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 walk. If 65% of the total residents engage in two or more of these three activities and 10% residents engage in all the three, what percentage of residents engage in none of the three activities?

A. 5
B. 10
C. 15
D. 25
E. 30

A. 5
B. 10
C. 15
D. 25
E. 30

Dereno · Answer

Consider a three set Venn diagram,

I+II+III + none =100

We have assumed the number of residents to be 100.

Yoga = 40

Cycling = 50

Walk = 60

Given that 65 engage in 2 or more activities.

10  engage in only three activities (  Exactly 3  )

So, Two or more includes = Exactly 2 + Exactly 3 = 65

Exactly 2 + Exactly 3 = 65

Exactly 2 + 10 = 65

Exactly 2 = 55

When we consider the over lapping equation,

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

I + 2II + 3III = 150

substitute the values of II=55 and III =10 in both equations, we get

I + 55+10+ none = 100

I + None = 35

Similarly, I + (2*55) + (3*10) = 150

I + 110 + 30 = 150

I + 140 = 150

I = 10

We get, None =  25

Option D

Bunuel · Answer

Experts' Global Video Explanation:
 
 https://www.youtube.com/watch?v=NJKBcMuUXoI

roshaun25 · Answer

I used the formula |A U B U C|=|A|+|B|+|C|-(|A ^ B|+|A ^ C|+|B ^ C|)+|A ^ B ^ C| -  , but somehow I got 5. I turned the percentages into real numbers but setting the total to 100, and based on the information given, determined that the number of people doing exactly 2 activities (|A ^ B|+|A ^ C|+|B ^ C|) is 55 (65-10). From there, I did 40 + 50 + 60 - 55 +10 - x = 100, which solves to x (the number who do no activities) = 5. Can someone explain what I did wrong?

Bunuel · Answer

Your mistake is in how you used |A and B| + |A and C| + |B and C|.

So that sum is not 55. It should be:

If you plug that in:

40 + 50 + 60 - 85 + 10 = 75

So 75 do at least one activity, and
100 - 75 = 25 do none.

That’s why your result came out wrong.

Bunuel · Answer

You can solve this question without complicated formulas.

Let Y be yoga, C cycling, W morning walk.

Given
Y = 40
C = 50
W = 60

All three = 10

Residents who do two or more = 65
So residents who do exactly two = 65 - 10 = 55

Now count total participation using inclusion–exclusion.

Y + C + W
= 40 + 50 + 60
= 150

This total counts
• people in exactly one activity once
• people in exactly two activities twice
• people in all three activities three times

Let x = percentage doing exactly one activity.

So
150 = x + 2 * 55 + 3 * 10
x = 10

So
Exactly one = 10
Exactly two = 55
Exactly three = 10

Total doing at least one
= 10 + 55 + 10
= 75

Therefore, doing none
= 100 - 75
= 25

Answer: D.

wleyes · Answer

Total = A+ B + C - Ex2 - 2Ex3 + None
100 = 40+50+60 - 55 - 20 + None
None = 25

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

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