At a community center, a total of 31 people enrolled in at least one

Question

At a community center, a total of 31 people enrolled in at least one of three weekend workshops: pottery, photography, and creative writing. If 17 people enrolled in pottery, 13 enrolled in photography, and 12 enrolled in creative writing, how many people enrolled in all three workshops?

(1) 24 people enrolled in exactly one workshop.
(2) A total of 5 people were enrolled in both the pottery workshop and the photography workshop.

Bunuel · Accepted Answer

GMAT Club Official Solution:

At a community center, a total of 31 people enrolled in at least one of three weekend workshops: pottery, photography, and creative writing. If 17 people enrolled in pottery, 13 enrolled in photography, and 12 enrolled in creative writing, how many people enrolled in all three workshops?

Let

a = number enrolled in pottery and photography only
b = number enrolled in pottery and creative writing only
c = number enrolled in photography and creative writing only
t = number enrolled in all three workshops

If we add the enrollment counts for the three workshops, we get

17 + 13 + 12 = 42

But there are only 31 people in total. So the extra 11 counts must come from people who were counted more than once because they enrolled in more than one workshop.

When we add 17 + 13 + 12, each person enrolled in exactly two workshops, that is, each person in a, b, or c, is counted twice and therefore contributes 1 extra count. Each person enrolled in all three workshops is counted three times and therefore contributes 2 extra counts. Therefore:

a + b + c + 2t = 11

The question asks for t.

(1) 24 people enrolled in exactly one workshop.

If 24 people enrolled in exactly one workshop, then the remaining 31 - 24 = 7 people must be the ones who enrolled in either exactly two workshops or all three. So:

a + b + c + t = 7

Now compare the two equations:

a + b + c + 2t = 11
a + b + c + t = 7

Subtracting the second equation from the first gives:

t = 4

Sufficient.

(2) A total of 5 people were enrolled in both the pottery workshop and the photography workshop.

This implies a + t = 5. But together with a + b + c + 2t = 11 this still does not fix t.

For example, if t = 1, then a = 4, and we get
b + c = 5

If t = 4, then a = 1, and we get
b + c = 2

Not sufficient.

Answer: A.

Adit_ · Answer

We get an equation like:
x+2y = 11

Statement 1:
24 in exactly 1:

We get:
24+Two+Three = 31.
x+y = 7.
Thus we can calculate for All 3 using two equations.
SUFFICIENT.

Statement 2:
Here we see 5 that is part of two as well as 3. 
We dont have any additional equation to calculate intersection of two items and 3 items.
INSUFFICIENT.

Answer:  Option A

rajdeep212003 · Answer

Let:
P = Pottery 
Ph = Photography 
C = Creative Writing

Total = 31

Given:
P = 17, Ph = 13, C = 12

---

17 + 13 + 12 = 42

So extra count = 42 − 31 = 11

This extra comes from people counted more than once.

---

Let:
a = number of people in all three 
b = number of people in exactly two

Then:
b + 2a = 11 ...(1)

(because people in exactly two are counted once extra, and people in all three are counted twice extra)

---

Statement (1):

Exactly one = 24

So,
people in more than one = 31 − 24 = 7

That is:
b + a = 7 ...(2)

---

From (1) and (2):

b + 2a = 11 
b + a = 7

Subtract:

a = 4

Unique value → sufficient

---

Statement (2):

P ∩ Ph = 5

This alone does not give enough to form both equations

→ Not sufficient

---

Final Answer: A

---

Hope this helps. 
Feel free to tag me if you want to go through it together.

– Rajdeep

thisisakr · Answer

Using the inclusion-exclusion formula:
Total = A + B + C − (exactly two) − 2(all three)
Where A=17, B=13, C=12, Total=31
So: 31 = 17 + 13 + 12 − (sum of exactly two groups) − 2(all three)
31 = 42 − (sum of two-group overlaps) − 2(all three)
Sum of two-group overlaps + 2(all three) = 11 ——— a
Let x = all three, and let the pairwise overlaps be known or unknown

Statement 1:
24 people enrolled in exactly one workshop.
Using: Total = (exactly one) + (exactly two) + (exactly three)
31 = 24 + (exactly two) + x
(exactly two) + x = 7
Also from (a): (exactly two) + 2x = 11
Subtracting: x = 4
Statement 1 alone is  sufficient

Statement 2:
5 people enrolled in both pottery and photography (including possibly all three).
This gives one pairwise overlap but we still don't know the other two pairwise overlaps. Multiple solutions are possible.
 not sufficient

Answer A

BongBideshini · Answer

Let:
x = number enrolled in all three workshops
a = number enrolled in exactly one workshop
b = number enrolled in exactly two workshops
a+b+x=31
31+b+2x=42
b + 2x = 11

Statement (1) 
a = 24
24 + b + x = 31
b + x = 7
We already have,
b + 2x = 11
 x = 4 
 Statement (1) is sufficient.

Statement (2) 
5 people were enrolled in both pottery and photography.
This includes those in all three.
So this does not uniquely determine the number in all three, because overlaps with writing are unknown.
 Statement (2) is not sufficient.

Ans A

At a community center, a total of 31 people enrolled in at least one

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GMAT Data Sufficiency (DS) Questions

At a community center, a total of 31 people enrolled in at least one

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