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There are 68 children in the cafeteria of a school and all of the chil

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There are 68 children in the cafeteria of a school and all of the chil  [#permalink]

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New post 31 Mar 2017, 02:54
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E

Difficulty:

  85% (hard)

Question Stats:

56% (03:10) correct 44% (03:13) wrong based on 172 sessions

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There are 68 children in the cafeteria of a school and all of the children have something for lunch. Thirty-four of the children brought lunches from home, 23 of the children bought a drink from the cafeteria beverage machine, and 32 of the children bought fruit in the cafeteria. If 18 children did at least 2 of these things, how many children did exactly two of these things?

A. 3
B. 6
C. 9
D. 13
E. 15

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Re: There are 68 children in the cafeteria of a school and all of the chil  [#permalink]

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New post 01 Apr 2017, 08:07
There are 68 children in the cafeteria of a school and all of the children have something for lunch. Thirty-four of the children brought lunches from home, 23 of the children bought a drink from the cafeteria beverage machine, and 32 of the children bought fruit in the cafeteria. If 18 children did at least 2 of these things, how many children did exactly two of these things?

A. 3
B. 6
C. 9
D. 13
E. 15

34 - Home lunch
34 - No home lunch

Scenario 1:
Out of 34 without Home lunch, lets say 32 with fruit from cafeteria & remaining 2 out of 23 drink from cafeteria. Therefore, 34 home lunch includes 21 with drink from cafeteria. Therefore, atleast 2 will be 34-21=13.

Scenario 2:
Out of 34 without Home lunch, lets say 23 with drink from cafeteria & remaining 11 out of 32 fruit from cafeteria. Therefore, 34 home lunch includes 21 with fruit from cafeteria. Therefore, atleast 2 will be 34-21=13.

Hence, answer is 13 (D).
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Re: There are 68 children in the cafeteria of a school and all of the chil  [#permalink]

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New post 13 Apr 2017, 10:04
I m confused. Can somebody explain this using Venn diagram?
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Re: There are 68 children in the cafeteria of a school and all of the chil  [#permalink]

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New post 13 Apr 2017, 10:40
As per the question stem total number of children having only one type of meal and having at least two meals would be equal to 50 and 18 respectively. Now, 50 =(34+23+32)-2(at least two)- all three and that implies all three = 3. Subsequently, only two = 18-3 =15 (AT LEAST TWO = ONLY 2 + ALL 3)
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Re: There are 68 children in the cafeteria of a school and all of the chil  [#permalink]

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New post 13 Apr 2017, 10:45
34+23+32=89
89-68=21
21-18=3
So there are 3 children doing all 3 things
Leaves us with 21-(3×2)=15 children doing exactly 2 things

E

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Re: There are 68 children in the cafeteria of a school and all of the chil  [#permalink]

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Re: There are 68 children in the cafeteria of a school and all of the chil  [#permalink]

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New post 13 Apr 2017, 21:09
@parulM Please check The Venn Diagram ...According to me E will be the answer


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Re: There are 68 children in the cafeteria of a school and all of the chil  [#permalink]

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New post 01 Aug 2018, 04:17
So, this is a three set venn diagrm question
Let students with lunches = A
Let students with Drinks = B
Let students with Fruit= C

Using a nice formula (Which you can easily derive) :
Total = [A+B+C] - [sum of 2-group overlaps] + all three common + Neither

Here we are told that "18 children did at least 2 of these things" which mean the SUM common area of two and three overlaps is 18.
This includes AnB, BnC, AnC which is the combination of three 'two-set' overlap area and one 'three-set' overlap area.
Let 'center(three-set overlap area)' be x
So putting numbers in the formula we get : 68=[34+23+32] - 18 + x + 0
x=3

So, only two set overlap area = number of children did exactly two of these things = 18-3 = 15
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Re: There are 68 children in the cafeteria of a school and all of the chil  [#permalink]

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New post 10 Aug 2018, 17:39
1
Bunuel wrote:
There are 68 children in the cafeteria of a school and all of the children have something for lunch. Thirty-four of the children brought lunches from home, 23 of the children bought a drink from the cafeteria beverage machine, and 32 of the children bought fruit in the cafeteria. If 18 children did at least 2 of these things, how many children did exactly two of these things?

A. 3
B. 6
C. 9
D. 13
E. 15


We can use the following formula:

Total = n(L) + n(D) + n(F) - n(exactly two) - 2 * n(all three) + n(none)

68 = 34 + 23 + 32 - x - 2 * (18 - x) + 0 [Note: n(all three) = n(at least two) - n(exactly two)]

68 = 89 - x - 36 + 2x

68 = 53 + x

x = 15

Answer: E
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Re: There are 68 children in the cafeteria of a school and all of the chil  [#permalink]

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New post 15 Aug 2018, 03:08
RishiQV wrote:
So, this is a three set venn diagrm question
Let students with lunches = A
Let students with Drinks = B
Let students with Fruit= C

Using a nice formula (Which you can easily derive) :
Total = [A+B+C] - [sum of 2-group overlaps] + all three common + Neither

Here we are told that "18 children did at least 2 of these things" which mean the SUM common area of two and three overlaps is 18.
This includes AnB, BnC, AnC which is the combination of three 'two-set' overlap area and one 'three-set' overlap area.
Let 'center(three-set overlap area)' be x
So putting numbers in the formula we get : 68=[34+23+32] - 18 + x + 0
x=3

So, only two set overlap area = number of children did exactly two of these things = 18-3 = 15


Using the above calculations the value of x is x=-3. @bunnel: Are we supposed to take the positive value in this case? Please help.
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Re: There are 68 children in the cafeteria of a school and all of the chil &nbs [#permalink] 15 Aug 2018, 03:08
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