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

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Bunuel
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Of a group of 50 households, how many have at least one cat or at [#permalink] New post   26 Jun 2017, 02:27
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Of a group of 50 households, how many have at least one cat or at least one dog, but not both?

(1) The number of households that have at least one cat and at least one dog is 4.
(2) The number of households that have no cats and no dogs is 14.
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   15 Nov 2017, 17:26
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Bunuel wrote:
Of a group of 50 households, how many have at least one cat or at least one dog, but not both?

(1) The number of households that have at least one cat and at least one dog is 4.
(2) The number of households that have no cats and no dogs is 14.


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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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   07 Apr 2020, 01:37
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sjung92 wrote:
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


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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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   03 Jul 2017, 01:04
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As in the attached picture.

Answer should be C.
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Of a group of 50 households, how many have at least one cat or at [#permalink] New post   26 Jun 2017, 03:00
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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)
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   06 Jan 2021, 08:15
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Video solution from Quant Reasoning:
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   30 Mar 2019, 23:23
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Bunuel wrote:
Of a group of 50 households, how many have at least one cat or at least one dog, but not both?

(1) The number of households that have at least one cat and at least one dog is 4.
(2) The number of households that have no cats and no dogs is 14.


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).

Attachment:
Screenshot 2019-03-31 at 11.50.25.png
Screenshot 2019-03-31 at 11.50.25.png [ 44.06 KiB | Viewed 43100 times ]


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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Of a group of 50 households, how many have at least one cat or at [#permalink] New post   26 Oct 2018, 15:42
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Hi sakuac and pks02

Please refer to the attached image I hope it helps.

Posted from my mobile device
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   27 Aug 2017, 19:47
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pushpitkc wrote:
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 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 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)


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.
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Of a group of 50 households, how many have at least one cat or at [#permalink] New post   05 Oct 2019, 07:28
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Salsanousi wrote:
Hi sakuac and pks02

Please refer to the attached image I hope it helps.

Posted from my mobile device


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.
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   08 Aug 2017, 09:43
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Bunuel wrote:
Of a group of 50 households, how many have at least one cat or at least one dog, but not both?

(1) The number of households that have at least one cat and at least one dog is 4.
(2) The number of households that have no cats and no dogs is 14.


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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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   13 Oct 2018, 23:23
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How to solve this question with a double set matrix ?
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   25 Feb 2019, 08:09
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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.
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   06 Apr 2020, 14:06
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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
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   02 Sep 2020, 22:07
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INSEADIESE wrote:
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,



Neither A nor B means NONE of the two should be there.
Not A and Not B also means both should not be there.....Since AND is used we consider cases where BOTH condition exist, that is NO A and NO B.

In these questions there is nothing known as NOT (BOTH A and B), it is simply NOT (A and B), that is Neither A nor B

Quote:
Of a group of 50 households, how many have at least one cat or at least one dog, but not both?

(1) The number of households that have at least one cat and at least one dog is 4.
(2) The number of households that have no cats and no dogs is 14.


We can draw a Venn diagram or 2*2 matrix.
........D......n(D)...Total
C......a.......b.....
n(C)..x......y.....
TOTAL.............a+b+x+y=50

We are looking for at least one cat or at least one dog, but not both => b+x

(1) The number of households that have at least one cat and at least one dog is 4....=> a=4
........D......n(D)...Total
C......4.......b.....
n(C)..x......y.....
TOTAL.............a+b+x+y=50=4+b+x+y...We require to know y.
Insuff

(2) The number of households that have no cats and no dogs is 14.....=> y=14
........D......n(D)...Total
C......a.......b.....
n(C)..x......14.....
TOTAL.............a+b+x+y=50=a+b+x+14...We require to know a.
Insuff

Combined..
........D......n(D)...Total
C......4.......b.....
n(C)..x......14.....
TOTAL.............a+b+x+y=50=4+b+x+14...
b+x=50-18=32
Suff

C
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   06 May 2021, 23:11
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Bunuel wrote:
Of a group of 50 households, how many have at least one cat or at least one dog, but not both?

(1) The number of households that have at least one cat and at least one dog is 4.
(2) The number of households that have no cats and no dogs is 14.


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Answer: Option C

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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   02 Sep 2020, 20:05
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"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.

INSEADIESE wrote:
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,
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   08 Sep 2020, 05:45
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dave13 wrote:
VeritasKarishma wrote:
sjung92 wrote:
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


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.


VeritasKarishma could you pls slightly tweak :grin: this DS question into the one where we would need to use Total = A + B - Both + Neither


Normally, the questions ask for number of households having at least one cat or at least one dog.
That is n(C or D)

"Total - Neither" is "at least one cat or at least one dog".

Total = n(C or D) + Neither
n(C or D) = n(C) + n(D) - Both

Look at what this question is asking and that is what makes it special:

Of a group of 50 households, how many have at least one cat or at least one dog, but not both?

It just wants you to remove Both from n(C or D).

n(C or D) - Both = n(C) + n(D) - Both - Both
n(C or D) - Both = n(C) - Both + n(D) - Both
n(C or D) - Both = n(Only C) + n(Only D)

That is how the two are equivalent.

Use your std formula if you wish.

Total = n(C or D) + Neither
50 = n(C or D) + 14
n(C or D) = 36
This includes those households that have both. So remove Both
n(C or D) - Both = 36 - 4 = 32
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   18 Feb 2021, 01:23
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VeritasKarishma wrote:
Bunuel wrote:
Of a group of 50 households, how many have at least one cat or at least one dog, but not both?

(1) The number of households that have at least one cat and at least one dog is 4.
(2) The number of households that have no cats and no dogs is 14.


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).

Attachment:
Screenshot 2019-03-31 at 11.50.25.png


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


Quote:
hello expert,

do't we need to deduct 4 two times in order to get number of household only in blue and orange?


50 - 14 = 36 gives you n(C or D)

Note that n(C or D) = n(C) + n(D) - Both = 36

So already 1 Both has been subtracted out and we got 36. Now to remove Both, all we need to do is subtract it out once more only.
You get 36 - 4 = 32
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink] New post   18 Feb 2022, 12:00
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AntonioGalindo wrote:
could you please explain me why Statement I means that 4 is the value for both cats and dogs?

When I first attempted the question, I placed it as the sum of (Both Cat and Dog) + (Cat, but not Dog) +( Dog, but not Cat).



AntonioGalindo consider these two statements:
1. The number of households that have at least one cat and at least one dog is 4.
2. The number of households that have at least one cat or at least one dog is 4.

For the first statement (the original statement from the problem), in order to be counted, you have to have:
at least one cat and at least one dog
For the second statement, in order to be counted, you have to have:
at least one cat or at least one dog is 4

Please let me know if you need further explanation.
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Re: Of a group of 50 households, how many have at least one cat or at [#permalink]
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