In a community, 30% residents practice yoga, 50% residents go for

Question

In a community, 30% residents practice yoga, 50% residents go for cycling, and 70% residents go for morning walks. 30% of the total residents are involved in exactly two of the three activities. If the number of residents involved in all three activities is thrice the number of residents involved in none, what percentage of residents are involved in all the three activities?

A. 0
B. 4
C. 12
D. 18
E. 24

A. 0
B. 4
C. 12
D. 18
E. 24

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Adit_ · Answer

30+50+70- -2 +None=100
Both=30 therefore,
120-2 +None=100
Solving for None:
None=4
Therefore Three activities=3*4=12

Answer:  Option C

amansoni5 · Answer

Let total residents be 100.
Y - 30
C - 50
MW - 70
Let n be the residents which go for neither.
By the inclusion and exlcusion formula
Total = Y + C + MW - (sum of exactly 2 groups) - 2 * (common group for all 3) + Neither
100 = 30 + 50 + 70 - 30 - 2 * (3*n) + n
100 = 120 - 6n + n
5n = 20
n = 4
therefore, number of residents common for all the three activties is 3*4 = 12.

Edskore · Answer

The key concept here is Three-Set Inclusion-Exclusion — one of the most consistently tested PS topics on the GMAT Focus Edition.

The formula is:
Total = A + B + C − (exactly two groups) − 2(all three) + None

Note the common trap first: many students write "−(at least two)" instead of "−(exactly two)." The question gives you exactly two (30%), so you can plug directly. If you mix these up, you'll get a wrong equation every time.

Let total residents = 100 (picking 100 makes percentages clean to work with).

Step 1: Write down what we know.
A (yoga) = 30, B (cycling) = 50, C (walks) = 70
Exactly two activities = 30
All three = x, None = n, and we're told x = 3n

Step 2: Plug into the Inclusion-Exclusion formula.
100 = 30 + 50 + 70 − 30 − 2x + n
100 = 120 − 2x + n

Step 3: Substitute x = 3n.
100 = 120 − 2(3n) + n
100 = 120 − 6n + n
100 = 120 − 5n
5n = 20 → n = 4

Step 4: Solve for x.
x = 3n = 3 × 4 = 12

Answer: C (12%)

The trap here is the formula version — some students subtract "2 × (at least two)" which double-counts the "exactly two" group. Once you're clear that the formula uses exactly two, this question solves cleanly in under 90 seconds.

Takeaway: Whenever a three-set problem gives you the "exactly two" overlap directly, you can plug it into the Inclusion-Exclusion formula without any additional decomposition.

Dereno · Answer

I + II + III + none = 100

I + 2II + 3III = (30+50+70) = 150

Subtracting both, we get

II + 2III - none = 50

We are given that II=30

2III - none = 20

More over , III = 3*none.

2*(3*none) - none = 20

5* none = 20

none = 4

III = 3* none = 3*4 =  12

Option C

ExpertsGlobal5 · Answer

Video explanation:

https://www.youtube.com/watch?v=8UFlJsI91_8

In a community, 30% residents practice yoga, 50% residents go for

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