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

Question

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

A 2
 B 4
 C 8
 D 12
 E 32

BrushMyQuant · Accepted Answer

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

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

x = 4

So, answer will be B
Hope it helps!

Carcass · Answer

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

apoorvarora · Answer

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.

MacFauz · Answer

Total = C + D + R - Exactly two - 2*All three

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

2x = 8

x = 4

Bunuel · Answer

3 overlapping sets with theory, diagrams and examples:  ADVANCED OVERLAPPING SETS PROBLEMS

shreerajp99 · Answer

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?
Please help.

Thanks,
Shreeraj

apoorvarora · Answer

@ 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...

MacFauz · Answer

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.

MacFauz · Answer

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.

ScottTargetTestPrep · Answer

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

Answer: B

Samuelboyle96 · Answer

Here is a good way to think about it:

rbane7 · Answer

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

Bunuel · Answer

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.

HasnainAfxal · Answer

cats+dogs+rabit-(exactly 2)-2(exatctly 3)=Total 
 30+40+10-12-2(T)=60 
 68-2T=60 
 2T=8 
 T=4

AashiS · Answer

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

HasnainAfxal · Answer

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

AashiS · Answer

Oh Okay.. Thanks!!

HasnainAfxal · Answer

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

KarishmaB · Answer

(Total - Neither) gives you the number of people with at least one of the three pets (which is 60 here). 
30 owned cats includes those who own (cats + one more pet) and those who own (cats + two other pets) 
Same for dogs and rabbits. 
So 30 + 40 + 10 = 80 gives us instances of pets owned, not people. So people who own 2 pets are counted twice in instances and people who own 3 pets are counted 3 times in instances.

So to get the number of people from instances, we need to subtract people who own 2 pets once because they are counted twice in instances and people who own all three pets twice because they are counted 3 times in instances.

60 = 80 - (people who own 2 pets) - 2*(people who own all three pets)

People who own all three pets = 4

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards