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

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In a village of 100 households, 75 have at least one DVD [#permalink]  27 May 2008, 18:41
In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. Every village has at least one of these three devices. If x and y are respectively the greatest and lowest possible number of households that have all three of these devices, x-y is:

a.65
b.55
c.45
d.35
e.25
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Re: Venn Diagram/Overlapping Set [#permalink]  27 May 2008, 18:58
downtobiz wrote:
In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. Every village has at least one of these three devices. If x and y are respectively the greatest and lowest possible number of households that have all three of these devices, x-y is:

a.65
b.55
c.45
d.35
e.25

C

1.x=55
2. y=
2.1 min(DVD, cell phone)=80+75-100=55
2.2 min (cell, MP3)=55+55-100=10
2.3 y=10
x-y=45
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Re: Venn Diagram/Overlapping Set [#permalink]  27 May 2008, 21:03
I dint get the solution - I think it is 55 (B)

100=75+80+55-ab-bc-ac-2abc

Now abc is max when ab+bc+ca=0 hence x=55

Min value of abc = 0 (for instance when ab=40, bc=50, ca=20 which is very much possible)
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Re: Venn Diagram/Overlapping Set [#permalink]  28 May 2008, 07:48
iamcartic wrote:
I dint get the solution - I think it is 55 (B)

100=75+80+55-ab-bc-ac-2abc

Now abc is max when ab+bc+ca=0 hence x=55

Min value of abc = 0 (for instance when ab=40, bc=50, ca=20 which is very much possible)

min value of abc can not be 0. coz in that case the ab+bc+ca = 110...which is not possible. So the min value of abc can be 10 and hence x-y = 45 ....

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Re: Venn Diagram/Overlapping Set [#permalink]  28 May 2008, 10:13

In order to arrive at the maximum value look at the question in the following way:

210= 75+ 55+ 80
also,
210= A (Households with only one equipment) X1 + B(Households with two equipments) X2 + D (Households with three equipments) X3-----------------[1]

In order to have the max. value lets put D=55, in the above equation (Higher values are obviously not possible)

210-165= 45
A=45, B=0, D=55 in equation [1] meets all the criteria. Thus 55 is the max value.

By arriving at 55, two options can be at least opted out.
In order to arrive at the minimum value, D in equation [1] should have the minimum value

now let us think of the possible minimum values of D (as per the options)
D=10 (55-10=45)
D=20 (55-20=35)
D=30 (55-30=25)

Going back to equation [1], and trying the different values of D, we'd arrive at option C) as the answer.
Re: Venn Diagram/Overlapping Set   [#permalink] 28 May 2008, 10:13
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