In a community, 40% residents practice yoga, 50% residents go for

Question

In a community, 40% residents practice yoga, 50% residents go for cycling, and 60% residents go for morning walks. If 65% of the total residents engage in two or more of these three activities and the number of residents engaging in all the three activities is half the number of residents engaging in none of the three, what percentage of residents engage in none of the three activities?

A. 10
B. 15
C. 20
D. 25
E. 30

A. 10
B. 15
C. 20
D. 25
E. 30

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monker1231 · Answer

Lets say 100 residents in the community:

No activity: N residents
1 Activity : O residents
2 Activities: T residents
3 Activities: TR residents

Now we know:
N + O + T + TR = 100
T + TR = 65
2TR = N

Additionally we know:
O + 2T +3TR = 40 + 50 + 60 = 150

Now we subtract the formulas to get a result:
O + 2T + 3TR = 150
(-) N + O + T + TR = 100
=> -N + T + 2TR = 50

Now we substitute T with = 65-TR
and substitute N with = 2TR

=> -2TR + (65 - TR) + 2TR = 50
=> Solved for TR = 15

N = 2TR so 2*15 = 30
Result is E (30)

Are there quicker ways to solve it? Please give Kudos if this helped

Adit_ · Answer

40+50+60=150
This total is equal to:
1 activity repeated one
2 activities repeated twice
3 activities repeated thrice
OR
In order to remove all overlaps and equate to 100
We need to subtract 2 activities once
We need to subtract 3 activities twice 
This sum is subtracted from the total 150 and Nothing is added to it

In other simple words:

150-(2 activities)-2(3 activities)+nothing=100
Now 3 activities=1/2(nothing)  
Therefore 
150-(2 activities)-2*1/2(nothing)+nothing=100
150-(2 activities)=100
2 activities = 50

Now that we know this 
65-50=15   (This is because we know 2 or more activities done is 65% in total)

Therefore we find that nothing = 2*15 = 30

Answer:  Option E

Adit_ · Answer

If you are strong with overlapping sets you can do mental math.
As soon as you see these numbers 
its 40+50+60=150
you will realise that no activities and 3 activities get cancelled out coz one is half of the other and to equate to 100 you need to subtract 3 activities done twice
Meaning 150-(2 activities)-2(3 activities)+nothing =100 (like you did urself)
Once you see the pattern of cancelling out 2 activities becomes 50, 3 activities is 65-50=15 and 15*2=30 is no activities

Dereno · Answer

I denote the people doing only 1 activities.

II denote the people doing only 2 activities.

III denote the people doing only 3 activities.

so,  I + II + III + None = 100

I + 2II + 3III = 40+50+60 = 150

I + 2II + 3III = 150

Moreover, we know that III = (none /2)

subtracting both equations, we get

II + 2III - None = 50

substitute None = 2*III

we get II =50.

Given, II + III = 65, if II=50, then  III = 15.

If III =15, then none =  30

Option E

Navdeep2000 · Answer

Since we have all the info in percentage form, it's safer to assume an integer to make it easier for us.

Let the total residents in the community be 200. 
 Then, Yoga (Y) = 80, Cycling (C) = 100, Morning walks (W) = 120. 
 Two or more of three activities mean Atleast 2 = 130

Also, Let N denote the total number of residents who do none of these.  
 Then, as per the given, residents engaging in all three activities would be (N/2)

Let x denote people in 2 activities that is ((Y∩C)+(Y∩W)+(C∩W)) 
 Plugging into our formula

At least 1 = (A+B+C) - ((A∩B)+(B∩C)+(A∩C)) + (A∩B∩C) + Neither 
 so, At least 1 = (Y+C+W) - (x) + (N/2) + N 
 200 = (80+100+120) - (x) + (N/2) + N 
 Solving this, we will arrive at : 2x = (200 + 3N) OR x = (200+3N)/2

At least 2 = ((A∩B)+(B∩C)+(A∩C)) - 2(All three) 
 So, 130 = ((200+3N)/2) - 2(N/2) 
 Solving for N, we will get N = 60

Now, we are asked what percentage of this N represents all the residents.  
 Therefore, (60/200) x 100 OR 30%.  
  Option E

egmat · Answer

Step 1: Set up variables 
Let  x  = % of residents in NONE of the activities
Then: residents in ALL THREE =  x/2  (given)

Step 2: Use what "two or more" means 
"Two or more" = (exactly 2) + (all 3) =  65% 
So: (exactly 2) = 65 - x/2

Step 3: Find "only one" 
All regions must add to  100% :
(only one) + (exactly 2) + (all 3) + (none) = 100
(only one) + (65 - x/2) + (x/2) + x = 100
(only one) =  35 - x

Step 4: Apply the key formula 
 Sum of individual sets = (only one) + 2(exactly two) + 3(all three)

Why this formula works:  When you add up 40% + 50% + 60%, people in exactly 2 activities get counted twice, and people in all 3 get counted three times. This formula accounts for that.

Plugging in:
40 + 50 + 60 = (35 - x) + 2(65 - x/2) + 3(x/2)
150 = 35 - x + 130 - x + 3x/2
150 = 165 - x/2
 x/2 = 15 
 x = 30

Answer: E (30%)

In 3-set Venn problems, think in terms of 4 regions: (only one), (exactly two), (all three), and (none). They always sum to 100%, and this often gives you the equation you need.

danimals · Answer

I think it is  A: 10%

Total = A + B + C - (at least 2 groups) + (3 groups) + N
A = 40 
B = 50 
C = 60 
at least 2 groups = 65 
3 groups = 0.5N 
N = N 
solve for N

100 = 40 + 50 + 60 - 65 + 1.5N 
15 = 1.5N 
N = 10

ExpertsGlobal5 · Answer

Video explanation:    https://www.youtube.com/watch?v=kkJip8MCz6E

In a community, 40% residents practice yoga, 50% residents go for

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