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# In a village of 100 households, 75 have at least one DVD

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Intern
Joined: 30 Apr 2016
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Location: India
Concentration: Entrepreneurship, General Management
Schools: Stern '19 (S)
GMAT 1: 640 Q44 V35
GMAT 2: 700 Q49 V37
GPA: 3.8
Re: In a village of 100 households, 75 have at least one DVD [#permalink]

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07 Nov 2016, 10:21
navigator123 wrote:
Well its way too straight forward.

The max value(x) must be 100.
Of all 3 the least value will be min possible value = 55

X-Y = 100-55 = 45.

Bunuel VeritasPrepKarishma Does this approach work? This was what I though of too because :

100 = No. of households
55 = No. of MP3s for sure (so assuming that min of 55 households have all 3)

Kudos [?]: 37 [0], given: 45

Veritas Prep GMAT Instructor
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Re: In a village of 100 households, 75 have at least one DVD [#permalink]

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07 Nov 2016, 20:50
vedantkabra wrote:
navigator123 wrote:
Well its way too straight forward.

The max value(x) must be 100.
Of all 3 the least value will be min possible value = 55

X-Y = 100-55 = 45.

Bunuel VeritasPrepKarishma Does this approach work? This was what I though of too because :

100 = No. of households
55 = No. of MP3s for sure (so assuming that min of 55 households have all 3)

This is incorrect.

How can 100 households have all 3 when only 55 households have MP3 players?
55 is the MAXIMUM number of households that can have all 3, not minimum.
and 10 is the minimum number of households that must have all 3 (explained in solutions on first page)
So 55 - 10 = 45
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In a village of 100 households, 75 have at least one DVD [#permalink]

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11 Dec 2016, 04:52
[quote="VeritasPrepKarishma"]hiredhanak: I am assuming you are looking for a venn diagram solution to this question..

It is pretty simple.
First of all maximum number of households: We want to bring the circles to overlap as much as possible.
80 - Cell phone
75 - DVD
55 - MP3
Lets take Cell phone and DVD circles since they will have maximum overlap. They must overlap in 55 households so that total number of households is 100. Now put the MP3 households in a way to maximize all three overlap.
Attachment:
Ques1.jpg

So at most 55 households can have all 3.
hi karishma

i do not understand. how did you derive at most 55 households can have all 3 by Van diagram

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Re: In a village of 100 households, 75 have at least one DVD [#permalink]

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10 May 2017, 19:05
How would one solve this problem if a portion of the 100 households did not own ANY of the electronics mentioned in the question?

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Re: In a village of 100 households, 75 have at least one DVD [#permalink]

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11 May 2017, 08:26
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Expert's post
hoopsgators wrote:
How would one solve this problem if a portion of the 100 households did not own ANY of the electronics mentioned in the question?

The question does not mention whether there are some households that own no electronics. There could be some.

Look at the figure here: https://gmatclub.com/forum/in-a-village ... ml#p825632

When discussing the maximum overlap case, none NEEDN'T be 0. It may be, it may not be.

Put the three circles within each other. The 75 circle within the 80 circle and the 55 circle within the 75 circle. The overlap will be 55 in that case and none = 20. The figure only shows one of the possible ways of obtaining the maximum.

In the case of minimum, you would want the circles to lie as far apart as possible. If none is anything other than 0, the circles would need to overlap more. Say none = 10, the circles of 80 and 75 would need to have an overlap of 65. So the 55 circle can occupy 25 but an overlap of 30 will be needed. Hence minimum overlap will increase. To minimize the overlap, we will need None = 0.
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In a village of 100 households, 75 have at least one DVD [#permalink]

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24 May 2017, 06:29
1
This post was
BOOKMARKED
I could not load the Venn Diagram, Please try to visualize it. ( The diagram is posted below)

a, b, and c one product each.

x, y, and z two products each.

d all three products.

a + b + c + d + x + y + z = 100
If you assume there is no one who uses two products each, then x = y = z = 0.
So, a + b + c + d = 100 ----------------(1)
a + d = 80 ------------------------------(2)
b + d = 75 ------------------------------(3)
c + d = 55 ------------------------------(4)

From 2, 3, and 4,
a + b + c + 3d = 210 ----------------(5)
a + b + c + d + 2d = 210

From 1 and 5,
100 + 2d = 210
d = 55 --------------Maximum.

If you assume there is no one who uses one product each, then a = b = c = 0.
So, d + x + y + z = 100 ----------------(6)
x + y + d = 80 ------------------------------(7)
x + z + d = 75 ------------------------------(8)
y + z + d = 55 ------------------------------(9)

From 7, 8, and 9,
2x + 2y + 2z + 3d = 2(x + y + z + d ) + d = 210 ----------------(10)

From 6 and 10,
2(100) + d = 210
d = 10 --------------Minimum.

So Maximum – minimum = 55 – 10 = 45.
Attachments

Sets Club.PNG [ 10.03 KiB | Viewed 295 times ]

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Re: In a village of 100 households, 75 have at least one DVD [#permalink]

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16 Aug 2017, 16:30
I often just don't get the complexity of such kind of questions
max overlap is 55 (as this is the max number of 3 types of households that can have all three devices)
min overlap for 3 devices is 75+55+80 - 100*2 (max number for households that can have 2 devices) = 210 - 200 = 10

so 55 - 10 = 45
Am i missing something here?

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Re: In a village of 100 households, 75 have at least one DVD   [#permalink] 16 Aug 2017, 16:30

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