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.

lets consider a sitaution where remining the households let be x are then ones where there is no dog or cat.
and also all families that have cat, also have a dog.

usingg this we can create the below table

	dog	dog'
cat	95	0	95
cat'	55	x	55 + x
	150	x	150 + x

so a possible answer consitent with this information is
atleast 1 - > 150 + x - x = 150
both -> 95

150, 95

ledzep10

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18 Dec 2025, 13:46

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From the information we can have the following range 150<= h <= 169. Therefore we can eliminate the last option of 170.
We know the at least 1 column needs to be >=150, which gives us the option of 150 and 160.
If we have "at least 1" to be 150, then all the dog owners also have cats so both becomes 95.
If we have "at least 1" to be 160, then we have 150 dog owners and an extra 10 as only cat owners, that leaves us with both to be 85 which doesn't have its value in the options.

Therefore "at least 1" = 150 and "both" = 95

Bunuel

canopyinthecity

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18 Dec 2025, 14:12

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only D + both = 150
only C + both = 95

Total <170

Say, at least one = 170. This will be inconsistent will given input that total <170. Hence invalid case.

Say, at least one = 160
=> only D + both + only C = 160
=> 150 + only C = 160
=> only C = 10
=> both = 95 - 10 = 85. Not part of the option hence invalid.

Say, at least one = 150
On similar lines as above, both = 75. Valid case.

at least one cannot be less than 150 as only D + both = 150.

Hence, the answer is at least one = 150, and both = 75

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18 Dec 2025, 15:08

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Bunuel

$dogs+cats-both<170$

$both > 75$

Both is greater than 75, but it can't be more than the # of cats surveyed, which was 95. Since there's no option between 75 and 95, it must mean that both is 95.

"At least one" is basically asking us to find the amount of houses without the overlap of "both". So that's a straightforward calculation of $dogs + cats - both = 150 + 95 - 95 = 150$.

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18 Dec 2025, 20:31

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Let (DC) represents atleast one of the households own atleast of a dog or cat , (BDC) represents households that own both cat and dog. So , applying two Overlapping sets formula ,

(DC) = (D) +(C) - (BDC)

(D) = 150 , (C) =95

(DC) = 150 +95-(BDC)

(DC) +(BDC) = 245

few constraints to be observed , (DC) >=150 (DC) <170 , (BDC) <=95 . Now from options jointly consistent with these equations , one pair is satisfying such that (DC) =150 , (BDC) =95 , which will be required answer.

Bunuel

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18 Dec 2025, 21:05

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So we need to find the total number of households owning pet and the number of households owning both.
Total = Dog + Cat - Both < 170
=> Total = 150 + 95 - Both < 170 => 75 < Both <= 95

Try Both = 95 => Total = 150

Answer: At least one: 150, Both: 95

Bunuel

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18 Dec 2025, 21:07

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Atleast one = H + D - Both
Atleast one = 150 + 95 - Both
Atleast one = 245 - Both

now both <= 95

Substituting both = 95
Atleast one = 245-95
Atleast one = 150

works!

Ans: 150, 95 or D, C

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18 Dec 2025, 22:04

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Solution:

	Dog	No Dog
Cat	75		95
No Cat	75
	150

So, No. of households that own atleast one dog is 75 and No. of households that own both is also 75.

Bunuel

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19 Dec 2025, 00:58

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At least one = 150 + 95 - both
At least one = 245 - both

Now testing the options,
For both = 75, At least one = 170, but the total itself is <170, so not the answer we are looking for.
For both = 95, At least one = 150, works.

Bunuel

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19 Dec 2025, 01:39

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Total Households (T) < 170
Dog owners (D) = 150
Cat Owners(C)= 95
let A = Owning at least 1 , Dog or cat
let B = Both
A = Dog + Cat - Both
150+95 - B = 245 - B
A≤ 169
245- B ≤169
B≥76
B≤95
From the available choices B = 95 (Both)✅

At least one
A= 245 - B
245 - 95 = 150 (At least one )✅

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19 Dec 2025, 01:41

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Let's Look at the Table

	Dog	No Dog	Total
Cat	a	b	95
No Cat	c	d	<75
Total	150	<20	<170

For at least 1

a+b+c can't be less than 150, so let's check with a+b+c=150.

Since a+c is already given 150 then b=0, and a=95

from above we already got a=95.

These 2 values are available in table.

Bunuel

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19 Dec 2025, 01:46

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Yikes, I SO got this wrong. As I re-read the question after 'solving' it last evening, the "FEWER than 170 households were surveyed", which I clearly forgot about in answering the question, means I made the rookiest of mistakes. Anyway, my method seems okay enough and what else is the explanation but an opportunity to undo the wrong?

We know that there fewer than 170 households surveyed regarding pet ownership. Also, we know that if 150 households reported owning one of the pets (dog), that about the lowest the total number of surveyed pet owners can be. So, 150 - 169 is our range.

95 households report owning a cat. Cool then, let's draw a table.

Pet & Ownership	Cat - Yes	Cat - No	Totals
Dog - Yes	x	y	150
Dog - No	z
Totals	95		150 - 169

Now, the question wants us to answer - the number of households who own both a dog and a cat; that is x in the table above.

And, the number of households who own AT LEAST ONE of a dog or a cat. What does this mean? At least dog, at least cat, or dog and cat (at least 1, 1 or both). Hence, this wants us to find x + y + z in the table above.

Now, looking at the options, I'd first tackle the 'boths', as that is a constituent in the 'at least ones'. Naturally, I'd go with the smallest number - 65.

The table then looks like:

Pet & Ownership	Cat - Yes	Cat - No	Totals
Dog - Yes	x = 65	y = 150-65 = 85	150
Dog - No	z = 95-65 = 30
Totals	95		150 to 169

Clearly, we now have the values for x, y, and z.

x + y + z = 65 + 85 + 30 = 180.

But do we have 180 in the options? No. Is it less than 170? No.

Now, we clearly know that by increasing the number of people in the "Boths", we will be reducing the total number of people - as now, the people removed from the "one pet only" categories, when added to the "both pet" category, will cover 2 pets in 1.

This will make sense when you change the value of "both" to 75. To 65, we are adding 10 then, but from the overall count of 180, we are subtracting 10 - and not 20 - because the 10 when in "one pet" was already contributing for "10", but in "both pets" contributes "20", so a deduction of 20 - 10 = 10, is enough.

However, if the total is 170 - we already know the total is less than that, so while 170 and 75 would be consistent, the condition of the question prevents us from picking those (this is exactly when I forgot this very fact and marked the incorrect answer

Now, moving on to the next bigger number in the list for both - 95.

Now, this means an addition of 20 to the 75, and hence, a deduction of 20 from the total. The total, with 75, was 170, so the deduction will be 20, hence, we get the answers as 95 (for both), and 170 - 20 = 150 (for at least one).

I'll draw the table again to visualize this better. Also, note that we cannot go below the 150 total any way because "dogs only" is 150.

Pet & Ownership	Cat - Yes	Cat - No	Totals
Dog - Yes	95	55	150
Dog - No	0
Totals	95

Here, 0 + 95 + 55 = 150. And at least one is the one for cats = 95.

Bunuel

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19 Dec 2025, 02:52

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Let dogs be d=150
Let cats be c =95
Let both be b
So the max value of both has to be the largest of the two which is 95
At least one = d+c-b<170
150+95-b<170
245-170<b
b>75 and less or equal to 95
Given that 95 is among the options we can it as the value for both
That means dogs only is 150-95 =55
And cats only is 95-95=0
So at least one = 55+95+0 = 150
Ans 150, 95

Bunuel

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19 Dec 2025, 03:18

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Dog owner = 150
Cat owner = 95
Both = x
Atleast one = 150 + 95 - x = 245 - x
Step 1:
Use the total household constraints
Fewer then 170 household were surveyed so:
245 - x < 170
x > 75
So Both = 95
-->150 (not possible can't exceed cat owners)

Step 2:
Find atleast one: 245 - 95 = 150

Atleast one: 150
Both: 95

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19 Dec 2025, 03:41

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We learn that the maximum number of people surveyed is 169. Then, 150 have a dog and 95 - a cat.

Surely, it's impossible to not have any overlap, since $150+95>169$. Even if we have the maximum number of people, there are only 169-150=19 people 'free' from owning a dog, so this is the maximum number of people that can only own a cat, and therefore at least $95-19 = 76$ people must have both pets. The maximum number of people who own both pets can only be 95, since it's the maximum number of pet owners. Therefore, the only number from the answer options that fits between 76 and 95 is 95, so this is how many people own both pets.

Now, since in theory all cat owners can be also dog owners, then the whole 95 cat owners can be 'inside' the dog owners, but there cannot be fewer than 150 people owning some kind of pet.

So, the answer is 150 for 'at least one' and 95 for 'both'.

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19 Dec 2025, 03:50

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Dog owners= 150.
Cat owners = 95.
Surveyed households = 170.
To know atleast 1 we have:
Dogs + Cats - Both.
150+95 - Both
= 245 - Both.
Lets test out the answers from increasing value:
If Both = 65
Atleast 1 = 245 - 65 = 180
Not feasible.

Atleast 1 : 75:
=> 245-75 = 170
but condition says fewer than 170 house holds.

Atleast 1: 95.
=> 245-95 = 150.
This fits.

Hence Atleast one = 150 and both = 95.

Bunuel

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19 Dec 2025, 03:55

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Number of dog owners = 150
Number of cat owners = 95
Number of owners who own either pet = 150 + 95 - k, where k is the number of owners who have both dog and cat.
Therefore, number of households that own atleast one of dog or cat = 245 - k
Given that 245 - k < 170
so k > 75
Also, since number of cat owners is 95, k can at most be 95.

From the given options, k can only take the value 95. This gives the number of households that own atleast one of dog and cat as 245 - 95 = 150.

Bunuel

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19 Dec 2025, 04:41

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C Cnot
D x y 150
Dnot z
95 <170
First selection: x+y+z
Second selection: x
Since we have no information on C not D not, take total as 150. in that case Z =0. x=95 and Y=55
Atleast cat or dog= 150
Both = 95

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19 Dec 2025, 04:47

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Bunuel

Given total <170, let's take it as $170^-$

Owning Dog = 150
Owning Cat = 95

\\

	D	D'	T
C			95
C'			\(75^-[/m
T	150	[m]20^-\)	$170^-$

We need to find something that fits this table.

Both Needs to be less than atleast one, as D and C have different count.

Trying options
Atleast one cannot be 170 as total is < 170

Atleast one is 160 which means neither is < 10 or $10^-$ table will be

\\

	D	D'	T
C	85	10	95
C'	65	$10^-$	\(75^-[/m
T	150	[m]20^-\)	$170^-$

We don't have 85 as a option for both.

Next option,
Atleast one is 150, table will be
\\

D	D'	T
C	95	0	95
C'	55	$20^-$	\(75^-[/m
T	150	[m]20^-\)	$170^-$

Both is 95, which we do have in option

Correct Answer

At least one = 150
Both = 95

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19 Dec 2025, 06:05

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Bunuel