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

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.
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.

x - y = 55 - 10 = 45

Check out this video discussion on how to handle max-min in Sets: https://youtu.be/oLKbIyb1ZrI

Well everything is on the diagram but I'll try to elaborate:

-----(-----------)---- 80 phones (black dashes represents # of people who don't have phone, so 20 people don't have phone);
-----(-----------)---- 75 DVD's (black dashes represent # of people who don't have DVD, so 25 people don't have DVD);

So you see that overlap of phone and DVD owners is 55 and 45 people don't have either phones or DVD's.

To have min overlap of 3 let 20 people who don't have phone to have MP3 and also 25 people who don't have DVD's to have MP3. So we distributed 45 MP3 and there is no overlap of 3 yet. But 10 DVD's are still left to distribute and only people to have them is the group which has both MP3 and phone.

-----(--)-------------;
-----(--)-------------;
-----(--)-------------.

Hope it's clear.

My imagination about minimum overlapping percentage.

For mainhoon and others who still didn't understand, you have to create the situation of minimum possible overlap. This is done in the way Bunuel explains it:

After the first step of distributing phones and DVDs among the 100, you get 55 as the minimum possible overlap for these 2.
Now you bring in the MP3s, of which there are 55. Remember, you are trying to make sure as few people as possible get all 3 - if possible, none. So, what do you do when you have 55 MP3s, and 45 people left who don't have an DVD and a phone? Distribute it among them first, naturally!

After doing that, you still have 10 left over. You are left with no choice but to distribute these among those who already have a DVD and phone. Hence the minimum overlap you can possibly achieve is 10.

This gives you the answer of 55 - 10 = 45.

Nice approach by Bunnel.

Even if we have to use Venn diagram.

For minimum overlap, 80 and 75 will have maximum overlap and 55 will be distributed among three sections : namely 80,75 and all three

Reason : If 55 have elements with no overlapping then 75 and 80 can not have max overlap.
To have minimum overlap, 75 and 80 must overlap to the max. So those elements which are not part of intersection of all the three can be shared between 75 and 80.

a+b+c = 80 .............1
b+c+d = 75...............2
a+c+d = 55...............3
=> 55 is distributed in 3 sections.... 80 intersection 55 ; 74 intersection 55; 55 intersection 75 intersection 80.

a+b+c+d = 100............4
using 4 and 1 we get d = 20
using 4 and 2 we get a = 25
using the above values of a,d and equation 3
we get c = 55 -a-d = 55- 20-25 = 10

Hence minimum is 10.

For max :
75 will have 55 inside
80 will have the intersection of 75 and 55 i.e. 55

thus total becomes 55 + 20 + 25 = 100 where 55 = intersection of all the three
20 = 75 outside the intersection of 75 and 55
25 = 80 outside the intersection of 75,55,80

thus 55 is the max.

x-y = 55-10 = 45

Fantastic explanation. Thanks Bunuel for your explanation and Hussain15 for posting this question.

(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.

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.

A more algebraic approach:

If we denote by \(N_2\) the number of households with exactly two types of devices, and by \(N_3\) the number of those with all three types, than we can write the following equation:

\(100=75+80+55-N_2-2N_3,\) from which we get that \(N_2+2N_3=110.\)
Households with exactly two types of devices we counted twice, so we have to subtract them once. Households with exactly three types of devices we counted three times, therefore we have to remove them twice.

Maximum for \(N_3\) is 55, in which case \(N_2=0\), meaning there are no households with exactly two types of devices, but only either with just one type or all three of them. Everybody who has an MP3 player, also has a DVD player and a cell phone.

In order to determine the minimum number of households with exactly three types of devices, let's take a look at the possible combinations of two devices. If \(N_3\) is minimum, there should be households with exactly two types of devices but not three.

All types of devices, but also those having exactly two types, explicitly DVD players and Cell Phones - If \(N_{DC}\) is the number of households that have exactly these two types of devices, then \(N_{DC}+N_3\geq75+80-100=55\).
All types of devices, but also those having exactly two types, explicitly DVD players and MP3 players - If \(N_{DM}\) is the number of households that have exactly these two types of devices, then \(N_{DM}+N_3\geq75+55-100=30\).
All types of devices, but also those having exactly two types, explicitly Cell phone and MP3 players - If \(N_{CM}\) is the number of households that have exactly these two types of devices, then \(N_{CM}+N_3\geq80+55-100=35\).

Adding the above three inequalities, we obtain that \(N_{DC}+N_{DM}+N_{CM}+3N_3\geq120.\) Since \(N_{DC}+N_{DM}+N_{CM}=N_2\), we get that \(N_2+3N_3\geq120.\) Taking into account that \(N_2+2N_3=110\), we obtain \(110+N_3\geq{120}\) or \(N_3\geq{10}.\)

\(55-10=45\)

Answer C

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.

I have given the solution to a similar problem. Here's the solution for this problem.

1. Let C, D and M be who have cell only, dvd only and mp3 only and CD be who have both cell and dvd only and so on.
2 CDM is minimum when C+D+M=0 and maximum when CD+CM+DM=0.
3. C+D+M+CD+DM+CM+CDM=100 -- (1) because the total of all the mutually exclusive has to sum up to 100 which is the number of households.
4. (C+CD+CM+CDM) + (D+CD+DM+CDM) +(M+CM+DM+CDM) = 80+75+55 or ,
C+D+M+2CD+2CM+2DM+3CDM =210 ----(2)
5. (2) - (1) = CD+CM+DM+2CDM=110 --- (3)
6. (3) - (1) => CDM-(C+D+M) =10 ---(4)
We can find the minimum value when we put (C+D+M) = 0 above giving minimum CDM=10
7. To find maximun value equation (1) can be changed as
C+D+M+CDM=100 -- (5) by putting CD+CM+DM=0
From (4) and (5) we have CDM=55 which is maximun CDM
8.The answer is max CDM - min CDM i.e., 55-10=45

As a shortcut for this type of problems where there are 3 items , the minimum possible for all the 3 can be found out by adding the percentages of the 3 i.e., 55+75+80=210 . and subtracting 200 i.e, 210-200=10 . the maximum can be found out by adding the minimum to 100 and dividing by 2 i.e., (100+10)/2 = 55. Both the answers are percentages.

Responding to a pm:

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.

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.

Stay away venn diagram for problems of this kind..
The maximum number is 55, you don't need any computing for this.

For finding the minimum possible household, just find out the number of households that would not have these gadgets

Number of houses that don't have a cellphone, DVD and MP3 are, 20, 25 and 45 respectively

When there is minimum overlap the number of households that cannot have all the three gadgets together is the sum of these three numbers which is 90. So the minimum possible number of households that can have all three gadgets is 10

55-10 = 45

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!!