12 Days of Christmas GMAT Competition - Day 5: On a farm with a total

Question

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

On a farm with a total of 72 animals, some are goats, and the rest are cows. If there is at least one goat and at least one cow, what is the positive difference between the number of vaccinated goats and vaccinated cows?

(1) One-fourth of the goats and one-ninth of the cows are vaccinated.
(2) The ratio of vaccinated goats to vaccinated cows is 9 to 4.

Bunuel · Accepted Answer

GMAT Club's Official Explanation:

On a farm with a total of 72 animals, some are goats, and the rest are cows. If there is at least one goat and at least one cow, what is the positive difference between the number of vaccinated goats and vaccinated cows?

(1) One-fourth of the goats and one-ninth of the cows are vaccinated.

Assuming there are g goats and 72 - g cows, the positive difference between the number of vaccinated goats and vaccinated cows would be |g/4 - (72 - g)/9| = |13g/36 - 8|. Since the difference must be an integer, 13g/36 must be an integer, meaning that g, the number of goats, must be a multiple of 36. Given that 0 < g < 72, the only possible value of g is 36, making the difference equal to |13g/36 - 8| = 5. Sufficient.

Alternatively, assuming the number of vaccinated goats is m and the number of vaccinated cows is n, we’d get 4m + 9n = 72. From this, we can deduce that m must be a multiple of 9 (for 4m + (a multiple of 9) to equal a multiple of 9). The only non-negative value of m that works is m = 9, giving n = 4, and the difference of m - n = 9 - 4 = 5. Sufficient.

(2) The ratio of vaccinated goats to vaccinated cows is 9 to 4.

If the number of vaccinated goats and cows are 9 and 4, respectively, the difference would be 5. However, if those numbers are 18 and 8, the difference would be 10. We already have two different answers; hence, this statement is not sufficient.

Answer: A

Kinshook · Answer

On a farm with a total of 72 animals, some are goats, and the rest are cows.

If there is at least one goat and at least one cow, what is the positive difference between the number of vaccinated goats and vaccinated cows?

Goats 
 Cows 
 Total 
 
  Vaccinated

Not vaccinated

Total

72

(1) One-fourth of the goats and one-ninth of the cows are vaccinated.

Goats 
 Cows 
 Total 
 
  Vaccinated 
 x/4 
 (72-x)/9 
 x/4 + (72-x)/9 
 
  Not vaccinated 
 3x/4 
 8(72-x)/9 
 3x/4 + 8(72-x)/9 
 
  Total 
 x 
 72-x 
 72

Since all numbers in the table are integers, x/4 is an integer (72-x)/9 is also an integer
x=72-y is a multiple of 4, 72-x=y is a multiple of 9

y 
 9 
 18 
 27 
  36  
 45 
 54 
 63 
  
 36 is a multiple of 9 
 
  x=72-y 
 63 
 54 
 45 
  36  
 27 
 18 
 9 
  
 36 is a multiple of 4

x = 36
    
 Goats 
 Cows 
 Total 
 
  Vaccinated 
 9 
 4 
 13 
 
  Not vaccinated 
 27 
 32 
 59 
 
  Total 
 36 
 36 
 72

The difference between the number of vaccinated goats and vaccinated cows = 9 - 4 = 5
SUFFICIENT

(2) The ratio of vaccinated goats to vaccinated cows is 9 to 4.

Goats 
 Cows 
 Total 
 
  Vaccinated 
 9k 
 4k 
 13k 
 
  Not vaccinated 
 x-9k 
 72-x-4k 
 72-13k 
 
  Total 
 x 
 72-x 
 72

Case 1: k = 1
    
 Goats 
 Cows 
 Total 
 
  Vaccinated 
 9 
 4 
 13 
 
  Not vaccinated 
 x-9 
 68-x 
 59 
 
  Total 
 x 
 72-x 
 72

The difference between the number of vaccinated goats and vaccinated cows = 9 - 4 = 5

Case 2: k = 2
    
 Goats 
 Cows 
 Total 
 
  Vaccinated 
 18 
 8 
 26 
 
  Not vaccinated 
 x-18 
 64-x 
 46 
 
  Total 
 x 
 72-x 
 72

The difference between the number of vaccinated goats and vaccinated cows = 18 - 8 = 10
NOT SUFFICIENT

IMO A

GraCoder · Answer

Let's assume there are x goats and 72-x cows, where 71 => x >= 1:

Case 1:

Goats vaccinated = x/4
Goats unvaccinated = 3x/4
Cows vaccinated = (72-x)/9 = 8 - x/9
Cows unvaccinated = 8(72-x)/9 = 64 - 8x/9

Difference of vaccinated cows and goats = | 8 - 13x/36 |
Doesn't seem to go anywhere yet but note all the above numbers are integers...
=> 9 | x and 4 | x => 36 | x, possible values of x => 36 only... (0 and 72 are not possible)
Hence this is suff to answer it.

Case 2:

vaccinated goat = 9y
unvaccinated goat = x - 9y
vaccinated cow = 4y
unvaccinated cow = 72 - x - 4y

Difference between vaccinated cows and goat = 5y
Notice x >= 9y => possible values of y are  
Not unique.

Hence  (A)

hr1212 · Answer

Total 72 animals, and atleast one cow and goat. Calculate positive difference b/w unvaccinated cows and unvaccinated goats?

Statement 1: 1/4 of the goats and 1/9 of the cows are vaccinated.

This means that goats are multiples of 4 and cows are multiples of 9.

ie. 4x + 9y = 72

Only positive integer solution of this equation when x!= 0 and y!=0 is (x,y) = (9,4). So in total we have 36 goats and 36 cows resulting into 9 vaccinated goats and 4 vaccinated cows.

This statement is sufficient.

Statement 2: vaccinated goats/vaccinated cows = 9/4

which means we can have (vaccinated goats, vaccinated cows) = (9, 4) or (18, 8) or (27, 12) ....

This statement is insufficient.

Answer: A

RahulPMT · Answer

Ans is  option A

St 1) One-fourth of the goats and one-ninth of the cows are vaccinated. i.e 
# of goats should be a multiple of 4 (let 4a) and # of cows should be a multiple of 9 (let 9b)

4a + 9b = 72, here a and b should be +ve integers as well.
if you try to solve b should be a multiple of 4. if you notice Only b=4 possible, because if b=8 then a=0 BUT it is given there is  atleast 1 goat and 1 cow

Turns out equation has unique solution a=9 and b=4. Hence number of Vacc. goat = 9 and vacc cow = 4.
SUFFICIENT.

St 2) ratio of Vacc G : Vacc cows is given here we cant be sure differece maybe 5, maybe 10 NOT SUFFICIENT

Krunaal · Answer

We are given C + G = 72, and there is atleast 1 C and 1 G

Statement 1.

Vaccinated = G/4 + C/9

We know that G/4 and C/9 need to be integer as they are animals and G + C = 72. When we substitute G = 4, C will be 68, but C won't be divisible by 9. Upon checking other terms (8, 64); (12, 60); (16, 56);..................(36, 36); (40, 32) and so on. We know that only  (36, 36)  is possible for G/4 and C/9 to be an integer and satisfy G + C = 72 .  Therefore, 9 - 4 = 5   SUFFICIENT

Statement 2.

VG / VC = 9/4, vaccinated goats and vaccinated cows could be 9 and 4, or 18 and 8. Multiple possibilities.  INSUFFICIENT

Answer. A

missionmba2025 · Answer

(1) One-fourth of the goats and one-ninth of the cows are vaccinated.

The number of goats is a multiple of 4, and the number of cows is a multiple of 9.

Cows 
 Goats 
 
  9 
 63 
 
  18 
 54 
 
  27 
 45 
 
  36 
 36 
 
  45 
 27 
 
  54 
 18 
 
  63 
 9

Out of all possible combinations, only one value 36 is a multiple of 4.

Hence, we can conclude that the number of goats = the number of cows = 36.

With this information we can find the number of vaccinated goats and cows and determine the difference.

(2) The ratio of vaccinated goats to vaccinated cows is 9 to 4.

Number of vaccinated goats = 9
Number of vaccinated cows = 4
Difference = 5

Number of vaccinated goats = 18
Number of vaccinated cows = 8
Difference = 10

The statement is not sufficient.

Option A

nishantswaft · Answer

The stem tells us that;
 
  On a farm with a total of 72 animals, some are goats, and the rest are cows   
  there is at least one goat and at least one cow    
 Then it asks us;  
 
  what is the positive difference between the number of vaccinated goats and vaccinated cows?

We are given that G+C=72 and G,C ≥ 1

Let's look at the statements;

(1) One-fourth of the goats and one-ninth of the cows are vaccinated.

GOAT 
 COW 
 TOTAL 
 
  VACCINATED 
 G/4 
 C/9 
 V 
 
  NOT VACCINATED

TOTAL 
 G 
 C 
 72

Here, G/4+C/9=V or  \frac{9G+4C}{36} =V
and, also  G+C=72

On substitution;
  \frac{9G+4(72-G)}{36} =V
  \frac{9G+288-4G}{36} =V
  \frac{5G}{36} +  \frac{288}{36}  =V
  \frac{5G}{36} + 8 =V

Since V should be an integer so  \frac{5G}{36}  should also be an integer.
I.e. G should be a multiple of 36
But since,  G,C ≥ 1 value of G can't be 72 and hence G=36.
 So, C=36.

And we have our answer.

Statement 1 is sufficient
 
 (Eliminate BCE)

Let's look at statement 2;

(2) The ratio of vaccinated goats to vaccinated cows is 9 to 4.

If this is the case, then

GOAT 
 COW 
 TOTAL 
 
  VACCINATED 
 9x 
 4x 
 13x 
 
  NOT VACCINATED

TOTAL 
 G 
 C 
 72

Here x can take any value.

Statement 2 is not sufficient.  
 
 (Eliminate D)

Answer is A

HarshaBujji · Answer

Wow!! This is a tricky one. Need to pay key attention to the details mentioned in the question, and I almost got it wrong.

Let there be g goats and c cows. 
Vg, Vc are vaccinated goats and cows.

We know that g+c = 72 and g,c are natural number and  g,c>=1( at least one goat and at least one cow )

Stmt (1) One-fourth of the goats and one-ninth of the cows are vaccinated.

Vg = g/4, Vc = c/9 . We are looking for Vg - Vc = g/4 - c/9.

We also know c= 72 - g.

So then this  Vg - Vc = g/4 - 8 +g/9 => 13g/36 - 8.

We know that this count has to be an integer.. 0,1,2....

But for that to be true 13g/36 has to be interger. 13,36 are coprimes. Hence g has to be multiple of 36. (36,72)..

We know that g cannot be 72 as c has to be 1 or more. (Given).  
 
Hence g=36, c = 36. Vg - Vc = 5 
 
Stmt 1 is sufficient. 
 
 AD

(2) The ratio of vaccinated goats to vaccinated cows is 9 to 4.

Vc/Vg = 4/9.

Vc = 4x, Vg = 9x.

Vc = 8 , Vg = 18, Or 4,9 any thing can be possible as they are not dependent on c,g. 
 
Hence this is insufficient, D can be eliminated.  
 
 IMO A

AVMachine · Answer

Total Animals = 72; C, G>=1

(1) One-fourth of the goats and one-ninth of the cows are vaccinated.

G/4 & C/9 are Integer.

4G + 9C = 72; And only one condition satisfy this equation 36 + 36 = 72;

G/4 = 9; C/9 = 4; Sufficient

(2) The ratio of vaccinated goats to vaccinated cows is 9 to 4.

9T-4T = 5T; (T= Total Vaccinated Animals, which is unknown) Not Sufficient. 
Ans A

andreagonzalez2k · Answer

g=goats
c=cows

g+c=72

(1) As the number of cows must be a multiple of 9 possible values for c are:
9, 18, 27, 36, 45, 54, 63

As g+c=72 the corresponding number of goats are:
63, 54, 45, 36, 27, 18, 9

But, as the number of goats must be a multiple of 4, only g=36 works.

So c=36, g=36 and the requested difference is 5

SUFFICIENT

(2)
vg=vaccinated goats
vc=vaccinated cows

If vg:vc = 9:4, vg and vc can be 9 and 4 respectively and the answer would be 5.
But if vg and vc are 18 and 8 respectively the answer would be 10.

INSUFFICIENT

IMO A

Navya442001 · Answer

Statement 1 
Let the number of Goats be x, number of cows will then be 72-x
According to the statement, One-fourth of the goats and one-ninth of the cows are vaccinated.

Goats 
 Cows 
 Total 
 
  Vaccinated 
 x/4 
 (72-x)/9 
 8 + (5/36)x 
 
  Non-Vaccinated

x 
 72-x 
 72

Now 8 + (5/36)x should be an integer

Therefore x should be a multiple of 36
THerefore x could be equal to 0, 36 or 72 (cannot be greater than 72 since total animals are 72)
However, x cannot be 0 or 72 since there is at least one cow and goat.
Therefore x = 36. 
Hence statement 1 is sufficient to find the difference between the number of vaccinated goats and vaccinated cows

Statement 2 
The ratio of vaccinated goats to vaccinated cows is 9 to 4. This means:
 
 vg/vc = 9/4 
 vg = (9/4)vc  
This gives us a relationship between the number of vaccinated goats and cows, but it doesn't tell us how many goats or cows there are in total, or how many are vaccinated. Therefore there could be multiple solutions here. 
This statement is insufficient.

vijayram24 · Answer

Total animals =72

Option A: 1/4 goats and 1/9 cows are vaccinated => Goats =4x, cows =9y
4x+9y=72 (Only x=9 ,y=4 satisfies this) .So goats are 36,cows are 36. So vaccinated goats =9, VC= 4 . Therefore difference =5 .. Sufficient.

Option B : VG/VC =9/4
VG could be 9,VC could be 4 (Difference 5)
VG could be 18 , VC could be 8 (Difference 10) and so on .. 
Not sufficient

Hence Option A

crimson_king · Answer

On analyzing option B we find the ratio of vaccinated goats to vaccinated cows is 9:4 but it does not explicitly tell us about the total number of cows or goats as their difference would be equal to 5x if we assume the individual ratios would be 9x & 4x respectively. This is insufficient to answer this question in hand.

Hence we can eliminate option (B) & option (D)

On analyzing just option A, we are given that One-fourth of the goats and one-ninth of the cows are vaccinated.

To satisfy these conditions we need the number of goats to be a multiple of 4 & the number of cows to be a multiple of 9 along with their individual sums being 72.

4x+9y=72

Only one positive integral part of the solution satisfies this given equation which is (9,4). So the value of x=9 & y = 4 suffixes & their difference is equal to 5 which is what the question requires.

Hence the option ( A) is sufficient to answer this question & is the correct answer

bellsprout24 · Answer

Answer is  A. Statement 1 alone is sufficient

1) If we assume number of vaccinated animals must be integers since you cannot vaccinate a portion of an animal, then the only integers give us number of goats divisible by 4 and number of cows divisible by 9 is goats = 36 and cows = 36. Therefore, we know that 9 goats are vaccinated and 4 cows are vaccinated. Statement 1 alone is sufficient.
2) Knowing only the ratio of vaccinated animals without number of animals is insufficient, because we could have 9 vaccinated goats and 4 vaccinated cows, or multiples of 9:4, such as 18 vaccinated goats and 8 vaccinated cows. Statement 2 alone is insufficient.

IssacChan · Answer

(1) The LCM of 4 and 9 is 36
Possible combination may be
- 0 goat 72 cow (Not possible as at least one goat)
- 72 goat 0 cow (Not possible as at least one cow)
- 36 goat 36 cow (Valid)

1/4 of vaccinated goats = 36*1/4 = 9
1/9 of vaccinated cows = 36*1/9 = 4
Difference = 9-4 = 5  (Sufficient)

(2) The ratio of vaccinated goats to vaccinated cows is 9 to 4 meet the condition of at least one goat and one cow already.
Case 1: Vaccinated goats 9, Vaccinated cow 4 Others are non-vaccinated => Difference: 5
Case 2: Vaccinated goats 18, Vaccinated cow 8 Others are non-vaccinated => Difference: 10
As more than one solution,  insufficient

OmerKor · Answer

Hi everyone

This is a Overlap question (I will use Double Matrix method): Goats Cows Total Vaccinated 0< 0< NoVaccinated Total 72 GV-GC= 1 value = Sufficient 2 values = Insufficient (1) 1/4 of G = GV , 1/9 of C = CV This is tricky, because the only 2 numbers that will add together to 72 and can be divided by 4&9 are 36 so 36 Goats and 36 Cows. GV=9, CV=4 the difference: 9-4=5 Try some cases: 45 and 27 (no /4) 54 and 18 (no /4) and so on... Sufficient. (2) Ratio of = So VG= 9,18,27.. CG= 4,8,12.. Let's try the 2 first options and see if we get 2 values of VG-VC. Option 1: Goats Cows Total Vaccinated 9 4 13 NoVaccinated 4 55 59 Total 13 59 72 Difference: 9-4=5 Option 2: Goats Cows Total Vaccinated 18 8 26 NoVaccinated 8 38 46 Total 26 46 72 Difference: 18-8=10 We got 2 different values 5 & 10 = Insufficient. Answer A

LastHero · Answer

(1)
The number of goats (x) must be a multiple of 4.
And the number of cows (72-x) must be a multiple of 9. As 72 is a multiple of 9, this implies that x must be a multiple of 9 too.
The only multiple of 4 and 9 less than 72 is 36, which is the number of goats and cows.

36/4-36/9=5

Condition (1) is sufficient

(2)
There is no way to relate this information with the total number of goats (vaccinated and no vaccionated) and the total number of cows (vaccinated and no vaccinated).
So vaccinated goats and vaccinated cows can be 9 and 4, 18 and 8, 27 and 12... and the difference is distinct in all the cases.

Condition (2) is insufficient

Answer A

jkkamau · Answer

The number of goats have to be in multiples of 4 and cows multiples of 9.
From statement 1 There can be Goats= 63.54.45.36.27.18.9. Only 36 is a multiple of 4 so there must be 36 goats and 36 cows. From these we can get the answer
 Cows= 9.18. 27.36.45.54.63
Statement 2 is insufficient because we could have 9:4, 18:8 etc

A

12 Days of Christmas GMAT Competition - Day 5: On a farm with a total

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	Goats	Cows	Total
Vaccinated	x/4	(72-x)/9	x/4 + (72-x)/9
Not vaccinated	3x/4	8(72-x)/9	3x/4 + 8(72-x)/9
Total	x	72-x	72

y	9	18	27	36	45	54	63		36 is a multiple of 9
x=72-y	63	54	45	36	27	18	9		36 is a multiple of 4

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12 Days of Christmas GMAT Competition - Day 5: On a farm with a total

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