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Galiya
The reason is simple; you do not want to include any of the common elements. In this case there are three elements;
So when you add A and B you are counting the exactly 2 common elements twice once with A and once with B ; so considering other combinations we subtract 2gr overlaps twice and not thrice.
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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
Therefore, % experienced more than one = 5+35 = 40%
So, % experienced only one = 60% or 300*60% = 180
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.
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.
mysterygirl
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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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
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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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
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
A. 105
B. 125
C. 130
D. 180
E. 195
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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macjas
Given: 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.
Asked: 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?
Total = G1 + G2 + G3 - (2 out of 3) - 2* ( All 3) + Neither
300 = 120 + 90 + 225 - 105 - 2x + 0
2x = 435 - 300 - 105 = 30
x = 15
Number of the subjects experienced only one of these effects = Total - (2 out of 3) - (all 3) = 300 - 105 - 15 = 180
IMO D
Hi,
Is there a way to get to the answer quickly here?
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
I was using the traditional Venn Diagram Approach here, but that's time-consuming. I looked at the GMAT Club forum and there are some formulas given that might save time, but is that it or is there a smarter elimination strategy that I am missing here. ?
Is there a way to get to the answer quickly here?
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
I was using the traditional Venn Diagram Approach here, but that's time-consuming. I looked at the GMAT Club forum and there are some formulas given that might save time, but is that it or is there a smarter elimination strategy that I am missing here. ?
macjas
Here's flashier way to solve it
. Approach credits - Genoa2000| Categories | 0 | 1 | 2 | 3 | Total |
| Number of people in each category | 0 | a | 35 | b | 300 |
| Total Experiences | 0 | a | 70 | 3b | (0.4+0.3+0.75)*300 |
a + b = 300 - 35 = 265
a + 3b = 435
Solving, a = 180, b = 85
Answer D.
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macjas
Answer: Option D
Video solution by GMATinsight
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Is there a faster way of doing this? All of the methods discussed are really lengthy and take 3+minutes. I understand the logic perfectly, just want to know how we can do 3 way overlapping sets quickly. For now I am just thinking about guessing the answer and moving on on the actual test since I am aiming for Q47-49 only
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Expert reply
Create a triple Venn diagram. Start by solving for the triple overlap (5%), then pick numbers, and make it true!
Here's the tricky part: note that the triple overlap is 5%—not 10%—because even within just one circle, it is included in the "double overlap" twice. If this doesn't make sense, then just memorize it. 2x = what's left of the overlaps (in this case, 10 percent).
Another way to think of it: in the same way that we have to "double subtract" the triple overlap in the total calculation, to avoid triple-counting it, we have to intentionally "double-add" the overlap when summing up double and triple overlaps, because the triple overlap is in fact part of two different double overlaps per circle.
Thus, if x represents the triple overlap, then 45 - 35 - 2x = 0, 10 = 2x, x = 5.
Here's the tricky part: note that the triple overlap is 5%—not 10%—because even within just one circle, it is included in the "double overlap" twice. If this doesn't make sense, then just memorize it. 2x = what's left of the overlaps (in this case, 10 percent).
Another way to think of it: in the same way that we have to "double subtract" the triple overlap in the total calculation, to avoid triple-counting it, we have to intentionally "double-add" the overlap when summing up double and triple overlaps, because the triple overlap is in fact part of two different double overlaps per circle.
Thus, if x represents the triple overlap, then 45 - 35 - 2x = 0, 10 = 2x, x = 5.
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In this question, are we suppose to assume there will be people who fall in the "applies to all 3" category?
Rainman91
Hi Rainman91
Thanks for your query.
I would request you to carefully read each statement in the question again. In the solution below, observe that every next step is a logical inference from the previous step – without making any assumptions at all. The key process skills here are INFER, TRANSLATE, and CONSIDER ALL CASES – these are the skills that help you move on solid grounds from one step to the next.
Let’s dive right into the solution!
SKILLS USED:
Below is the question statement, in which I have highlighted and underlined just one phrase:
“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?”
So, let’s talk about how these skills help one understand this question in a better way.
INFER:
Observe that from the highlighted statement - “all subjects experienced at least one of the three effects” - we can first infer one important thing:
- ALL SUBJECTS – This word helps us infer that there is not a single subject who experienced NONE of the three effects. It tells us that everyone experienced at least one of the three effects. So, None = 0.
TRANSLATE and CONSIDER ALL CASES:
Now to completely translate this same statement - “all subjects experienced at least one of the three effects” - correctly, we MUST consider all cases:
- AT LEAST ONE – This word implies that everyone must have experienced either EXACTLY ONE or EXACTLY TWO or EXACTLY THREE of these effects (Beyond three is not even a possibility).
This translation and inference above clearly allow for some subjects experiencing all three effects. In fact, if you leave this out, you are ASSUMING that no one can experience all three effects. Remember, “exactly three” is also a possibility that comes under “at least one”.
TAKEAWAY:
If you have a solid command on your process skills, you will be able to comfortably apply concepts on even the hardest of questions. You will never have to depend on assumptions to take the next step in a solution. 😊
Best,
Aditi Gupta
Quant expert, e-GMAT
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Stick to the basics.
I found it easy to work with percentages.
Remember Total = A + B + C - (Intersection of exactly 2) - 2 * (Intersection of Exactly 3)
100 = 40 + 30 + 75 - 35 - 2x
x = 5
So, now you have 35 % of these people were in exactly 2 activities, 5 % were in exactly 3.
This implies that of 100%, 40% were in 2 or more activities.
Thus 60% were in one activity or less(none).
We know (none) is 0 from the stem,
So, number of people in one activity only was 0.6*300=180.
Answer is D.
I found it easy to work with percentages.
Remember Total = A + B + C - (Intersection of exactly 2) - 2 * (Intersection of Exactly 3)
100 = 40 + 30 + 75 - 35 - 2x
x = 5
So, now you have 35 % of these people were in exactly 2 activities, 5 % were in exactly 3.
This implies that of 100%, 40% were in 2 or more activities.
Thus 60% were in one activity or less(none).
We know (none) is 0 from the stem,
So, number of people in one activity only was 0.6*300=180.
Answer is D.
"Excuse me. I'm trying to find a book that contains the same practice problems as the ones listed above. Could you please provide me with the title of this book? Thank you."
Quote:
Let a be the number who experienced only one of the effects, b be the number who experienced exactly two of the effects, and c be the number who experienced all three of the effects. Then a + b + c = 300, since each of the 300 participants experienced at least one of the effects. From the given information, b = 105 (35% of 300), which gives a + 105 + c = 300, or a + c = 195 (Eq. 1).
Also, if the number who experienced sweaty palms (40% of 300, or 120) is added to the number who experienced vomiting (30% of 300, or 90), and this sum is added to the number who experienced dizziness (75% of 300, or 225), then each participant who experienced only one of the effects is counted exactly once, each participant who experienced exactly two of the effects is counted exactly twice, and each participant who experienced all three of the effects is counted exactly 3 times. Therefore, a + 2b + 3c = 120 + 90 + 225 = 435.
Using b = 105, it follows that a + 2(105) + 3c = 435, or a + 3c = 225 (Eq. 2).
Then solving the system defined by Eq. 1 and Eq. 2 will give you C=15
Sustitute 15 for C and 105 for B in our first equation a + b + c = 300 to get the final answer A=180
Also, if the number who experienced sweaty palms (40% of 300, or 120) is added to the number who experienced vomiting (30% of 300, or 90), and this sum is added to the number who experienced dizziness (75% of 300, or 225), then each participant who experienced only one of the effects is counted exactly once, each participant who experienced exactly two of the effects is counted exactly twice, and each participant who experienced all three of the effects is counted exactly 3 times. Therefore, a + 2b + 3c = 120 + 90 + 225 = 435.
Using b = 105, it follows that a + 2(105) + 3c = 435, or a + 3c = 225 (Eq. 2).
Then solving the system defined by Eq. 1 and Eq. 2 will give you C=15
Sustitute 15 for C and 105 for B in our first equation a + b + c = 300 to get the final answer A=180
Discussed how to do this quickly using a venn diagram here: https://youtu.be/18ZDLj7KMkc
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Rohanx9
Joined: 15 Jul 2024
Last visit: 15 Apr 2026
Posts: 48
Location: Australia
Schools: Anderson (UCLA) (WD)
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Schools: Anderson (UCLA) (WD)
Posts: 48
Another way to solve this question intuitively is to distribute the number of votes to the number of people.
There are a total of 40+30+75 =145 votes given by 100 people.
35 people gave 2 = 70 votes. Which means 145-70 = 75 votes left and 100-35 = 65 people left.
This means we need to give 75 votes to 65 people.
If each person gets one vote, there are still 10 votes left. We need to give 5 more people 2 votes each (for those 5 to have a total of 3 votes). This means there are 65 - 5 people who gave one vote = 60 people.
No equations.
There are a total of 40+30+75 =145 votes given by 100 people.
35 people gave 2 = 70 votes. Which means 145-70 = 75 votes left and 100-35 = 65 people left.
This means we need to give 75 votes to 65 people.
If each person gets one vote, there are still 10 votes left. We need to give 5 more people 2 votes each (for those 5 to have a total of 3 votes). This means there are 65 - 5 people who gave one vote = 60 people.
No equations.
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