On a farm, there are chickens, cows, and sheep. The total number of

Question

On a farm, there are chickens, cows, and sheep. The total number of chickens and cows is three times the number of sheep. If there are more cows than either chickens or sheep, and the combined total of heads and feet of chickens and cows is 100, how many sheep are on the farm?

A. 5
B. 8
C. 10
D. 14
E. 17

Bunuel · Accepted Answer

Official Solution:

On a farm, there are chickens, cows, and sheep. The total number of chickens and cows is three times the number of sheep. If there are more cows than either chickens or sheep, and the combined total of heads and feet of chickens and cows is 100, how many sheep are on the farm?

A. 5
B. 8
C. 10
D. 14
E. 17

Let's denote the number of chickens as  c , cows as  w , and sheep as  s .
 
We know that the combined total of heads and feet for chickens and cows is 100, so we can construct the equation:  100 = (1c + 1w) + (2c + 4w) . In this equation, 1 represents the number of heads per animal, while 2 and 4 denote the numbers of legs for a chicken ( c ) and a cow ( w ) respectively. Simplifying, we get:  100 = 3c + 5w .
 
Since both 100 and  5w  are multiples of 5, to satisfy  100 = 3c + 5w ,  c  must also be a multiple of 5. If it isn't, then  3c + 5w  would become a sum of a non-multiple of 5 and a multiple of 5, which cannot be a multiple of 5. Considering multiples of 5 for  c  and keeping in mind the constraint of  w > c  (it's specified that there are more cows than either chickens or sheep), we find two fitting pairs:  c=5 ,  w=17  and  c=10 ,  w=14 .
 
Finally, since the total number of chickens and cows is three times the number of sheep, we have  c + w = 3s . The pair  c=5 ,  w=17  doesn't satisfy this equation, as  5 + 17 = 22  isn't divisible by 3. However,  10 + 14 = 24  is divisible by 3, making this pair valid. As a result, for  10 + 14 = 3s , we find that  s = 8 .

Answer: B

KarishmaB · Answer

Using options, in my opinion!

When I read the question, I note the following:

chickens + cows = 3 * sheep

Chickens have 2 legs and one head. Cows have 4 legs and one head.
Total legs + heads = 100
If chicken = cows, average number of legs would be 3 for every head. So out of 100, we would have 75 legs and 25 heads (i.e. 25 = chickens + cows)
But we know cows > chickens. So average number of legs must be more than 3. Then, we must have fewer than 25 animals (chickens + cows).

If number of sheep = 10/14/17, three times would be 30/42/51 but this cannot be the number of (chickens + cows).
So only possible options are 5 and 8. 
If number of sheep = 5, number of (chickens + cows) = 15. Even if they are all cows, they would have only 60 legs and total legs + heads = 15+60 = 75. We cannot make 100 in this case.

So answer has to be 8.

lansac · Answer

My answer is 8

Chicken - ch
Cows - C
Sheep -S

ch+C=3S
C> ch and c>s
Each cow has 4 legs and 1 head
Each chicken has 2 legs and 1 head
So 5c+3ch=100 (sum of legs and head)
There are 2 possible solutions to this equation
c=11 and ch=9 or
c=14 and ch=10

Since from first equation where ch+c=3s

The sum of ch and c should be divisbile by 3. 20 is not so the only possible solution is c=14 and ch=10. So s=8

mau5 · Answer

Just another approach :

Cows+Chicken = 3*Sheep.

Also, Cows>Chicken and Cows>Sheep.

Also, Cows+Chicken+4*Cows+2*Chicken = 100

or 5Cows+3Chicken = 100 --> 2Cows + 3*(Cows+Chicken) = 100 --> 2Cows + 9*Sheep = 100. Replace the options for Sheep,

A= odd , thus discard
C = # of Cows<# of Sheep, thus discard
D and E = # of Cows coming as negative,discard

B.

DJ1986 · Answer

I was confused because I thought that there were 100 feet and (100) heads. Is the real GMAT this ambiguous in the wording?

KarishmaB · Answer

In my opinion, the wording is not ambiguous. It says there are a total of 100 feet and heads. 
If there were 100 feet and 100 heads, it would have been written as such. Also, then the animals would not have been cows and chickens since per head there need to be at least 2 feet. We certainly cannot have equal number of heads and feet until and unless we are talking about molluscs.

DJ1986 · Answer

Now that you mention it, what I was thinking really doesn't make sense. I was imagining fried chicken and someone ate all the legs!

KarishmaB · Answer

Don't study on an empty stomach!

arpithamouli · Answer

A farm has chickens, cows and sheep. There are three times the number of chickens and cows than sheep. If there are more cows than chickens or sheep, and together, cows and chickens have a total of 100 feet and heads, how many sheep live at the farm?

5
8
10
14
17

what is the quickest way for this? help please

A. 5
B. 8
C. 10
D. 14
E. 17

Solving this question with a common sense approach,

considering 100 feet and the fact cows>Chicken( 20 cows and 10 chickens which exactly adds upto 100 feet and 30 heads)

Pluging in the answers, when sheep's 5.. the cow+chicken= 15(doesnt make sense)

Sheep's 8 Cow+chicken=24(makes sense because the numbers possible)

Sheep's 10 cow +chicken=30( Values much higher than derived) and hence no need to check for other answers
and the answer is B)8.

hellosanthosh2k2 · Answer

chickens - a, cows - b , sheep - c

a + b = 3c
3a + 3b = 9c ----(1)

3a + 5b = 100 (heads and feet)
9c - 3b + 5b = 100
b = 50 - (9c/2) => to make b integer, c must be even

given cows > sheeps => b > c => 50 - (9c/2) > c
= (11c/2) < 50
= c < (100/11) = c <= 9, even number less than 9 in the options is 8 (B)

jabhatta2 · Answer

Hi  VeritasKarishma   Bunuel  - i was wondering if i can do this problem via ratio's specifically

I laid down the following as ratio's.

Just wondering, am i allowed to assume the unknown multiplier is common across cows / chicken and sheep ?

KarishmaB · Answer

There isn't enough information to use ratios meaningfully.

Why did you start with 2k and 6k? 
How about 3k and 9k - Now I can split 9k into '5k and 4k' or '6k and 3k' or '7k and 2k' or '8k and k'.
and so on...

Vitthal19 · Answer

the line "   If there are more cows than chickens or sheep"  because the use of OR means that either cows> chickens or cows>sheeps. I got confused becaused of this that this is a either or this condition should also follow.  
 what are your thoughts on this because I think this is a small but a genuine error that adds one additional condition which can't be satisfied with any option

Krunaal · Answer

The line means, there are more cows than chickens as well as more cows than sheep (cows > chickens and cows > sheep). If it were an  either..or..  statement it would have been written as such.

mcepeci · Answer

The problem states that there are more cows than chickens or Sheep. You wrote Cow>Chickens and Cow>Sheep. However, if it is Or then we know that at least one of those statements are 
true. So we really don't know for sure that the number of Cows is more than the the number of sheeps.
Please correct me if I am wrong. Thank you for your help

sujoykrdatta · Answer

Chicken = h, cow = w and sheep = s
h + w = 3s
w>h, w>s
h+w+2h+4w = 100
=> 3h + 5w = 100
Since 100 and 5w are multiples of 5, 3h must be a multiple of 5

Possible cases: 
h=5,w=17 => s=(h+w)/3 - not possible 
h=10,w=14 => s=8 - possible 
h=15,w=11 => s=8.6 - not possible

Other values won't work since w has become smaller than h

Only solution 
h=10,w=14,s=8

Ans B

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