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# In a village of 100 households, 75 have at least one DVD player, 80

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Veritas Prep GMAT Instructor
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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18 Dec 2013, 02:44
cumulonimbus wrote:
noboru wrote:
I have done it in 1:41 doing this:

x= (75+55+80-100-z)/2 for z=0 ->55
y=the same for z=100 ->10

Therefore x-y=45

PS: z is obviously the people who have 2 devices.

Hi Karishma,
is this correct? I am not able to get this approach.

100=80+75+55-z-2a

where z - no. of people with exactly 2 devices
a – no of people with 3 devices
a(max) = x, a(min) = y, (x-y) ?
100 = 210-z-2a
2a = 110-z
a = (110-z)/2

Now, how can we ever take z = 0, as even in case of maximum overlap i.e. when maximum no. of people have 3 devices,
z = 20, which is the overlap b/w C(mobile device) and D(DVD).

Also how is z = 100, all 100 cannot have exactly 2 devices.

This solution hasn't considered 'None'. Even if we assume that they saw that None = 0 works for both cases and hence None is immaterial, notice that when z = 100, you get a as 5. That is not correct.

The maximum value of z is 90 (the number of people with exactly 2 devices). This gives the minimum number of people with 3 devices as 10.
The minimum value of z is 0 so max value of a is 55.
Either way, to find the max/min value of z you will need to use some logic. You might as well use it for max/min value of a.
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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11 May 2014, 22:54
mitulsarwal wrote:
shs0145 wrote:
Am I missing something here??? it seems straightforward......

The obvious maximum that have all 3 is 55, because you are limited by the SMALLEST number.

The minimum is simply the sum of the max of each people who DONT have the product, so:

100-80 = 20 don't have Cell
100-75 = 25 don't have DVD
and 100-55 = 45 don't have MP3

SO a total of 20+25+45 = 90 combined who might NOT have SOME combination of the 3 products. So subtract that from 100, to give you the minimum of the people who COULD have all 3 and you get 100-90 = 10.

55-10 = 45

i think this is the simplest way to solve this, even i did it the same way.

This looks the simplest approach of the lot. I wonder if it will be true for all scenarios. Bunuel any comments ?
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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19 May 2014, 06:22
himanshujovi wrote:
mitulsarwal wrote:
shs0145 wrote:
Am I missing something here??? it seems straightforward......

The obvious maximum that have all 3 is 55, because you are limited by the SMALLEST number.

The minimum is simply the sum of the max of each people who DONT have the product, so:

100-80 = 20 don't have Cell
100-75 = 25 don't have DVD
and 100-55 = 45 don't have MP3

SO a total of 20+25+45 = 90 combined who might NOT have SOME combination of the 3 products. So subtract that from 100, to give you the minimum of the people who COULD have all 3 and you get 100-90 = 10.

55-10 = 45

i think this is the simplest way to solve this, even i did it the same way.

This looks the simplest approach of the lot. I wonder if it will be true for all scenarios. Bunuel any comments ?

This is a wrong approach. Imagine you have number like 80, 75, 65 instead of 55. In this case the maximum would not be 65 as suggested but would be 60.

Even for minimum case this will not work. You have to use the approached specified above.

Hope it helps!!!

Kudos if you like!!
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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19 May 2014, 20:52
1
mittalg wrote:
This is a wrong approach. Imagine you have number like 80, 75, 65 instead of 55. In this case the maximum would not be 65 as suggested but would be 60.

Even for minimum case this will not work. You have to use the approached specified above.

Hope it helps!!!

Kudos if you like!!

Actually the method used above is fine. You will get a max of 65, not 60. Imagine the circles one inside the other. 75 inside the 80 and 65 inside the 75. All 65 will have all 3 products.
For minimum, you need to spread them as far away as possible. 80 and 75 will have an overlap of 55. So after spreading 65 on 45, you will be left with 20 which will need to overlap with the 55. Hence minimum will be 20.

In the method used above, people who do not own at least one product will be 20, 25 and 35. Spread them out as far apart as possible, you get 20+25+35 = 80
Minimum number of people who must have all 3 = 100 - 80 = 20

So you can go with people who have products or those who don't. Either way, you get the same answer.
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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19 May 2014, 20:57
VeritasPrepKarishma wrote:
mittalg wrote:
This is a wrong approach. Imagine you have number like 80, 75, 65 instead of 55. In this case the maximum would not be 65 as suggested but would be 60.

Even for minimum case this will not work. You have to use the approached specified above.

Hope it helps!!!

Kudos if you like!!

Actually the method used above is fine. You will get a max of 65, not 60. Imagine the circles one inside the other. 75 inside the 80 and 65 inside the 75. All 65 will have all 3 products.
For minimum, you need to spread them as far away as possible. 80 and 75 will have an overlap of 55. So after spreading 65 on 45, you will be left with 20 which will need to overlap with the 55. Hence minimum will be 20.

In the method used above, people who do not own at least one product will be 20, 25 and 35. Spread them out as far apart as possible, you get 20+25+35 = 80
Minimum number of people who must have all 3 = 100 - 80 = 20

So you can go with people who have products or those who don't. Either way, you get the same answer.

Yes, you are right Karishma. When I did the analysis, I mis-assumed that all of the 100 people have at least one of the 3 products which is not the case. Sorry for the confusion.
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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19 May 2014, 23:15
Its funny how complicated your are thinking. Here was my approach:

All the figure say they have AT LEAST ... (so maybe more)

all of them could have 100. which is the maximum.

the lowest possible number is 55(55 have at least one mp3 player)

100-55=45
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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20 May 2014, 09:53
dansa wrote:
Its funny how complicated your are thinking. Here was my approach:

All the figure say they have AT LEAST ... (so maybe more)

all of them could have 100. which is the maximum.

the lowest possible number is 55(55 have at least one mp3 player)

100-55=45

This is not correct. The question does not say "at least 75 people have one DVD player". It says "75 people have at least one DVD player" which means 75 people have one or more DVDs. It doesn't mean that number of people having a DVD is 75 or more.
The maximum is 55 and minimum is 10. Check the solutions given above.
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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26 Jan 2015, 07:06
For the minimum people seem to be having some difficulty. Look at it logically, 45 will not receive an MP3 player, 25 will not receive a DVD and 20 will not receive a cell phone. The total that no one could have all would be 90. The total of people is 100, 100-90 is 10. Y is 10

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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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26 Jan 2015, 07:06
For the minimum people seem to be having some difficulty. Look at it logically, 45 will not receive an MP3 player, 25 will not receive a DVD and 20 will not receive a cell phone. The total that no one could have all would be 90. The total of people is 100, 100-90 is 10. Y is 10

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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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19 Jun 2015, 05:43
Hussain15 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. 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

The maximum value for all three is 55 and the minimum value for it is 10. Therefore the answer is C.
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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21 Jun 2015, 18:50
VeritasPrepKarishma wrote:
suchoudh wrote:
VeritasPrepKarishma wrote:
Now, minimum number of households: We want to take the circles as far apart from each other as possible. Now put the MP3 households in a way to minimize all three overlap. So put it in the shaded region. You will need to put 10 of the MP3 households in the common area. So minimum overlap is 10.

Ok, and how did you get to the number 10?

(25 + 20 =) 45 households have either only Cell or only DVD Player. Out of the 55 households who have MP3 Players, put 45 in these areas so that all three do not overlap. But the rest of the (55 - 45 =)10 households that have MP3 players need to be put in the common region consisting of 55 households that have both Cell and DVD Player. Hence overlap of all three will be 10.

________________________________

Where does it say that every household in the village have at least one of these three devices?
There could be none of these three devices in some households.
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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22 Jun 2015, 05:29
1
wrightvijay wrote:
________________________________

Where does it say that every household in the village have at least one of these three devices?
There could be none of these three devices in some households.

Hi wrightvijay,

It doesn't matter whether they have given the highlighted fact above or not because it's the question of maximizing and minimizing the overlapping portion which will be maximized when everything else is minimized and vice versa.
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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22 Jun 2015, 20:29
2
wrightvijay wrote:

Where does it say that every household in the village have at least one of these three devices?
There could be none of these three devices in some households.

It's not given. It is something we have derived using logic.

Take a simpler case:

Say you have 3 siblings (A,B and C) and 5 chocolates which you want to distribute among them in any way you wish. Now you want to minimise the number of your siblings who get 3 chocolates. No one gets more than 3. What do you do?

Will you leave one sibling without any chocolates (even if he did rat you out to your folks!)? No. Because if one sibling gets no chocolates, the other siblings get more chocolates and then more of them will get 3 chocolates. So instead you give 1 to each and then give the leftover 2 to 2 of them. This way, no sibling gets 3 chocolates and you have successfully minimised the number of siblings who get 3 chocolates. Basically, you spread out the goodies to ensure that minimum people get too many of them.

This is the same concept. When you want to minimise the overlap, you basically want to spread the goodies around. You want minimum people to have all 3. So you give atleast one to all of them.
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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28 Jul 2015, 23:04
The most (x) number of households with all three is the least of the given, so 55. That one's straightforward.

The least (y) is when you add up all of the single households and assume as many households with at least two of the three items. 75 + 80 + 55 = 210. There are 100 households and if we are to maximize the number of households with at least two of the three items, that's 200 of the 210 items. Then, 10 are still remaining which tells us that there must be at least 10 households which have three items once every household doubles up. y = 10

x-y= 55-10=45
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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29 Aug 2015, 15:19
Hussain15 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. 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

To The Point Answer with Simpler Approach and in Relatively Quicker Time.

Approach

:-

x = greatest possible number of households that have all three gadgets.
[Max. no. of household who does have all three gadgets]

y = lowest possible number of households that have all three gadgets.
[Max. no. of household who does have all three gadgets] - [Max. no. of household who does NOT have all three gadgets]

Calculations

:-

55 <--- Max. no. of household who does have all three gadgets.
(..because among the total 100 households, 55 MP3 owners holds the lowest share when compared to other gadget owners. Thus a max. of 55 households, in total, are eligible to have all three gadgets).

45 <--- Max. no. of household who does NOT have all three gadgets.
(..because among the total 100 households, 45 does NOT own MP3 players and it is the maximum when compared to same scenario of other gadgets. Thus a max. of 45 households, in total, are NOT eligible to have all three gadgets).

Result

:-

x = 55 = greatest possible number of households that have all three gadgets.

y = 10 = (55 - 45) = lowest possible number of households that have all three gadgets.

Answer = x - y = 55- 10 = 45
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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19 Mar 2016, 04:01
VeritasPrepKarishma wrote:
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.

Now, minimum number of households: We want to take the circles as far apart from each other as possible. Now put the MP3 households in a way to minimize all three overlap. So make the MP3 households occupy the shaded region i.e. region occupied by DVD players alone and cell phone alone. You will be able to adjust 45 MP3s outside the common area but you will need to put 10 of the MP3 households in the common area. So minimum overlap is 10.
Attachment:
Ques2.jpg

x - y = 55 - 10 = 45

Hi Karishma

Total-Neither = A+B+C - (common in two) - 2(common in three)
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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20 Mar 2016, 23:39
VeritasPrepKarishma wrote:
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.

Now, minimum number of households: We want to take the circles as far apart from each other as possible. Now put the MP3 households in a way to minimize all three overlap. So make the MP3 households occupy the shaded region i.e. region occupied by DVD players alone and cell phone alone. You will be able to adjust 45 MP3s outside the common area but you will need to put 10 of the MP3 households in the common area. So minimum overlap is 10.
Attachment:
Ques2.jpg

x - y = 55 - 10 = 45

Hi Karishma

Total-Neither = A+B+C - (common in two) - 2(common in three)

In this question, you cannot just plug in the numbers in a formula and get the answer. You will need to do a bit of analysis for the possible overlap since you need the maximum and minimum value of overlap of all 3.

Common in three = A + B + C - (common in two) - Total + Neither
Common in three = (75 + 80 + 55 - (common in two) - 100 + Neither)/2
Common in three = (110 - (common in two) + Neither)/2
You need to maximise "common in three". For that, imagine the 75 circle inside the 80 circle and the 55 circle is inside the 75 circle. In that case, neither = 100 - 80 = 20 and common in two = 75 - 55 = 20.
So Common in three = 55

Similarly, we will minimise Common in three.

Effectively, we have used the venn diagram only to answer the question.
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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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21 Jul 2016, 08:46
VeritasPrepKarishma wrote:
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.

Now, minimum number of households: We want to take the circles as far apart from each other as possible. Now put the MP3 households in a way to minimize all three overlap. So make the MP3 households occupy the shaded region i.e. region occupied by DVD players alone and cell phone alone. You will be able to adjust 45 MP3s outside the common area but you will need to put 10 of the MP3 households in the common area. So minimum overlap is 10.
Attachment:
Ques2.jpg

x - y = 55 - 10 = 45

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Re: In a village of 100 households, 75 have at least one DVD player, 80  [#permalink]

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03 Nov 2016, 14:40
1

Venn diagram representation.
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Re: In a village of 100 households, 75 have at least one DVD player, 80 &nbs [#permalink] 03 Nov 2016, 14:40

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