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

Question

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.

ScottTargetTestPrep · Accepted Answer

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

KarishmaB · Answer

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.

pushpitkc · Answer

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)

amanvermagmat · Answer

As in the attached picture.

Answer should be C.

Salsanousi · Answer

Hi  sakuac  and  pks02

Please refer to the attached image I hope it helps.

Posted from my mobile device

KarishmaB · Answer

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). 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

shabuzen102 · Answer

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.

sjung92 · Answer

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

goaltop30mba · Answer

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,

chetan2u · Answer

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

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

dave13 · Answer

VeritasKarishma could you pls slightly tweak

this DS question into the one where we would need to use Total = A + B - Both + Neither

KarishmaB · Answer

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

ScottTargetTestPrep · Answer

"not A and not B" is not the same thing as "not(A and B)". In order for "not A and not B" to hold, both "not A" and "not B" must hold; in other words, your element should not belong to A and should not belong to B. In a two-set Venn diagram, this would correspond to the area outside the two sets, which is usually denoted as "Neither". So, "not A and not B" is actually the same thing as "Neither A nor B".

On the other hand, "not(A and B)" is true when your element is not in "A and B", meaning your element either isn't in A or isn't in B. As long as the element misses at least one of the sets, your element is not in "A and B" (which is the same thing as saying your element is in "not(A and B)"). In a two-set Venn diagram, this would correspond to the area outside the overlap of the two sets.

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=O3Zmh_pR9Sc

GMATinsight · Answer

Answer: Option C

Video solution by  GMATinsight

https://www.youtube.com/watch?v=Sq11u4NcQas

AntonioGalindo · Answer

avigutman , 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).

Will appreciate your input!

avigutman · Answer

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.

Borrat · Answer

What you want:
 “cat or dog but not both” =  exactly one  = A + B − 2×BOTH

either one =  A + B − BOTH

T = A + B - BOTH + NEITHER

1) 50 = A + B - 4 +  NEITHER  ,  MISSING NEITHER

2) 50 = A + B -  BOTH  + 14 ,  MISSING BOTH

1,2)  50 = A + B -   4  +  14   =>>> a+ b = 40 , AND BOTH = 4 so solvable

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

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