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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
A. 105
B. 125
C. 130
D. 180
E. 195
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
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
100 = 40 + 30 + 75 - 35 - 2 x ALL ----(standard formula)
ALL = 5%
Exactly 3 = 5% Of 300 = 15
Exactly 2 = 35% of 300 = 105
Total = Exactly 1 + Exactly 2 + Exactly 3
300 = Exactly 1+ 15 + 105
Exactly 1= 180 Ans.
ALL = 5%
Exactly 3 = 5% Of 300 = 15
Exactly 2 = 35% of 300 = 105
Total = Exactly 1 + Exactly 2 + Exactly 3
300 = Exactly 1+ 15 + 105
Exactly 1= 180 Ans.
ADVANCED OVERLAPPING SETS PROBLEMS
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macjas
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
Answer is D.
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Quote:
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
Answer is (D)
Regards,
Attachments
Venn.jpg [ 21.47 KiB | Viewed 269130 times ]
x+y+z+p+q+r+w = 300 ---- (a)
x+p+w+q = 120 (40% of 300) ----(1)
p+q+w+r = 90 (30 % of 300)----(2)
similarly q+w+r+z = 225----(3)
Need to find x+y+z=?
Adding equations (1), (2) and (3)
we get x+y+z+2(p+q+r+w)+w=435
subtract equation (a) from above equation
we get p+q+r+2w = 135
given p+q+r = 105 (35% of 300)
so w =15 and p+q+r+w = 120
substitute value of above equation in (a) gets x+y+z = 180
x+p+w+q = 120 (40% of 300) ----(1)
p+q+w+r = 90 (30 % of 300)----(2)
similarly q+w+r+z = 225----(3)
Need to find x+y+z=?
Adding equations (1), (2) and (3)
we get x+y+z+2(p+q+r+w)+w=435
subtract equation (a) from above equation
we get p+q+r+2w = 135
given p+q+r = 105 (35% of 300)
so w =15 and p+q+r+w = 120
substitute value of above equation in (a) gets x+y+z = 180
Attachments
Venn.JPG [ 13.05 KiB | Viewed 260681 times ]
Posts: 6,727
macjas
40 percent experienced sweaty palms, 30 percent experienced vomiting, and 75 percent experienced dizziness.
We get:
35 percent of the subjects experienced exactly two of these effects
35% of 300 = 105
So, a + b + c = 105
If 120 experienced experienced sweaty palms, we can write: X + a + b + d = 120
If 90 experienced experienced vomiting, we can write: Y + a + c + d = 90
If 225 experienced experienced dizziness, we can write: Z + b + c + d = 225
ADD all three equations to get: X + Y + Z + 2a + 2b + 2c + 3d = 435
Rewrite as follows: X + Y + Z + 2(a + b + c) + 3d = 435
Since a + b + c = 105, we can substitute to get: X + Y + Z + 2(105) + 3d = 435
Simplify to get: X + Y + Z + 210 + 3d = 435
Subtract 210 from both sides to get: X + Y + Z + 3d = 225
300 subjects participated in the experiment
We can write: X + Y + Z + a + b + c + d = 300
Substitute: X + Y + Z + 105 + d = 300
Subtract 105 from both sides: X + Y + Z + d = 195
We now have two different equations:
X + Y + Z + 3d = 225
X + Y + Z + d = 195
Subtract the bottom equation from the top equation to get: 2d = 30, which means d = 15
Replace d with 15 to get: X + Y + Z + 15 = 195
Subtract 15 from both sides: X + Y + Z = 180
So, 180 subjects experienced only one effect.
Answer: D
Posts: 2
people who experienced
1 symptom only - a
2 symptom only- b =35% (given)
3 symptom only- c
no symptoms- 0
a+b+c=100%
a+35%+c = 100% -----> (1)
also
Group 1= 40%
Group 2= 30%
Group 3= 75%
Total = Group1 + Group 2 + Group 3 - (people with 2 symptoms only) - 2*(people with 3 symptpoms only) + Neither
Total = Group1 + Group 2 + Group 3 - (b) - 2*(c) + 0
Total = 40% +30%+75%-35% - 2*(c) + 0= 100%
110%-2c=100%
c=5% -----> (2)
from (1) and (2)
a + 35% + 5% = 100%
a= 60%= 60%(300)= 180. Answer D
1 symptom only - a
2 symptom only- b =35% (given)
3 symptom only- c
no symptoms- 0
a+b+c=100%
a+35%+c = 100% -----> (1)
also
Group 1= 40%
Group 2= 30%
Group 3= 75%
Total = Group1 + Group 2 + Group 3 - (people with 2 symptoms only) - 2*(people with 3 symptpoms only) + Neither
Total = Group1 + Group 2 + Group 3 - (b) - 2*(c) + 0
Total = 40% +30%+75%-35% - 2*(c) + 0= 100%
110%-2c=100%
c=5% -----> (2)
from (1) and (2)
a + 35% + 5% = 100%
a= 60%= 60%(300)= 180. Answer D
sujoykrdatta
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macjas
Please find the answer in the diagram attached:
Attachment:
IMG_20210418_235303__01.jpg [ 1.79 MiB | Viewed 106336 times ]
Answer D
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macjas
100%=40%+30%+75%-35%-2*x
or, 2x=10%
or, x=5%
Experienced only one of these effects=100%-35%-5%=60%
By the way, 100%=300
or, 1%=300/100
or, 60%=300*60/100=180
So, the best answer is (D). posted By mannan mian
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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
--------------------------------------------
Adding above 3 equations
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
Answer = D
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
--------------------------------------------
Adding above 3 equations
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
Answer = D
Attachments
Venn.jpg [ 23.99 KiB | Viewed 113732 times ]
Posts: 4,849
Shekhar89
Hello Shekhar89,
Here’s how you could solve this with minimum effort and time. Of course, the Venn diagram is a necessity, you can’t think of a smart strategy without drawing the Venn diagram.
The Venn diagram and the following solution are as follows:
Attachment:
29th April 2020 - Reply 4.jpg [ 59.64 KiB | Viewed 108190 times ]
Hope that helps!
Posts: 31
Updated on: 07 Oct 2018, 17:16
Originally posted by EMPOWERMathExpert on 03 Oct 2018, 04:08.
Last edited by EMPOWERMathExpert on 07 Oct 2018, 17:16, edited 1 time in total.
Last edited by EMPOWERMathExpert on 07 Oct 2018, 17:16, edited 1 time in total.
First, let’s find the number of participants that experience the listed symptoms.
Since \(40\%\) reporting sweaty palms, that’s \(\frac{40}{100} \times 300=120\) participants experiencing this symptom. Similarly, there are
\(\frac{30}{100} \times 300=90\) participants reporting vomiting and \(\frac{75}{100} \times 300=225\) who reported experiencing dizziness.
Also, \(\frac{35}{100} \times 300=105\) have experienced two of the effects.
We can construct a Venn diagram to better visualize the problem.
venn 1.jpg [ 17.68 KiB | Viewed 107491 times ]
The intersections between two circles represent the participants who have experienced two symptoms. I’ve shaded it in purple to better understand it.
venn 2.jpg [ 17.58 KiB | Viewed 107343 times ]
If we let \(x\) be the number of participants who experienced all three, we have
venn 3.JPG [ 16.58 KiB | Viewed 107241 times ]
If you add up all the number of reports, we have \(225+120+90=435\). That’s clearly more than the number of participants and the reason behind this is that people who reported two effects were also accounted for in the individual effect report.
For example, participants who reported vomiting AND dizziness are accounted for twice since their report will be counted in the group reporting Dizziness and in the group reporting vomiting. In addition, people who reported three symptoms were counted thrice.
This means that there is an extra \(105\) and extra \(2x\) in the count. So, if this gets subtracted from the total, then it must now be \(300\).
\(435-105-2x=300\)
\(330-2x=300\)
\(30=2x\)
\(x=15\)
So, we have exactly \(105\) people reporting two symptoms and \(15\) people who have reported all three.
This means that there are \(300-105-15=180\) participants who only experience one symptom.
Hence, The final answer is .
Since \(40\%\) reporting sweaty palms, that’s \(\frac{40}{100} \times 300=120\) participants experiencing this symptom. Similarly, there are
\(\frac{30}{100} \times 300=90\) participants reporting vomiting and \(\frac{75}{100} \times 300=225\) who reported experiencing dizziness.
Also, \(\frac{35}{100} \times 300=105\) have experienced two of the effects.
We can construct a Venn diagram to better visualize the problem.
Attachment:
venn 1.jpg [ 17.68 KiB | Viewed 107491 times ]
The intersections between two circles represent the participants who have experienced two symptoms. I’ve shaded it in purple to better understand it.
Attachment:
venn 2.jpg [ 17.58 KiB | Viewed 107343 times ]
If we let \(x\) be the number of participants who experienced all three, we have
Attachment:
venn 3.JPG [ 16.58 KiB | Viewed 107241 times ]
If you add up all the number of reports, we have \(225+120+90=435\). That’s clearly more than the number of participants and the reason behind this is that people who reported two effects were also accounted for in the individual effect report.
For example, participants who reported vomiting AND dizziness are accounted for twice since their report will be counted in the group reporting Dizziness and in the group reporting vomiting. In addition, people who reported three symptoms were counted thrice.
This means that there is an extra \(105\) and extra \(2x\) in the count. So, if this gets subtracted from the total, then it must now be \(300\).
\(435-105-2x=300\)
\(330-2x=300\)
\(30=2x\)
\(x=15\)
So, we have exactly \(105\) people reporting two symptoms and \(15\) people who have reported all three.
This means that there are \(300-105-15=180\) participants who only experience one symptom.
Hence, The final answer is .
Posts: 167
Hi everyone,
Here's my video explanation of the question. Hope you enjoy!
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Here's my video explanation of the question. Hope you enjoy!
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Updated on: 06 Dec 2023, 21:14
Originally posted by CrackverbalGMAT on 03 Jun 2021, 23:05.
Last edited by CrackverbalGMAT on 06 Dec 2023, 21:14, edited 1 time in total.
Last edited by CrackverbalGMAT on 06 Dec 2023, 21:14, edited 1 time in total.
Solution:
Given Total = 300
Sweaty palms (S) =0.4 * 300 = 120
Vomiting (V) =0.3 * 300 =90
Dizziness (D) = 0.75 * 300 = 225
We know that Total = S + V + D -(sum of 2-group overlaps) - 2*(all three) + Neither
=> 300 = 120 + 90 + 225 - 105 - 2*(all three) + 0
=> 300 = 330 - 2*(all three) + 0
=> 2*(all three) = 30
=>all three = 15
=>People with only one of these effects = S + V+ D - 2-group overlaps * 2 - 3-group overlaps * 3
=120 + 90 + 225 - 105*2 - 15*3
= 435 - 210 - 45
= 180 (option d)
Given Total = 300
Sweaty palms (S) =0.4 * 300 = 120
Vomiting (V) =0.3 * 300 =90
Dizziness (D) = 0.75 * 300 = 225
We know that Total = S + V + D -(sum of 2-group overlaps) - 2*(all three) + Neither
=> 300 = 120 + 90 + 225 - 105 - 2*(all three) + 0
=> 300 = 330 - 2*(all three) + 0
=> 2*(all three) = 30
=>all three = 15
=>People with only one of these effects = S + V+ D - 2-group overlaps * 2 - 3-group overlaps * 3
=120 + 90 + 225 - 105*2 - 15*3
= 435 - 210 - 45
= 180 (option d)
Posts: 31
This question can be solved using a Venn diagram.
vvv.png [ 15.94 KiB | Viewed 106704 times ]
Number of:
Sweaty \(= 40\% = A\)
Vomiting \(= 30\% = B\)
Dizziness \(= 75\% = C\)
Now we need to use formula for Venn diagram.
\(15\) people had \(3\) symptoms. Now we can use this to find the number of people with 2 symptoms.
The final answer is .
Attachment:
vvv.png [ 15.94 KiB | Viewed 106704 times ]
Number of:
Sweaty \(= 40\% = A\)
Vomiting \(= 30\% = B\)
Dizziness \(= 75\% = C\)
Now we need to use formula for Venn diagram.
\(A∪B∪C=A+B+C-(A∩B+B∩C+C∩A)+2 \times A∩B∩C\)
\(A+B+C→one \ symptom\)
\(A∩B+B∩C+C∩A→total \ number \ of \ patients \ with \ 2 \ symptoms\)
\(A∩B∩C→3 \ symptoms\)
\(300=300(40\%)+300(30\%)+300(75\%)-300(35\%)-2(x) \\\\
300=120+90+225-105-2x \\\\
x=15\)
300=120+90+225-105-2x \\\\
x=15\)
\(15\) people had \(3\) symptoms. Now we can use this to find the number of people with 2 symptoms.
\(300=120+90+225-y+3(15) \\\\
y=180\)
y=180\)
The final answer is .
General Discussion
Posts: 3
vandygrad11
I'm having trouble understanding this formula. Why is the sum of all three overlaps multiplied by two?
Thanks in advance.
Quote:
i dont understand why Vandygrad multiplies 2 gr overlaps by 2 and 3 gr overlaps by 3?
dont we need to minus 3 times "exactly 2 gr" overlaps and once "3 gr" overlaps?
macjas
exactly two = A+B+C-2(A n B n C)-(A u B u C)
OR, 35 = 40+30+75 - 2(A n B n C) - 100
OR, (A n B n C) = 5% = 5% OF 300 = 15
Exactly 3 = 15
Exactly 2 = 35% of 300 = 105
So exactly one = 300 -(15+105) = 180 (Answer)
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