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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 09:00
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At least one: 150 Both: 95
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95% (hard)

Question Stats:

58% (02:25) correct 42% (02:31) wrong based on 210 sessions

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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.

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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At least one Both
65
75
95
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170
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At least one: 150 Both: 95
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 10:00
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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.

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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GMAT Club Official Explanation:



Given that 150 households own a dog and 95 own a cat, we have:

At least one = 150 + 95 - Both = 245 - Both.

Thus, we need two options for At least one (245 - Both) and Both, such that their sum is 245.

From the options, two pairs satisfy this condition for (At least one, Both): (170, 75) and (150, 95). However, since fewer than 170 households were surveyed, At least one cannot be 170. Therefore, the only valid pair is (At least one, Both) = (150, 95).
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 09:27
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Let the no. owning both be x
Household owning 1 dog or cat = 150-5-x = 245-x
Since surveyed < 170, therefore, 245-x<170
x<75
Also, No. owning both<(or equal to) no. owning Cat
only value satisfies is x=95
And atleast 1 = 245-95 = 150

Hence, Both = 95 & Atleast 1 =150
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 09:56
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Total peope surveyed = fewer than 170
150 own dogs and 95 own cats.
Let, the people who both own cats and dogs = x
People who own atleast 1 = Dog+Cat-Both
Atleast 1 = 150+95-x = 245-x
Because fewer than 170 households were surveyed... 245-x<170 ; x>75
And also the njmber of people who own both cannot exceed the smaller group (cats) 95..... so, x<=95
We get, 75<x<=95
From the guven options, only possible value is 95
If x=95, then atleast 1= 245-95=150

Atleast 1= 150
Both = 95
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 09:59
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Number of householdsOwn dogDon't own dogTotal
Own catboth95-both = T-150-none95
Don't own catT-95-none = 150 - bothnoneT-95
Total150T-150T<170

T - 95 - none = 150 - both
At least one = T - none
At least one + both = 245

245 - 65 = 180; Not an option
245 - 75 = 170; Not feasible since T<170
245 - 95 = 150; Feasible

At least one > both
At least one = 150
Both = 95

At least one150
Both95
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 10:12
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Total < 170 (thus you can immediately eliminate the option 170 in both columns)

Cats + Dogs - Both + Neither = Total
Total - Neither = At least one
At least one must be < 170 (It cannot be higher than the total)
Cats + Dogs - Both = At least one
Cats + Dogs - Both < 170
95 + 150 - Both < 170
Both > 75
Also Both must be ≤ 95 (because only 95 people have cats, so at MOST 95 can have both cats and dogs).
The only option for Both is 95.
Using the formula above: At least one = 95+150-95=150

So the answers are 150 and 95
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 10:16
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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.

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.

use 2x2 matrix

plug in values and check which satisfies the condition given:

---D---ND----total
C--- 75---20---95
NC--75--0------75
-----150---20---170

at least one is 20+75 ; 95 and both is 75
95;75
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i would draw a Venn diagram, with C,B,D. With B indicating the part in the middle, (meaning both Cats and dogs), and C and D indicating only cats or only dogs respectively.
C+B+D=<170
And we know that C+B=95, and that D+B=150.

Combing the above information we get:
(C+B)+(D+B)-B=170. (here we subtract B as we have taken it into consideration 2 times).
This gives: (95)+(150)-B=170. -> B=95+150-170 -> B=75. This gives us the "BOTH" part of the question.

Since C+B=95, and we know that B=75, we get C+75=95 -> C=20. (This is the number of people who own just cats).
Since D+B=150, and we know that B=75, we get: D+75=150 --> D=75. (This is the number of people who own just dogs).
Therefore, C+D= 75+20= 95. This is the total number of people who own at least a dog or a cat.
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 10:25
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T<170
D=150
C=95
Let be atleast 1 and both be A and B

T = D+C-B+N
150+95-B+N<170
75<B-N
So B min = 76
Also max B = min(95,150) = 95
Out of the options B=95 works

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

Answer 150,95

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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.

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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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 10:48
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At least one can be either 150 or 160. 170 is not possible as total no. of households shall be less than 170.

Now suppose at least one =160
Then both will be =95-(160-150)=85
Both don't have option 85. so at least 150 is not possible.

So at least one possible =150
So Both =95-(150-150)=95
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 11:00
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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.

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.
Show more
N<170; Dog owners (D)=150; Cat owners (C)=95
x=households that own both dogs and cats
"At least one"= households that own a dog or a cat
At least one = D+C-x = 150+95-x = 245-x
The maximum overlap cannot exceed the smaller group: ≤95
245-x<170
x>75
So, 75<x≤95
Only possible value for Both = 95
At least one = 245-95=150
At least one = 150.
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 11:12
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Let x be the number that holds both cats and dogs and y be the number with neither holding.

We have 150+95-x +y < 170

x-y> 75

We can deduce that x has to be greater than 75. We are asked to find appropriate combination of 245-x and x

The only combination that satisfies the above constraints are

Atleast one - 150 and Both -95
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 11:28
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Bunuel
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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.

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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Let only Dog as pet = d households
only Cat as pet = c households
both Dog & Cats as pet = x households

We need to find "at least one" = dog or cat = d + c +x
and "Both" = x

Given , d+x =150 and x+c =95
Add, d+x+x+c =150+95
or (d+x+c) + (x) = 150 +95 .............................(a)

Also given , d+x+c < 170 , add x to both
245 < 170 +x or x > 75. .....................(b)

As per (a) and (b) and given options. both =95 and atleast one = 150
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 11:33
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total household fewer than 170
if 160, 160=150+95-both ..both = 85 ..not in option
if 150, 150=150+95-both...both = 95 ...YES

Ans 150 & 95.
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 12:04
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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 , 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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A=dog owner
B= cat owner
AnB= both
AUB = atleast 0ne
As per question
150<=AUB<170
AnB <=95
Only Atleast one= 150
Both=95 satisfies the situation
AUB= A+B-AnB <=169
150+95-AnB,=169
AnB>=76
So,
76<=AnB<=95
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Dog owners= 150
cat= 95
Both dog $ cat= x
Atleast one= y
y=150+95-x=245-x
245-x<170
x=75 so x must be greater than 75 so its 95
y=245-95=150
atleast oe (dog or cat) is 150
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 12:17
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Total household number < 170

House with Dogs, D = 150
House with Cats, C = 95
Let houses with both D & C be B

Therefore, house with at least one should be D + C - B (removing the repetitions)
=>245 - B

Also, the number should be <170,

245 - B < 170
B > 75

This eliminates the first 2 options of 65 and 75.

Now, best way would be to check with the options.

Putting third option,
When B = 95
D = 55
C = 0
Therefore, at least one = 245 - 95 = 150

It is consistent with the given options. Hence the choice.
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 12:45
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Bunuel
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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.

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.
Show more
Let the total number of household is LESS THAN 170.

The number of dogs = 150

Number of Cats = 95

Let us assume variable I and II which contains only one set, and elements in 2 sets.

I + II < 170

I + 2II = 245

Subtracting the first equation from second equation, we get

Subtracting results in change of inequality.

(I+2II) - ( I+ II) > 245-170

II > 75.

Since, CAT has only 95.

The max value of overlapping region can be 95.

So, 75 < II <= 95

If , II = 95, then I+ 2* 95 = 245

I = 55

At least I = I+ II = 95 + 55 = 150

So,

Both = 95

At least one = 150
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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 12:54
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150 at least one, 95 both


150 dogs, 95 cats, most that have both is 95 (limited by cats) 245 animals - 95 with both equals 150 who have at least one animal


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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.

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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12 Days of Christmas GMAT Competition 2025 - Day 4: In a recent 18 Dec 2025, 13:35
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given fewer than 170 households surveyed Dogs=150, cats=95
union value=150+95 - intersection value
union value of 169 corresponds to intersection value of 76,which further increases as the no of households surveyed decreases.
so 65 and 75 must not be the intersection value.
taking intersection value (both) 95 the corresponding union value (at least one) is 150
are the right answers.
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