Of a group of 50 households, how many have at least one cat or at

We have a group of 50 households and need to determine how many of those households have at least one cat or at least one dog, but not both.

We can use the following formula:

total = # with at least one dog only + # with at least one cat only + # with both + # with neither

50 = # with at least one dog only + # with at least one cat only + # with both + # with neither

So, we need to determine # with at least one dog only + # with at least one cat only

Statement One Alone:

The number of households that have at least one cat and at least one dog is 4.

So, we have:

50 = # with at least one dog only + # with at least one cat only + 4 + # with neither

46 = # with at least one dog only + # with at least one cat only + # with neither

Since we don’t know the # with neither, we can not determine # with at least one dog only + # with at least one cat only. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

The number of households that have no cats and no dogs is 14.

So, we have:

50 = # with at least one dog only + # with at least one cat only + # with both + 14

36 = # with at least one dog only + # with at least one cat only + # with both

Since we don’t know the # with both, we cannot determine # with at least one dog only + # with at least one cat only. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using statements one and two, we have:

50 = # with at least one dog only + # with at least one cat only + 4 + 14

50 = # with at least one dog only + # with at least one cat only + 18

32 = # with at least one dog only + # with at least one cat only

Answer: C
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A different formula is being used here:

Total = Only A + Only B + Both + Neither

In the formula given by you, A includes Both and B also includes Both so we subtract out Both once.
In the formula being used in this question, we are considering those who have dogs only and those who have cats only. So we add to them those who have both and those who have neither to give us the total number of people.
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P(Total) = 50
P(at least one Dog) = x
P(at least one Cat) = y
P(Both at least one dog and at least one cat) = z
P(Neither) = Households which have neither a dog nor a cat.

P(Total) = P(at least one Dog) + P(at least one Cat) + P(Both at least one dog and at least one cat) + P(Neither)

1) The number of households that have at least one cat and at least one dog is 4 - z = 4
Since we do not have any information about houses with neither, we cannot clearly
tell how many households have at least one cat or at least one dog(but not both). (Insufficient)

2) The number of households that have no cats and no dogs is 14 - P(Neither) = 14
Since we do not have any information about houses with both, we cannot clearly
tell how many households have at least one cat or at least one dog(but not both). (Insufficient)

Combining information from both the statements, 50 = x + y + 4 + 14 | x+y = 32 (Sufficient - Option C)

As in the attached picture.

Answer should be C.

Total = 50
Households with at least 1 cat OR 1 dog = ?

1) Households with at least 1 cat AND 1 dog = 4
Remaining Households = 46
We cannot determine the number of households with at least 1 cat OR 1 dog.
Insufficient.

2) No Cats and No Dogs = 14
Remaining Households = 36
We cannot determine the number of households with at least 1 cat OR 1 dog.
Insufficient.

1+2)
Total = 50
Households with 1 cat AND 1 dog = 4
Households with NO cats and dogs = 14
=> Households with at least 1 cat OR 1 god = 32

Sufficient. C is the answer.
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I'm not understanding the fact that, isn't the formula for this set problem is total=(a+b-both+neither)? Why are you adding "both" here? Could you please clarify me.

How to solve this question with a double set matrix ?

waiting for the same.

Hi sakuac and pks02

Please refer to the attached image I hope it helps.

Posted from my mobile device

Salsanousi: How are you getting 32? Because according to your picture -z + z = 0.

It should be P(Total) = P(at least one Dog) + P(at least one Cat)-P(Both at least one dog and at least one cat) + P(Neither)
Not P(Total) = P(at least one Dog) + P(at least one Cat) + P(Both at least one dog and at least one cat) + P(Neither)

But anyway, you don't need to get an accurate number, Im just saying.

chetan2u VeritasKarishma

This is wrong because X + Z + Y + 14 should always be equal to total sum i.e 50

moreover, Z will cancel out and which variable is 32 then ??

I guess the first statement is incorrect, it should rather say

"The number of households that have at least one cat and at least one dog but not both is 4"

because in that scenario

X+Y+Z+14 = 50

we know, X+Y = 4

which gives Z= 32.


please correct me, experts

It is fairly simple.

You have 50 households. No of households with no cats and no dogs = 14.
So 50 - 14 = 36 households have at least one cat or at least one dog or both.

So 36 households lie in the two overlapping circles, some in yellow region (only at least 1 cat), some in blue (only at least 1 dog) and 4 in green (both).



Question: "how many have at least one cat or at least one dog, but not both?"
There are 4 households in both region. So number of households in yellow + blue only = 36 - 4 = 32
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Are we not double counting the middle of the diagram? Is it not included when you calculate x or z since those houses are counted? For example, if z = 20 then wouldnt counting the 4 houses that have cats as well as the 4 that have dogs be double counting since we have already included them in the calculation for z? I am reading the equation as this.

P(total) = z+x-y+neither

Salsanousi sakuac pks02
This is wrong. The correct one is in the image attached.

The first statement said: The number of households that have at least one cat AND at least one dog is 4.
That means the box where there's both Dog and Cat is 4. Take 50 minus Neither, minus Both, then we have Cats (not dogs) + Dogs (not cats), which is what the question is asking.

This is a easy q.

we need number of houses that have only one animal .
stmnt 1 : says about all houses with both animals. so these are substracted from 50. but it doesnt mention about houses with no animals. so, NS
stmnt 2: mentions houses with no animals. so these are substracted from 50. but doesnt mention about houses with animals.. so, NS

both stmnts, mentions houses with 2 animals, and no animals. so left ones are houses with one animal.

hence, answer is 'C'

Why is everybody adding both instead of subtracting? Official GMAT also adds both and I'm confused as to why.

I thought the formula was Total = A + B - Both + Neither

VeritasKarishma Bunuel chetan2u GMATBusters nick1816

ScottTargetTestPrep

Hi Experts!

Hope you all are doing well

Although i am able to understand that we need both "neither" and "both" so as to find what has been asked in this question, I have a doubt..

For point (2) of the question, Isn't "not A and not B" equal to "not(A and B)", which basically means "not(Both A and B)", which definitely isn't the same as "Neither A nor B" ????

Looking forward to hearing from you

Best Regards,

"not A and not B" means there should not be any A

so only A is also not acceptable

similarly, only B is not acceptable.

Hence "not A and not B" = "not(A or B)"


(when you used "not A and not B" = "not(A and B)", you are accepting only A or only B which is NOT right.


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