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# In the city of San Durango, 60 people own cats, dogs, or rab

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Intern
Joined: 30 Jun 2012
Posts: 11
In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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02 Dec 2012, 00:45
3
1
18
00:00

Difficulty:

15% (low)

Question Stats:

77% (01:25) correct 23% (02:00) wrong based on 495 sessions

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In the city of San Durango, 60 people own cats, dogs, or rabbits. If 30 people owned cats, 40 owned dogs, 10 owned rabbits, and 12 owned exactly two of the three types of pet, how many people owned all three?

A 2
B 4
C 8
D 12
E 32
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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Updated on: 02 Dec 2012, 01:23
2
6
apoorvarora wrote:
In the city of San Durango, 60 people own cats, dogs, or rabbits. If 30 people owned cats, 40 owned dogs, 10 owned rabbits, and 12 owned exactly two of the three types of pet, how many people owned all three?

A 2
B 4
C 8
D 12
E 32

Total = C + D + R - (CD + DR + CR) - 2CDR

60 = 30 + 40 + 10 - (12) - 2x

x = 4

Hope it helps!
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Originally posted by BrushMyQuant on 02 Dec 2012, 01:08.
Last edited by BrushMyQuant on 02 Dec 2012, 01:23, edited 1 time in total.
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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Updated on: 09 Dec 2012, 09:30
5
2
$$@apoorvarora$$

when a problem is like you posted here or similar, like this one

Quote:
Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

is much better to use the formula Total = C + D + R - Exactly two - 2*All three (refers also to the post linked by Bunuel (the second one)

$$IF$$ the problem is like this

Quote:
Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?

or this

Quote:
The 38 movies in the video store fall into the following three categories: 10 action, 20 drama, and 18 comedy. However, some movies are classified under more than one category: 5 are both action and drama, 3 are both action and comedy, and 4 are both drama and comedy. How many action-drama-comedy movies are there?
(souce MGMAT)

you can use either $$venn diagram$$ or the formula $$Total= A+B+C - (AnB+AnC+BnC) +AnBnC + Neither$$ is the same thing: it depends on which you are more comfortable

In the first case, that is the case of the question of the post at stake, is much better use the formula because is faster ans because set up a Venn diagram is more convoluted and prone of errors. Try to stay flexible.

Hope this is useful
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Originally posted by carcass on 09 Dec 2012, 09:19.
Last edited by carcass on 09 Dec 2012, 09:30, edited 1 time in total.
##### General Discussion
Intern
Joined: 30 Jun 2012
Posts: 11
Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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02 Dec 2012, 01:16
No the ans is 4...

You are considering a case when cat dogs and rabbit are not exactly equal to 12 .

The solution shall be 60=30+40+10-12-2x

and hence x shall be 4.
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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02 Dec 2012, 01:19
2
Total = C + D + R - Exactly two - 2*All three

60 = 30 + 40 + 10 - 12 - 2x

2x = 8

x = 4
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Joined: 02 Sep 2009
Posts: 62498
Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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02 Dec 2012, 03:12
apoorvarora wrote:
In the city of San Durango, 60 people own cats, dogs, or rabbits. If 30 people owned cats, 40 owned dogs, 10 owned rabbits, and 12 owned exactly two of the three types of pet, how many people owned all three?

A 2
B 4
C 8
D 12
E 32

3 overlapping sets with theory, diagrams and examples: ADVANCED OVERLAPPING SETS PROBLEMS
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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08 Dec 2012, 23:19
Can any1 show it by using venn diagrams?
I tried to set up the equation as follows:
People owning all 3 pets = 'X'.

People owning cats and dogs only = a

Owning rabbits and cats only = b

Owning dogs and rabbits only = c

Then i set up eqn as follows to find X

30-(a+b+x) + 40-(a+x+c) +10 - (b+x+c) = 60

and got the equation:

80-2(a+b+c)-3X = 60

also,can we select either a+c =12 or b+c =12 or a + b=12?

Thanks,
Shreeraj
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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09 Dec 2012, 09:27
1
@ shreeraj

Just check u cant take any two values to be 12

i.e a+ c =12 is not possible coz we are given exact value as 12 and not either of the 2 value as 12...a+b+c-x will be 12...
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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09 Dec 2012, 11:26
2
shreerajp99 wrote:
Actually i dont want to use the formulas.

I've tried my best to put this into Venn form.... From the below figure, we can see that

Cats + Dogs + Rabbits - Total - Exactly Two = All Three + All Three

Dividing this by 2, we get what "All Three" is.

As you can see, venn diagramming in this case is quite tedious and when the question has "Exactly Two" in it, it is better to use the formula.
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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09 Dec 2012, 23:17
The problem with Venn diagramming in this case is that we do not know in what ratio the "exactly two" is divided into each group.
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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19 Apr 2017, 15:15
2
apoorvarora wrote:
In the city of San Durango, 60 people own cats, dogs, or rabbits. If 30 people owned cats, 40 owned dogs, 10 owned rabbits, and 12 owned exactly two of the three types of pet, how many people owned all three?

A 2
B 4
C 8
D 12
E 32

We can use the following formula:

total pet owners = # who own dogs + # who own cats + # who own rabbits - # who own exactly two types - 2 x # who own three types + # who own none

Thus:

60 = 40 + 30 + 10 - 12 - 2T + 0

60 = 68 - 2T

2T = 8

T = 4

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Posts: 15
Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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31 Jul 2017, 10:18
Here is a good way to think about it:
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Posts: 2
Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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03 Sep 2017, 20:05
Hi Bunuel,
This is my first post after learning and watching solutions by all experts.
I solved this and got the right answer, however I approached it in a different way but was quicker for me to understand. Please advise if this is also an alternate way?
Total 60.
Exactly 2 pets : CD,CR,DR = 12.
As we are asked to find common among total 3 group : I just divided 12/3 : 4.

I know the method was ridiculous. However please advise the correct way or if this can be used?

Best regards
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Joined: 02 Sep 2009
Posts: 62498
Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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03 Sep 2017, 22:18
rbane7 wrote:
Hi Bunuel,
This is my first post after learning and watching solutions by all experts.
I solved this and got the right answer, however I approached it in a different way but was quicker for me to understand. Please advise if this is also an alternate way?
Total 60.
Exactly 2 pets : CD,CR,DR = 12.
As we are asked to find common among total 3 group : I just divided 12/3 : 4.

I know the method was ridiculous. However please advise the correct way or if this can be used?

Best regards

No, this is not correct. The number of people who owned exactly two of the three types of pet is not always thrice the number of people who owned all three.

Foe example, if the number of people who owned exactly two of the three types of pet were 18 instead of 12, the answer would be 1, not 18/3 = 6 (from Total = C + D + R - Exactly two - 2*All three + Neither).

There are several correct approaches above and here is a link which explains the theory behind the formula used: 3 overlapping sets with theory, diagrams and examples: ADVANCED OVERLAPPING SETS PROBLEMS.

Hope it helps.
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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07 Oct 2018, 07:25
$$cats+dogs+rabit-(exactly 2)-2(exatctly 3)=Total$$
$$30+40+10-12-2(T)=60$$
$$68-2T=60$$
$$2T=8$$
$$T=4$$
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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14 Nov 2018, 00:30
HasnainAfxal wrote:
$$cats+dogs+rabit-(exactly 2)-2(exatctly 3)=Total$$
$$30+40+10-12-2(T)=60$$
$$68-2T=60$$
$$2T=8$$
$$T=4$$

Hi,
I am not able to understand Why exactly 3 is multiplied by 2??
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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14 Nov 2018, 00:41
1
AashiS wrote:
HasnainAfxal wrote:
$$cats+dogs+rabit-(exactly 2)-2(exatctly 3)=Total$$
$$30+40+10-12-2(T)=60$$
$$68-2T=60$$
$$2T=8$$
$$T=4$$

Hi,
I am not able to understand Why exactly 3 is multiplied by 2??

Hey AashiS!
That's the basic formula for three overlapping sets. I don't know derivation of this formula but i memorized it.
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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14 Nov 2018, 00:43
HasnainAfxal wrote:
AashiS wrote:
HasnainAfxal wrote:
$$cats+dogs+rabit-(exactly 2)-2(exatctly 3)=Total$$
$$30+40+10-12-2(T)=60$$
$$68-2T=60$$
$$2T=8$$
$$T=4$$

Hi,
I am not able to understand Why exactly 3 is multiplied by 2??

Hey AashiS!
That's the basic formula for three overlapping sets. I don't know derivation of this formula but i memorized it.

Oh Okay.. Thanks!!
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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14 Nov 2018, 00:44
1
AashiS wrote:
HasnainAfxal wrote:
$$cats+dogs+rabit-(exactly 2)-2(exatctly 3)=Total$$
$$30+40+10-12-2(T)=60$$
$$68-2T=60$$
$$2T=8$$
$$T=4$$

Hi,
I am not able to understand Why exactly 3 is multiplied by 2??

There are following two formulas for three overlapping sets that you can use:
1) Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither
2) Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither
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Re: In the city of San Durango, 60 people own cats, dogs, or rab  [#permalink]

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Re: In the city of San Durango, 60 people own cats, dogs, or rab   [#permalink] 19 Nov 2019, 01:45
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