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# Of the 300 subjects who participated in an experiment using

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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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24 Jul 2014, 03:44
2
1
2
Refer diagram below:

Given that
p + q + r = 105

a + b + c + x + p + q+ r = 300

We require to find value of (a + b + c)

a + b + c = 195 - x .................... (1)

---------------------------------------------
a + p + x + q = 120

b + p + x + r = 90

c + q + x + r + = 225
--------------------------------------------

a + b + c + 2(p + q + r) + 3x = 435

a + b + c = 225 - 3x ................... (2)

Equating RHS of equation (1) & (2)

225 - 3x = 195 - x

2x = 30

x = 15

a + b + c = 195 - 15 = 180

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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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07 Aug 2014, 22:07
1
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

Thanks for this formula. I really like this way of working the problem, as it seems to be faster (at least for a guy like me, who avoids doing venn diagrams and hasn't practiced triple-venn-diagrams very much).
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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16 Aug 2014, 12:00
How are people finishing this in 3 min!?
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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21 Aug 2014, 02:11
JackSparr0w wrote:
How are people finishing this in 3 min!?

"Picture says thousand words"

Venn diagram made the work easy for me in this problem. Yes took 3 minutes; but seems to be OK for this lengthy problem
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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31 Aug 2014, 03:21
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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19 Jan 2015, 13:41
I also don't get why we multiply by 3... This is not in the formula, while multiplications by 2 is...
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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20 Mar 2015, 23:04
2
40 + 30 +75 -35 -2x +0 = 100 ==> x = 5
Therefore, % experienced more than one = 5+35 = 40%
So, % experienced only one = 60% or 300*60% = 180
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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25 Apr 2015, 09:12
I solved this question a little bit differently:

300 x 0.4 = 120 (S)
300 x 0.3 = 90 (V)
300 x 0.75 = 225 (D)

300: at least one effect
105: 2 effects
x: only one effect

How many subjects have all three effects:

120/300 x 90/300 x 225/300 = 2/5 x 3/10 x 15/12 = 1/20, which is equivalent to 15/300 (i.e. 15 subjects).

Subjects with only one effect:

total - number with exactly 2 effects - number with all three effects:

300 - 105 - 15 = 180
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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21 May 2015, 09:08
Hi guys
Could you please tell me whether my solution is right ???

300*40/100= 120 sweaty palms
300*30/100=90 vomiting
300*75/100=225 dizziness
300*35/100=105 (35%subjects experienced exactly two of these effects)
Neither=0
At first we need to find overlapping between 3 groups(AnBnC):

Total=A+B+C-(AnB+AnC+BnC)(or Exactly 2-group overlaps)- 2*(AnBnC)+Neither

300=120+90+225-105-2(AnBnC)+0
300=435-105-2(AnBnC)
300=430-2(AnBnC)
(AnBnC) = 15 overlaps of three groups

300-105(Exactly 2-group overlaps)-15(overlaps of three groups)=180
180= subjects experienced only one of these effects
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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23 Jul 2015, 00:56
hI,

I went through all the problems explained in the link. I did understand that 15 experienced all three.

Thanks.

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.
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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27 Aug 2015, 23:16
Hi,

Bunuel - I went through the link on advanced overlapping set, but I fail to understand the basis for the second formula used in the solution ->
A+B+C- (2-grp overlays)*2 - (3-grp overlays)*3

Please let me know how this formula was derived?

Thanks.
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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05 Oct 2015, 02:59
1
mysterygirl wrote:
Hi,
1:
Bunuel - I went through the link on advanced overlapping set, but I fail to understand the basis for the second formula used in the solution ->
A+B+C- (2-grp overlays)*2 - (3-grp overlays)*3

Please let me know how this formula was derived?

Thanks.

For this reply, I'll be referencing Paresh's brilliant answer, because he goes through the effort of breaking down the various components of our three sets.

My advice is don't start off with the formula as a base to understand the question. Key idea here is understanding the question, and parsing the information in the prompt to reach a valid solution. You read the question, you have an idea of what's being given, you've established that it relates to Overlapping Sets. Great, so what do we have?

The question supplies us with these following knowns
Number of Groups: 3
Group 1 (Sweaty Palms: SP): 40% of 300= 4*30=120
Group 2 (Vomiting: V): 30% of 300 = 3*30 = 90
Group 3 (Dizziness: D): 75% of 300 = 7.5*30=225

All people in the study experience atleast 1 of these symptoms, 35% experience exactly 2 of those symptoms.

So where are those people that fall in 35% of the study? Referring to Paresh's Venn Diagram:
They are sectors p+q+r, notice that x was not included? Because x represents people who experienced SP, V, and D. The sectors in bold, represent the people who exactly two symptoms, for clarity

p: Represents people who experienced both SP and V.
q: Represents people who experienced both SP and D
r: Represents people who experienced both D and V.

The unknowns:
1. People who experienced exactly one of these symptoms.
2. People who experienced all three of these symptoms.

You should get into the habit of not mechanically applying the formula, but take a look at the Venn Diagram .

From the first list of unknowns, where is that represented in the Venn Diagram? Well, obviously, it's going to be as follows:
If we only look at SP, this contains more regions than we would want, we want the part of SP the excludes the overlaps.
Here are the following totals for all three groups.

Total SP: A+p+q+x
Total V: B + p + r + x
Total D: C + q + r + x

Now you should be asking yourself, do we want all those sectors? As you probably assumed, the answer is no. Why don't we want all those sectors? Because the sectors of interest are:

Only SP: A
Only V: B
Only D: C

What's left over?
The intersection of SP and V: p + p + x = 2p + x -> P is in both SP and V
The intersection of SP and D: q + q + x = 2q + x -> Q is in both SP and D
The intersection of D and V: r + r + x = 2r + x -> D is in both D and V.

This is what's meant by exactly two groups overlapping, elements common to two sectors and not the third.

It helps to refer to the Venn Diagram, to see exactly how these intersections are derived. It's easy to see those letters and get lost in the learning process, but this is a way to help distill the information, before you intuitively apply the formula (and not mechanically).

Okay now we sum up what we know.

35% of 300 are in p + q + r, so that means means 105 people in the study had exactly two symptoms.

We know that we care to get the people that are in A, B, and C (Refer to Venn Diagram). So using the supplied in formation

Sweaty Palms has 120 people, both from those 120, there are people who fall in p or q, we don't want that.
Vomiting has 90 people, we obviously don't want all 90. Some fall in the Sweaty palms group (p), others fall in the dizziness group (r).
Dizziness has 225 people, I think you can spot the pattern by now. We don't want the entire set of 225, some are in Sweaty Palms (q), others are in Vomiting (r)

The Sectors we do want are in bold:
SP= A + p+q + x
V= B + p + r + x
D = C + q + r + x

It's like we made a full circle, but we'll start plugging in values.

SP + V + D = A + p+q + x + B + p + r + x + C + q + r + x

We know know SP=120, V=90, D=225. We want ONLY A from the Sweaty Palms group, ONLY B from the Vomiting Group, ONLY C from the Diziness group.

Referring the color coded equation above, we do some simple algebraic manipulation to get it in this form.
I moved all the sectors of interest as the first three terms on the RHS.

In the form of sectors:
SP + V +D= A + B + C + p+ p + q + q + r + r + x + x + x
In the form of actual numbers
SP + V+ D= 120 + 90+ 225
SP + V+ D = 435

Now combining the representation that's in the form of sectors, with the representation that's in the form of actual numbers, we get

435 = A + B + C + p+ p + q + q + r + r + x + x + x
You will noticed x is counted three times, because it's where all three groups intersect. x contains people who were dizzy, vomiting, and had sweaty palms. Those poor people

Also, p is counted twice, because p happens to fall in both Sweaty Palms and Vomiting. q happens to be the people the people who were dizzy, and had sweaty palms, and finally r are people who were dizzy and vomiting. While you're reading along, just tick these guys off in the Venn Diagram.

Okay, so we get it in the final form, we're almost there.

435= A + B + C + 2p + 2q + 2r + 3x
435 = A + B + C + 2(p+q+r) + 3x

Now we know Total = 300. We know p+q+r = 105, but it looks like we don't know A + B +C (our areas of interest) AND x. But we can solve for x

Total = A + B + C + x + p + q + r

You might be looking at this, and thinking why is that different from the one we just derived, this one
435 = A + B + C + 2(p+q+r) + 3x

Here were started off the SP, V, D, so within those, some of our sectors were counted twice, and three times as discussed above. The one we will work with now, contains all individual sectors, don't think about groups, just think of the areas they occupy.

Again,

54= A + B + C + x + p + q + r
300 = A + B + C + x + 105
A + B + C= 300-105-x
A + B + C = 195-x

Looks like we're getting somewhere. Back to this formula, were it contains groups that are counted more than once
435 = A + B + C + 2(p+q+r) + 3x
We will sub in the following
p+q + r = 105
A + B + C = 195-x

435= 195-x + 2*(105) + 3x
435=195 + 210 + 2x
435 = 405 + 2x
435- 405 = 2x
30 = 2x
x=15

We know A + B + C = 195-x
and we know x=15
so
A + B + C=195-15
A + B + C=180

Now I hope you can see why those some sectors were counted twice, and others three times. The post you quoted had:
Group 1 + Group 2 + Group 3 - 2-group overlaps * 2 - 3-group overlaps * 3

Group 1: SP; Group 2: V, Group 3: D
2-Group Overlaps: p+q+r
3-group overlaps: x

Taking the totals of of SP and V and D, subtracting the 2-Group Overlaps (p+q+r) two times and subtracting the three groups overlaps three times
leaves us with the unique regions A,B,C

To summarize, A, B, C were our areas of interest within SP, V, and D. To partition them we utilized totals of Sweaty Palms, which contained regions that fell in two other sectors, and one region that was shared by all three sectors. Same from the other Groups of Symptoms.

This should not be your process when solving it, but it helps to break it down while you're studying. So during the actual exam, you'll know how to parse the question, what information to utilize and HOW to utilize it to reach the answer.
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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17 Oct 2015, 11:21
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,

Hi Bunuel ,
Acc to him : $$A\cap B+B\cap C+C\cap A=$$Exactly two - 3x (assuming $$A\cap B\cap C=x$$)
Is this right?
I thought it the formulae is :$$A\cap B+B\cap C+C\cap A=$$Exactly two + 3x. If you have time please help me in this. I m a bit struggling to understand.
Thanks again!
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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18 Oct 2015, 10:12
Johnjojop wrote:
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,

Hi Bunuel ,
Acc to him : $$A\cap B+B\cap C+C\cap A=$$Exactly two - 3x (assuming $$A\cap B\cap C=x$$)
Is this right?
I thought it the formulae is :$$A\cap B+B\cap C+C\cap A=$$Exactly two + 3x. If you have time please help me in this. I m a bit struggling to understand.
Thanks again!

The following two posts should help:
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Of the 300 subjects who participated in an experiment using  [#permalink]

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10 Dec 2015, 06:12
Hi guys!

Is there any similar questions like this one? I would appreciate if you could share here.

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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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28 Jun 2016, 05:45
1
2
macjas wrote:
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

This is a 3-circle Venn Diagram problem. Because we do not know the number of unique items in this particular set, we can use the following formula:

Total # of Unique Elements = # in (Group A) + # in (Group B) + # in (Group C) – # in (Groups of Exactly Two) – 2 [#in (Group of Exactly Three)] + # in (Neither)

Next we can label our groups with the information presented.

# in Group A = # who experienced sweaty palms

# in Group B = # who experienced vomiting

# in Group C = # who experienced dizziness

We are given that 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.

We can solve for the number in each group:

# who experienced sweaty palms = 300 x 0.4 = 120

# who experienced vomiting = 300 x 0.3 = 90

# who experienced dizziness = 300 x 0.75 = 225

We are also given that all of the subjects experienced at least one of these effects and 35 percent of the subjects experienced exactly two of these effects.

This means the following:

# in Groups of Exactly Two = 300 x 0.35 = 105

Since all the subjects experienced at least one of the effects it means that the # in (Neither) is equal to zero. We can now plug in all the information we have into our formula, in which T represents # in (Group of Exactly Three).

Total # of Unique Elements = # in (Group A) + # in (Group B) + # in (Group C) – # in (Groups of Exactly Two) – 2 [#in (Group of Exactly Three)] + # in (Neither)

300 = 120 + 90 + 225 – 105 – 2T + 0

300 = 330 – 2T

30 = 2T

15 = T

Now that we have determined a value for T, we are very close to finishing the problem. The question asks how many of the subjects experienced only one of these effects.

To determine this we can set up one final formula.

Total = # who experienced only 1 effect + # who experienced two effects + # who experienced all 3 effects + # who experienced no effects

We can let x represent the # who only experienced 1 effect.

300 = x + 105 + 15 + 0

300 = x + 120

180 = x

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Of the 300 subjects who participated in an experiment using  [#permalink]

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10 Oct 2016, 08:12
Hi everyone,

Here's my video explanation of the question. Hope you enjoy!

Rowan
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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29 Jun 2017, 18:35
Using Venn diagram,

sweaty palms = a+e+g+d
vomiting = b+e+g+f
dizziness= c+g+d+f
------------
Total = a+b+c+2d+2e+2f+3g which is d+e+f+2g more than the desired form of a+b+c+d+e+f+g
40%+30%+75% all show some condition = 145%
we can see that we have counted some people more than once and they add up to the extra 45%

i.e d+e+f are people counted twice and make up the 35% as given in the question
The remaining 45%-35%=10% must represent people who are triple counted or 2g=10% or g=5%

hence, d+e+f+g=40 or 40%, remaining 60% are the ones who show only one of the three effects or .6*300 = 180
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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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08 Aug 2017, 22:35

100 = 40 + 30 + 75 - (35 +3x) + x

x = 5

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Re: Of the 300 subjects who participated in an experiment using  [#permalink]

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25 Oct 2017, 03:44
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?

Hi,
Can you please explain why did you multiply in 105*2 and 15*3?
Re: Of the 300 subjects who participated in an experiment using &nbs [#permalink] 25 Oct 2017, 03:44

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