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# Of the 300 subjects who participated in an experiment using

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Intern
Joined: 02 Nov 2017
Posts: 22
Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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14 Dec 2017, 04:49
geometric wrote:
The best way to tackle this question is probably the formula for three overlapping sets:

Total = Group1 + Group 2 + Group 3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Total = 300(.4) + 300(.3) + 300(.75) - 300(.35) - 2*(all three) + 0
300*.1 = 30
300 = 120 + 90 + 225 - 105 - 2*(all three)
2*(all three) = 30
:. 15 experienced all three effects

So Group 1 + Group 2 + Group 3 - 2-group overlaps * 2 - 3-group overlaps * 3 is our answer
= 120 + 90 + 225 - 105*2 - 15*3
= 435 - 210 - 45
= 180

I did not understand the 2nd part of the answer. how did you derive this formula and when is it supposed to be used? Any help would be appreciated.
Intern
Joined: 30 Jan 2017
Posts: 3
Of the 300 subjects who participated in an experiment using  [#permalink]

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02 Jan 2018, 00:47
N(a U b U c) = N(a) + N(b) + N(c) - N(a ∩ b) - N(b ∩ c) - N(c ∩ a) + N(a ∩ b ∩ c)

N(a ∩ b) = (only a, b) + N(a ∩ b ∩ c)
N(c ∩ b) = (only c, b) + N(a ∩ b ∩ c)
N(a ∩ c) = (only c, a) + N(a ∩ b ∩ c)

In percentages:
100 = 40+30+75-35-2N(a ∩ b ∩ c) ----> N(a ∩ b ∩ c) = 5

100 = (exactly one) + (exactly two) + (exactly three) ---> exactly one = 100-35-5 ---> exactly one = 60% ---> 180 people

Director
Joined: 09 Mar 2016
Posts: 884
Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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02 Jan 2018, 10:23
Bunuel wrote:
iwillbeatthegmat wrote:
The best way to tackle this question is probably the formula for three overlapping sets:

Total = Group1 + Group 2 + Group 3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Total = 300(.4) + 300(.3) + 300(.75) - 300(.35) - 2*(all three) + 0
300*.1 = 30
300 = 120 + 90 + 225 - 105 - 2*(all three)
2*(all three) = 30
:. 15 experienced all three effects

So Group 1 + Group 2 + Group 3 - 2-group overlaps * 2 - 3-group overlaps * 3 is our answer
= 120 + 90 + 225 - 105*2 - 15*3
= 435 - 210 - 45
= 180

I'm having trouble understanding this formula. Why is the sum of all three overlaps multiplied by two?

Hope it helps.

Hello Bunuel, can you please explain this 2*(all three) how do we get 15*3 ? I checked the link you provided, but still cant understand this moment. thank you
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Director
Joined: 09 Mar 2016
Posts: 884
Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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02 Jan 2018, 15:13
geometric wrote:
The best way to tackle this question is probably the formula for three overlapping sets:

Total = Group1 + Group 2 + Group 3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Total = 300(.4) + 300(.3) + 300(.75) - 300(.35) - 2*(all three) + 0
300*.1 = 30
300 = 120 + 90 + 225 - 105 - 2*(all three)
2*(all three) = 30
:. 15 experienced all three effects

So Group 1 + Group 2 + Group 3 - 2-group overlaps * 2 - 3-group overlaps * 3 is our answer
= 120 + 90 + 225 - 105*2 - 15*3
= 435 - 210 - 45
= 180

Why did you multiply 300 by 10% - where do you see such information that all three is 30 ....another confusion why here you use figures except of (all three) 300 = 120 + 90 + 225 - 105 - 2*(all three) if all three is unknown then we could use 3x = all three multiplied by 2 = 6x

and why are you using two equations ? where as Bunuel in his post provides one formula to apply when we have three groups ? https://gmatclub.com/forum/advanced-ove ... 44260.html which says: Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither
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Director
Joined: 09 Mar 2016
Posts: 884
Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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03 Jan 2018, 05:29
dave13 wrote:
geometric wrote:
The best way to tackle this question is probably the formula for three overlapping sets:

Total = Group1 + Group 2 + Group 3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Total = 300(.4) + 300(.3) + 300(.75) - 300(.35) - 2*(all three) + 0
300*.1 = 30
300 = 120 + 90 + 225 - 105 - 2*(all three)
2*(all three) = 30
:. 15 experienced all three effects

So Group 1 + Group 2 + Group 3 - 2-group overlaps * 2 - 3-group overlaps * 3 is our answer
= 120 + 90 + 225 - 105*2 - 15*3
= 435 - 210 - 45
= 180

Why did you multiply 300 by 10% - where do you see such information that all three is 30 ....another confusion why here you use figures except of (all three) 300 = 120 + 90 + 225 - 105 - 2*(all three) if all three is unknown then we could use 3x = all three multiplied by 2 = 6x

and why are you using two equations ? where as Bunuel in his post provides one formula to apply when we have three groups ? https://gmatclub.com/forum/advanced-ove ... 44260.html which says: Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither

these overlapping sets overlap both parts of my brain - totally confusing
_________________

In English I speak with a dictionary, and with people I am shy.

Manager
Joined: 28 Jan 2018
Posts: 54
Location: Netherlands
Concentration: Finance
GMAT 1: 710 Q50 V36
GPA: 3
Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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07 Feb 2018, 09:49
Don't bother with actual number until the very end, just keep in percentage form and multiply the number when you find the fraction.

We got: (keep in mind that everything is percentage based)
100 = Group 1 + group 2 + group 3 - (exactly 2) - 2*(exactly 3)
100 = 40 + 30 + 75 - 35 - 2*(exactly 3), solve for this we got (exactly 3) = 5

Here is a very useful formula for these kind of question:
Total number of people = (exactly 1) + (exactly 2) + (exactly 3), it's also equal 100%, hence:
100 = (exactly 1) + 35 + 5
(exactly 1) = 60%, total number of exactly 1 = 300 * 60% = 180
Senior Manager
Joined: 29 Jun 2017
Posts: 416
Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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09 Feb 2018, 05:03
Quote:
Of the 300 subjects who participated in an experiment using virtual-reality therapy to reduce their fear of heights, 40 percent experienced sweaty palms, 30 percent experienced vomiting, and 75 percent experienced dizziness. If all of the subjects experienced at least one of these effects and 35 percent of the subjects experienced exactly two of these effects, how many of the subjects experienced only one of these effects?

A. 105
B. 125
C. 130
D. 180
E. 195
Hi,

We know, $$A\cup B\cup C = A+B+C-A\cap B-B\cap C-C\cap A +A\cap B\cap C$$
where
$$A = 40%$$
$$B = 30%$$
$$C = 75%$$
As per the attached Venn diagram,
$$A\cup B\cup C=100%$$

$$A\cap B+B\cap C+C\cap A=$$Exactly two - 3x (assuming $$A\cap B\cap C=x$$)
$$=35-3x$$
Thus,
$$100= 40+30+75-(35-3x)+x$$
or $$x = 5%$$

Thus, subjects expriencing only one effect = 100% - (subjects expriencing only two effects) - (subjects expriencing all effects)
or subjects expriencing only one effect = 100 - 35 - 5 = 60%

60% of 300 = 180

Regards,

perfect solution.

but remember, there are many form of formulae for this kind. know what formalar is used for what condition
Intern
Joined: 04 Apr 2018
Posts: 2
Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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19 Apr 2018, 02:36
I found the easiest way to solve the problem is to first find AnBnC and then add it to the sum of EXACTLY two overlapping sets (sum of two overlapping sets ) and then subtract it from the total

So, Tot = A+B+C - (sum of exactly 2 groups overlaps) - 2 (sum of all three = x ) + None
300 = 120+90+225 - 105 -2x + 0 => x = 15

Thus the sum of two groups overlaps = 105 + 15 = 120
Therefore, the subjects experienced only one of these effects = Total - the subjects who experienced more than 1 of these effects
= 300 - 120 = 180
Intern
Joined: 27 May 2018
Posts: 8
Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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30 May 2018, 14:58
Bunuel wrote:
saintforlife wrote:
The best way to tackle this question is probably the formula for three overlapping sets:

Total = Group1 + Group 2 + Group 3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Total = 300(.4) + 300(.3) + 300(.75) - 300(.35) - 2*(all three) + 0
300*.1 = 30
300 = 120 + 90 + 225 - 105 - 2*(all three)
2*(all three) = 30
:. 15 experienced all three effects

So Group 1 + Group 2 + Group 3 - 2-group overlaps * 2 - 3-group overlaps * 3 is our answer
= 120 + 90 + 225 - 105*2 - 15*3
= 435 - 210 - 45
= 180

Can you please explain why 2 group overlap was multiplied by 2 and 3 group overlap multiplied by 3 ??
The formula stated in the link posted by you states "Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither".
Intern
Joined: 28 Apr 2016
Posts: 27
Location: United States
GMAT 1: 780 Q51 V47
GPA: 3.9
Of the 300 subjects who participated in an experiment using  [#permalink]

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11 Aug 2018, 17:36
kabirchaudhry92 wrote:
Can you please explain why 2 group overlap was multiplied by 2 and 3 group overlap multiplied by 3 ??
The formula stated in the link posted by you states "Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither".

While the above is not how I would have done the question, it is actually correct. The question is asking us for the number of people who experienced exactly one effect, which is the total minus the 2-group overlaps minus the 3-group overlaps. Because we can write the total as:

Total = Group 1 + Group 2 + Group 3 - 2-group overlaps - 3-group overlaps * 2

if we subtract from this total the 2-group overlaps and the 3-group overlaps, we get:

Group 1 + Group 2 + Group 3 - 2-group overlaps - 3-group overlaps * 2 - (2-group overlaps + 3-group overlaps)

which is equal to

Group 1 + Group 2 + Group 3 - 2-group overlaps * 2 - 3-group overlaps * 3

For those who are thinking about how to do questions like this most efficiently, here are my GMAT Timing Tips for this question (the links have growing lists of questions that you can use to practice these tips):

Three-set Venn Diagrams: I recommend starting three-set Venn Diagram questions by drawing the diagram. We can write in all of the numbers we have been given, realizing that we are told that there are 0 people who have experienced none of the side effects. Then, we need to recognize that the total percent (100%) is equal to the sum of the percents in each group (145%) minus the percent we double counted because there are in exactly two groups (35%) minus 2 times those we double counted because they are in all three groups (I would call this X3, because we need to solve for this). We would find that X3 = 5%.

Write down what the question is asking: While this applies to all Quant questions, it is especially important here. If we wrote down that we needed to solve for the total (100%) minus those in exactly two groups (35%) minus those in all three groups (X3) and convert this percent to a number of people (total number is 300), we can quickly answer the correct question when we finish solving for the number in all three groups. For example, I would have written what I was solving for as: (100% - 35% - X3) of 300 = ?

Please let me know if you have any questions about my tips, and if you would like me to post a video solution.
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Manager
Joined: 10 Apr 2018
Posts: 109
Of the 300 subjects who participated in an experiment using  [#permalink]

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13 Aug 2018, 22:51
1
Bunuel wrote:
saintforlife wrote:
The best way to tackle this question is probably the formula for three overlapping sets:

Total = Group1 + Group 2 + Group 3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Total = 300(.4) + 300(.3) + 300(.75) - 300(.35) - 2*(all three) + 0
300*.1 = 30
300 = 120 + 90 + 225 - 105 - 2*(all three)
2*(all three) = 30
:. 15 experienced all three effects

So Group 1 + Group 2 + Group 3 - 2-group overlaps * 2 - 3-group overlaps * 3 is our answer
= 120 + 90 + 225 - 105*2 - 15*3
= 435 - 210 - 45
= 180

Can someone explain how we get the formula highlighted in red above? Why do we multiply the 2-group overlaps and 3-group overlaps by 2 and 3 respectively? I didn't get that part. Thanks.

Hope it helps.

Hi dave13,
In case your doubt regarding this question is not cleared here it is .
First I will explain the derivation of both the formulas and then you can find its application.

refer to advanced-overlapping-sets-problems-144260.html for the pictorial representation.

Only A= a, only B=b, only C=c ,
only A&B =d, only A&C =e , only B&C=f, These are exactly two group overlaps.
A&B&C = g all three group over laps
Two group overlaps d+g , e+g,f+g.

Now Each element is of certain value that adds up to total of the group. ie
a+b+c+d+e+f+g+n=T, where n=none .
Now Set A= a+d+e+g, We will take into account all the elements the construct the Set which is the value of Only A+only A&B+only A&C+A&B&C. This is how set A is build.
Set B= b+d+f+g We will take into account all the elements the construct the Set which is the value of Only B+only A&B +onlyB&C+A&B&C
Set C=c+e+f+g We will take into account all the elements the construct the Set which is the value of Only C+only A&C +onlyB&C+A&B&C

Well there are two Basic Formulas for three overlapping sets

T= A+B+C- (sum of exactly two over-lapping sets)-2(all three) +none
lets understand how is it constructed

First part
A+B+C= (a+d+e+g)+(b+d+f+g)+(c+e+f+g )

A+B+C=a+b+c+2d+2e+2f+3g. To cancel out the additional (d,e,f and the two g's that we got subtract them so that our final result is T= a+b+c+ d+f+e+g+n

Second part Removing the additional d,e,f and 2g
This can be done in two ways where each way gives us a formula

Method 1

Remove the exactly two group elements and remove twice of element that belongs to all three groups )
Elements that belong to exactly two group are d,e,f and element that belongs to all three groups is "g"

T= a+b+c+2d+2e+2f+3g- d-f-e-2g
T=a+b+c+d+e+f+g
add the final element "n" to get the total T
T=a+b+c+d+e+f+g+n
Hence the formula
T= (a+d+e+g)+(b+d+f+g)+(c+e+f+g )-(d+e+f)-2(g)+n
T= A+B+C- (sum of exactly two over-lapping sets)-2(all three) +none

Method 2
Removing can also be done by removing the two overlap groups ,
So the elements of the this group are (d+g,e+g,f+g)
T= a+b+c+2d+2e+2f+3g -d-g-e-g-f-g
T= a+b+c+d+e+f We dont have element"g" which completes the total
So to undo this effecy we add an element "g" which is nothing but element belonging to all three sets
T=a+b+c+d+e+f+g+n
SO this gives us the second formula.
T= A+B+C- (sum of two over-lapping sets)+all three +none
T= (a+d+e+g)+(b+d+f+g)+(c+e+f+g ) - ( d+g+e+g+f+g)+(g)+n

No Coming back to the question

We are given certain information , is 40 % of 300 = 120, 30% of 300= 90 and 75%of 300=225 and 35 % of 300=105.
let Sweaty Palms be A= a+d+g+e=120,
Vomiting be B=b+d+f+g= 90,&
Dizziness be C=c+e+f+g=225,
also exactly two of the three effects be " sum of exactly two over-lap groups"= d+f+e= 105.
We are asked to find a+b+c=?

So first I will use the formula T= A+B+C- (sum of exactly two over-lapping sets)- 2(all three) +none and calculate all three.

Why do we need to calculate that since T=a+b+c+d+e+f+g, we are given the values of d+f+e but not g. If we can calculate value of g we can find a+b+c= T-(d+e+f+g)
Let all three symptoms be x then,
120+90+225-105-2(x)+0=300
2(x)=30
so x=15
Now to calculate a+b+c i will use the first formula
T=a+b+c+d+e+f+g+n
we know that d+e+f= 105, and g=15
300=a+b+c+105+15
a+b+c=300-(105+15)
a+b+c= 180.

Let me know if there is something you need more help on, will try my best.
Probus
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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14 Aug 2018, 00:08
hello Probus, many thanks for taking time to explain. Highly appreciated! Great explanation!
Have an awesome day!
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Re: Of the 300 subjects who participated in an experiment using &nbs [#permalink] 14 Aug 2018, 00:08

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