12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent

In a recent neighborhood survey, fewer than 170 households were surveyed regarding pet ownership. Of those, 150 households reported owning a dog, while 95 households reported owning a cat.

In the table, select for At least one the number of households that own at least one of a dog or a cat, and select for Both the number of households that own both a dog and a cat that would be jointly consistent with the given information. Make only two selections, one in each column.

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19 Dec 2025, 06:32

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In a recent neighborhood survey, fewer than 170 households were surveyed regarding pet ownership. Of those, 150 households reported owning a dog, while 95 households reported owning a cat.

Total number of people surveyed = less than 170.
Number of people who own dogs = 150
Number of people who own cats = 95
Number of people who own both cat and dogs = B

Number of people who own atleast one pet = Dog+Cat-Both
Atleast one = 150+95-B= 245- B
Less than 170 were surveyed, it means 245-B must be less than 179
245-B<170
B>75 and the number of people who own both pets cannot exceed the number of people who own smaller number of only pet (cats=95)
Hence, 75<B=<95
From the choices given in the options only 95 fits in this.
So, B=95 and atleast one = 245-95= 150

Atleast one = 150 & Both=95

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19 Dec 2025, 06:41

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Given: Dog owners = 150; Cat owners = 95
Let : Owning both be x then at least one will be 150+95-x = 245-x

Applying constraint:
Total surveyed < 170
=> This implies that at least one should be less than 170 (as it cannot exceed total) => 245-x <170 => x>75
=> x cannot exceed total dog owners or total cat owners i.e. 75<x<=95<150 => 75<x<=95
=> Based on options x can only be 95

At least one = 245 - x = 245 - 95 = 150

Hence the answers are Both = 95 , At least one = 150.

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19 Dec 2025, 06:49

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d=150
c=95
n: numer of households that own at least one of a dog or a cat
b: numer of households that own both a dog and a cat

n can be any number between 150 and 169 inclusive.

n=150 is in the options:
150 = d+c-b = 150+95-b
b=95, that is also in the options

At least one = 150
Both = 95

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19 Dec 2025, 07:20

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Total = Dog + Cat -Both = 245 - Both

Total is 169 at most, but it is not in the possible values.
Test with 160
160 = 245 - Both
Both = 85, not in the list of values

Total is 150 at least
150 = 245 - Both
Both = 95
The two options are in the list

At least one = 150 and Both = 95

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19 Dec 2025, 07:22

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Dogs = 150
Cats = 95

Starting with the options

We can't take 170 as the houses surveyed were fewer than 170

For 'At least one' we take 160
=> 160 = 150 + 95 - B
=> B = 10
We don't have any option for Both as 10

Next, we take 150 for 'At least one'
=> 150 = 150 + 95 - B
=> B = 95
We have both as 95 in options (This one works)

At least one = 150
Both = 95

Bunuel

In a recent neighborhood survey, fewer than 170 households were surveyed regarding pet ownership. Of those, 150 households reported owning a dog, while 95 households reported owning a cat.

In the table, select for At least one the number of households that own at least one of a dog or a cat, and select for Both the number of households that own both a dog and a cat that would be jointly consistent with the given information. Make only two selections, one in each column.

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19 Dec 2025, 07:22

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Let’s start with those who like at least 1 (Total-None). Because we have less than 20 participants who don’t like dogs (<170-150), those who don’t like dogs nor cats will be less than 20 (0-19). Thus, those who like at least 1 will be 151 to 169, hence 160.

For those who like both, the highest will be 95, whom those who like cats but not dogs=0. The lower band will be >75 (95-<20; when we allocate the maximum number possible (19) to those who like cats but not dogs. So answer for both will be 90.

Bunuel

In a recent neighborhood survey, fewer than 170 households were surveyed regarding pet ownership. Of those, 150 households reported owning a dog, while 95 households reported owning a cat.

In the table, select for At least one the number of households that own at least one of a dog or a cat, and select for Both the number of households that own both a dog and a cat that would be jointly consistent with the given information. Make only two selections, one in each column.

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19 Dec 2025, 07:32

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From what we are given here lets say

Number of households owning a dog = A = 150
Number of households owning a cat = B = 95
Number of households owning both a dog and a cat = C
Number of households owning atleast a dog or a cat = D

We know the general equaltion here, that is,
D = A + B - C
D = 245 - C

We are also given that,
D < 170
=> 245 - C < 170
=> C > 75
Its also given that, no of cats which is the smallest group = 95,
=> 76 <= C <= 95

Substituing these values in D = 245 - C

=> 150 <= D <= 169

Now we need the values which satisfy for C and D from the given values in the table,
and the values which which satisfy the above constraints are

C = Both = 95
D = Atleast one = 150

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19 Dec 2025, 07:38

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Bunuel

In a recent neighborhood survey, fewer than 170 households were surveyed regarding pet ownership. Of those, 150 households reported owning a dog, while 95 households reported owning a cat.

In the table, select for At least one the number of households that own at least one of a dog or a cat, and select for Both the number of households that own both a dog and a cat that would be jointly consistent with the given information. Make only two selections, one in each column.

Let's consider x number of households own both a dog and cat.

170 > 150+95-x ----> x > 245-170 ---> x>75

Also, since only 95 households own a cat, x cannot be greater than 95.

Only applicable value of x = 95 ----> for at least one = 245-95 = 150

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19 Dec 2025, 07:38

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Given
Dog owners (D) = 150
Cat owners (C)= 95
Let:
B= number who own both

[*]A= number who own at least one

A=D+C−B=150+95−B=245−B

constraints: 0<=B<=95

We’re told: A<170
Substitute A=245−B
75<B<=95

Only one choice falls in the range.
B=95

Now, A=245−B=245−95=150

So, A= 150 and B = 95 are the only option that satisfies the condition

Bunuel

In a recent neighborhood survey, fewer than 170 households were surveyed regarding pet ownership. Of those, 150 households reported owning a dog, while 95 households reported owning a cat.

In the table, select for At least one the number of households that own at least one of a dog or a cat, and select for Both the number of households that own both a dog and a cat that would be jointly consistent with the given information. Make only two selections, one in each column.

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19 Dec 2025, 07:43

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The number who own at least one dog or cat is:

D+C-B=150+95-B=245-B

Fewer than 170 households were surveyed, so 245-B<170 -> B>75.
The overlap cannot exceed the smaller group so B<=95.

75<B<=95

From the given options, the only possible value for B is 95.

245-B = 245-95 = 150

At least one = 150
Both = 95

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19 Dec 2025, 08:03

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The maximum overlap is when all households owning a cat also own a dog: 95

The minimum overlap is when the total numer of households is 169:
169=dogs+cats-both=150+95-both=245-both
both=76

Only Both=95 is in the options.
For Both=95, At least one=150 (because all households owning a cat also own a dog and there are 150 households reported owning a dog)

At least one = 150 and Both = 95

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19 Dec 2025, 08:04

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Bunuel

In a recent neighborhood survey, fewer than 170 households were surveyed regarding pet ownership. Of those, 150 households reported owning a dog, while 95 households reported owning a cat.

In the table, select for At least one the number of households that own at least one of a dog or a cat, and select for Both the number of households that own both a dog and a cat that would be jointly consistent with the given information. Make only two selections, one in each column.

Total number of people surveyed is fewer than 170

Fixed number which we know are

Dog reported - 150
Cat reported - 95

We need :
Number of people with at least one pet -- Both + only dog + only cat
Number of people with both pet

In TPA type of question, we always need to concentrate on options, because it helps in a massive way

So here as we know the total number of dog and cat owner so we can safely that owner with at least one > owner with both

So let's suppose there are 95 (from option) owner with both cat and dog so

	Dog	Not Dog	Total
Cat	95	0	95
Not cat	55
Total	150		<170

so 95 + 55 + 0 =150

So our answer is 95,150

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19 Dec 2025, 08:32

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fewer than 170, which means surveyed people are between 150 to 169
Dog owners: 150, cat owners: 95

There are two cases, either there is a complete overlap of dog & cat owners, then
Atleast one = 150 (as 95 cat owners are also those who own dogs)
Here those who own both are clearly 95

Or there is as minimum overlap between both the groups as possible, in that case
Atleast one = 169
Dog owners are 150, the 19 remaining are cat owners, hence both would be 95 - 19 = 76

We have two sets of ans for (Atleast, Both), (150, 95) & (169, 76)

Only first work as per given options.

Hence ans is (150, 95)

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19 Dec 2025, 09:00

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At least one = 150
Both = 95

Add 150 + 95 = 245 then deduct 169 (as remember fewer than 170 were surveyed) to identify the overlap = 76 hence from the table we can select 95. Once we have the overlap then we deduct from 245 to identify the at least one as 150.

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19 Dec 2025, 09:28

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	Dog	No Dog
Cat	x	y	95
No Cat	a	b	<75
	150	<20	<170

The table contains info based on the information provided.

For households with both (x), we can safely say that x needs to be less than or equal to 95. This leaves us with 65, 75 & 95. If we apply 75, we notice that a would be 75 too but that wouldn't work because a+b must be less than 75. Same applies for x = 65. But if we apply x = 95, we get a = 65 which satisfes a + b < 75

Therefore, both = 95

With x = 95, a becomes 65 and y becomes 0. We need to find x + a + y = 150

Therefore, at least one = 150.

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21 Dec 2025, 09:21

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Bunuel