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12 Days of Christmas GMAT Competition 2025 - Day 2: A survey was

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Bunuel

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A survey was conducted among the tenants of the Arconia building about their support for two propositions, X and Y. The ratio of the number of tenants who support proposition X to the number who support both X and Y is 3 to 2. The ratio of the number of tenants who support proposition Y to the number who support both propositions is twice that ratio. The ratio of the number of tenants who support neither proposition to the number who support both propositions is half the first ratio. What is the ratio of the number of tenants who support only proposition X to the number who support only proposition Y?

A. 1 : 4
B. 1 : 3
C. 1 : 2
D. 3 : 1
E. 4 : 1

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Bunuel

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Bunuel

GMAT Club Official Explanation:

The ratio of the number of tenants who support proposition X to the number who support both X and Y is 3 to 2:

X : (both X and Y) = 3 : 2

The ratio of the number of tenants who support proposition Y to the number who support both propositions is twice that ratio, so its numeric value is 2 * (3/2) = 6/2. Therefore:

Y : (both X and Y) = 6 : 2

Adjust the first two ratios, 3 : 2 and 6 : 2, so that the “both X and Y” term is 4, matching the third ratio:

X : (both X and Y) = 6 : 4
Y : (both X and Y) = 12 : 4

In this case, assuming X = 6, Y = 12, and (both X and Y) = 4, we get:

(Only X) = 6 - 4 = 2
(Only Y) = 12 - 4 = 8

Therefore, the ratio of those is 2 : 8 = 1 : 4.

Answer: A.

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Let x, y, b and n denote supporting X, Y, both and neither
x/b = 3/2
y/b = 3
n/b = 3/4

Taking the denominator as 4 as common for all these, we get
x:y:b:n = 6:12:4:3

We can also assume x = 6k, y=12k, etc, but since we are solving for ratio, we can ignore it as k will cancel out.
x only = x - b => 6 - 4 = 2
y only = y - b =>> 12 - 4 = 8

x only/y only = 2/8 = 1/4

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Let Only(X) = a
Only(Y) = b
Both (X&Y) = c
Neither = d

a+c : c = 3:2
a:c = 1:2

b+c : c = 3 : 1
b:c = 2:1

d:c = 3:4

On simplifying, you'll get the ratio of a:b = 1:4

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From the text we know that:
X : Both = 3 : 2
Y : Both = 6 : 2
Neither : Both = 3 : 4

Drawing a Venn diagram can help visualise the data, one circle represents X, one circle represents Y. Let's call their intersections 2x.

This means that Only X = X - Both = 3x - 2x = x
In a similar way Only y = Y - Both = 4x
(We don't actually need the last ratio)

Therefore X : Y = x : 4x -> X : Y = 1 : 4

Answer A

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	Number of tenants who support proposition X	Number of tenants who do not support proposition X	Total
Number of tenants who support proposition Y	2k	y = 4k	6k
Number of tenants who do not support proposition Y	x = 3k - 2k = k	1.5k
Total	3k

x/y = k/4k = 1/4
x:y = 1:4

The ratio of the number of tenants who support only proposition X to the number who support only proposition Y = 1:4

IMO A

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	x	x'
y	2m	4m	6m
y'	m
	3m

making a grid for sets usign given information

we need ration of xy' / x'y
m:4m
1:4

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X:(X&Y)=3:2, then, let x= 3a, then X&Y= 2a
Y:(X&Y) = 6:2 [since its twice the ratio]. Therefore, if X&Y= 2a, then Y= 6a
(Neither X not Y):(X&Y) =3/2:2 therefore, Neither X not Y = 3/2a
Only X = X-X&Y = 3a-2a=a
Only Y = Y-X&Y= 6a-2a= 4a
only X:Only Y = a:4a= 1:4

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Bunuel

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Deconstructing the Question
Let $B$ be the number of tenants who support Both propositions X and Y.
We are given ratios comparing other groups to $B$.

Step 1: Analyze Proposition X "
The ratio of tenants who support X to the number who support both is 3 to 2.

" Note: "Support X" usually refers to the Total X circle (Only X + Both).

$\frac{\text{Total X}}{B} = \frac{3}{2} = 1.5$

$\text{Total X} = 1.5 B$

To find "Only X":

$\text{Only X} = \text{Total X} - B$

$\text{Only X} = 1.5 B - 1 B = 0.5 B$

Step 2: Analyze Proposition Y
"The ratio of tenants who support Y to the number who support both is twice that ratio."
Original ratio = 1.5.

Twice the ratio = $1.5 \times 2 = 3$.

$\frac{\text{Total Y}}{B} = 3$

$\text{Total Y} = 3 B$

To find "Only Y":

$\text{Only Y} = \text{Total Y} - B$

$\text{Only Y} = 3 B - 1 B = 2 B$

Step 3: Calculate the Target Ratio
Target: Ratio of "Only X" to "Only Y".

$\frac{\text{Only X}}{\text{Only Y}} = \frac{0.5 B}{2 B}$

$= \frac{0.5}{2}$

$= \frac{1}{4}$

(Note: The information regarding "neither proposition" is extra information not needed to solve for this specific ratio).

Answer: A

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Let, the number of tenants who support both X&Y = B
X:B=3:2 or X=(3/2)B
Y:B=twice the first ratio= 2(3/2)=3:1 or Y=3B
Neither:B = half of first ratio= 3/4 or Neither= (3/4)B
Only X= X-Both = (3/2)B-B =(1/2)B
Only Y= Y-Both = 3B-B =2B
Only X:Only Y= (1/2)B:2B = 1:4

A

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GIVEN,
Take X&Y supporters as 2x
X/X&Y=3/2
Y/X&Y=2×3/2=6/2
only X= 3x-2x=X
only Y= 6x-2x=4x
Only X/ Only Y = 1:4

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Let,

X & Y be the tenants supporting the respective propositions
B be the tenants supporting both
N be the tenants that support neither
X_{only} & Y_{only} be the tenants supporting ONLY either X or Y

We need to find the ratio of $\frac{X_{only}}{Y_{only}}$

X_only = |X| - B
Y_only = |Y| - B

Already given:

|X|/B = 3/2 , so, |X| = 3/2 B
|Y|/B = 2 * 3/2 = 3 , so, |Y| = 3B
N/B = 1/2 * 3/2 = 3/4 , so, N = 3/4 B

X_only = |X| - B = 3/2 B - B = 1/2 B
Y_only = |Y| - B = 3B - B = 2B

X_only / Y_only = (1/2 B) / 2B = 1/4

Option A

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IMO ans C

X: X+Y =3:2
Y: X+Y = 6:2
T- (X+Y): X+Y = 1:2, where T is total
then, Assuming X+Y = 20, X= 30, Y= 60
So, X:Y = 1:2

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Bunuel

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Draw a 2 * 2 matrix

Number of people who support X = x
Number of people who support Y = y
Both = z
Neither = n

Given

x/z = 3/2
n/z= 3/4
y/z = 3

x = 1.5z
y = 3z

Asked
(x-z)/(y-z) = ?

= 1.5z-z / 2z = 1/4

Option A

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The ratio of teats who support oly x to those who support Y is 1:4

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Bunuel

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Let us assume variables to solve this problem.

Let a denote x only,

Let b denote both x and y.

Let c denote y only,

Let d denote neither x and y.

We need to find : a:c

The ratio of the number of tenants who support both x and y to those who support y only :

(a+b) / b = (3/2)

(a+b) : b = 3:2

The ratio of the number of tenants who support Y to those who support Y only : TWICE THE FIRST RATIO.

(b+c)/b = 2*(3/2)

(b+c) = 3b

c= 2b

c : b = 2:1

The ratio of neither to those who support Y only = HALF of the first ratio

d/b = (1/2)*(3/2). = 3/4

d: b = 3:4

If b = 4, and d = 3, then the ratio of c:b becomes 8:4 .

Thus the values are : b = 4, c=8, and d =3

(a+b) : b = 3:2 , the ratio becomes 6:4

a+b =6, and b=4, thus a = 2.

The ratio of those who like X only to those who like Y only =

a: c = 2:8 = 1:4

Option A

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X/Both= 3/2
Y/Both= 2/1
Neither/Both= 3/4
Taking LCM with Both=4, we have:
X:Y:Both:Neither = 6:8:4:3

Only X= X- Both that is, 6-4=2
Only Y= Y- Both that is, 8-4=4
Only X/ OnlyY= 2/4 that is 1:2.

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I selected Choice A. Here is my method:

Write out the given information. This one has a lot of text so its important to make sure you are reading exactly what the question is asking for.

X : Both X + Y = 3/2
Y : Both X + Y = 6/2 (or; 3/1)

Neither : Both X + Y = 3/4

The question is asking us for the ratio of Only X : Only Y.

Now select a number with a common denominator to calculate the Only X : Only Y ratio. I selected 20. For ratios its best to select a number that is a multiple of the whole (remember ratios are a part to part relationship, so the whole is actually the two parts added, in this case, 3 + 2 =5 and 3 + 1 = 4). LCM of 4,5 = 20. We can ignore the ratio with neither because it doesn't affect what we're looking for.

So, with our pick of 20,
20/5*3 = 12 who support X,
20/5*2 = 8 who support both,
8*3=24 who support Y (since ratio is 3/1 for Y : Both).

Subtract 8 from each to get the number who only support X and Y.
12-8= 4
24-8 = 16.

4 / 16 = 1 : 4, choice A.

This took me a very long time, if anyone has a faster way to do it, please let me know.

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Let X and Y denote the number of tenants who support the proposition X and Y respectively. Let N be the number of tenants who support neither. Let B be number of tenants who support both.
Given:
X/B=3/2
Y/B=6/2
N/B=3/4
Converting these fractions to same denominator to compare
X/B = 6/4
Y/B = 12/4
N/B = 3/4
Since denominator is now same, we can take same constant of proportionality for all three equations. Hence we can write them in terms of numbers as
X = 6k B=4k Y=12k N=3k where k is a positive number
X only = X - B = 6k - 4k = 2k
Y only = Y - B = 12k - 4k = 8k
Ratio of Xonly to Yonly = 2/8 = 1/4

Bunuel

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Let who support X = x, Who support = Y , supports both = Z, supports Neither = w

Supports only X= x-z, Y= y-z

Given x:z= 3:2 ---> x=(3/2)z
y:z = 6:2 ---> y =(3/1)z
w:z= 3:4

Required = (x-z)/(y-z)= (1/2)z/2z = 1/4 (A answer )