1
x= like 1 and 2
y= like 1 and 3
z= like 1,2 and 3

Group2= B+x+v+z
B= Like Only 2
v= like 2 and 3

Group3=C+y+v+z
C= Like Only 3

Thanks for taking your time to explain again to make me understand. Thanks a lot

[RoadtoIVY]

Mind boggling!!

[havok]

Can someone apply this to:

This table lists enrollment in an afterschool program by activity. There are 30 total students enrolled in the entire program. Students may participate in one, two, or three activities. How many students participate in all three activities?

Basketball: 19
Math: 12
Wrestling: 11

(1) 21 students only participate in one activity.

(2) 6 students participate in both basketball and math.

[Stuart@KaplanGMAT]

Edit: Posted, but might have changed my mind - need to think about it some more!

[Resp007]

Answer is: Only answer A is sufficient and B is not.
Any1 care to validate? I know its too old of a post but good que i think?!