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M06-12

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Intern
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Joined: 27 Mar 2017
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Re: M06-12  [#permalink]

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New post 18 May 2017, 03:01
It is always so time taking to arrive at the integer values of the equations like this by picking numbers! Is there any material on how we can quicken this step?
Thank you!
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New post 30 Oct 2017, 07:14
(5,25) clearly violates the condition that the total number of heads and feet between the cows and chickens equals 100, so C cannot be a possible answer. (5,25) = 5(3) + 25 (5)
= 15 + 125 = 140.

Sachin07 wrote:
A farm has chickens, cows and sheep. The number of chickens and cows combined is 3 times the number of 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?

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


Let \(C\) be a number of chickens, \(W\) - cows, and \(S\) - sheep.

Chickens and cows together have 100 feet and heads: \(100=1C+1W+4W+2C\), assuming that 1 is the number of heads per either unit and 4 and 2 are the number of legs per \(C\) and \(W\) accordingly. Therefore: \(100 = 3C + 5W\).

It is given that \(W \gt C\), and we also see that \(C\) should be a multiple of 5 for \(3C + 5W\) to add up to 100. Picking a few numbers, we get two pairs: (5, 17) and (10, 14).

\(S = \frac{(C+W)}{3}\); \(5 + 17 = 22\) which is not divisible by 3, but \(10 + 14 = 24\) which is divisible by 3, therefore only the second pair will work. So, \(S = 8\).


Answer: B



another pair of (5,25) is possible and as per this the answer can be C.10
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Re: M06-12  [#permalink]

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New post 17 Nov 2017, 18:58
First post here, so sorry if it is put in the wrong place.

This is a solution to - M06-12

Chicken + Cows = 3 times Sheep
=> Chicken + Cows + Sheep = 3 Sheep + Sheep = 4 Sheep

Therefore, the answer need to be a multiple of 4. Only one choice i.e. 8.

Thanks!
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New post 03 Mar 2018, 19:09
Hi Bunuel,

I was wondering could you please explain this concept one more time? You mentioned in your OE that, "3C + 5W = 3C + (a multiple of 5) = 100 = (a multiple of 5), thus 3C must also be a multiple of 5, which means that C must be a multiple of 5". I understood that 100 = (a multiple of 5) so we have, 3C + (a multiple of 5) = (a multiple of 5). What I couldn't understand was how did we deduce that 3C is also a multiple of 5 and then concluded that C must be a multiple of 5 as well? Could you please explain this part? Would greatly appreciate it?
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New post 21 Oct 2018, 06:54
seems impossible to solve under 2 minutes... It's unlikely to encounter something like this on the GMAT
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Re: M06-12  [#permalink]

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New post 11 Jun 2019, 23:59
Here's a potential alternative solution I used to solve this correctly.

Equation 1: Chicken + Cow = 3*Sheep

Cow > Chicken

Reverse plug-in the answer choices

Cow + Chicken = 100 feet and heads

Test (b) 8 for sheep in Equation 1

Ch + Cow = 3(8)
Ch + Cow = 24
Cow> Ch
Try Cow = 13, Ch = 11
13*4 legs on a cow + 11*2 feet on chicken + 24 heads = 52 + 22 + 24 = 98

Bit short here, but we can tweak it

Try cow = 14, ch = 10
14*4 + 10*2 + 24 = 56+20+24 = 100

Perfect


I tested AC (D) first and I wasn't able to get it within the desired 100 legs, feet and heads, so I moved straight to (B).
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New post 06 Oct 2019, 15:56
1
Bunuel wrote:
A farm has chickens, cows and sheep. The number of chickens and cows combined is 3 times the number of 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?

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


I am surprised more people in the comments above have not attempted an answer-first approach, given that this question is asking about a single unknown--i.e. what is the value of such-and-such? Such an approach can lead to a quick and assuredly accurate answer. (An aside: I agree that the phrasing of the question could be clarified a bit: instead of "a total of 100 feet and heads" which I myself interpreted as 100 feet and 100 heads (before realizing that such an interpretation proved untenable), the question could probably use a switch to "a total of 100 feet and heads combined," or the "and together" could be moved and tweaked, as in, "and cows and chickens have a total of 100 feet and heads altogether." No more room for confusion.) Why not start with a (B) or (D) answer to use as a gauge? I started with (D):

If there are 14 sheep, then 3 times the number of sheep means that there are 14 * 3, or 42 chickens and cows combined. Knowing that there must be more cows than chickens within this barnyard subgroup, then at a bare minimum, there will be 22 cows. 22 cows = 22 * 4 (legs) + 22 (heads) = 88 + 22 = 110 extremities. Too high. There is no need to even bring in the chickens. Scrap (D) and (E). Importantly, we can now test (B), knowing that the number in the middle of the remaining responses will either be the answer itself or point directly to what the answer needs to be (higher or lower). Assume there are 8 sheep and repeat the process:

If there are 8 sheep, then there will be 8 * 3, or 24 chickens and cows combined. At a bare minimum, there will be 13 cows. 13 * 4 (legs) + 13 (heads) = 52 + 13 = 65 extremities. Time for the chickens: 11 * 2 (legs) + 11 (heads) = 22 + 11 = 33 extremities. 65 (cow extremities) + 33 (chicken extremities) = 98 extremities. Our extremity count is a little low, but 98 is tantalizingly close, and if you can appreciate that the cow-to-chicken extremity ratio (perhaps the first time in the English language these words have been paired together) is 5:3 (from 4 legs + 1 head to 2 legs + 1 head), then you can see that all we need to do is swap out a chicken for a cow. For the must-have-proof nagging voice in your mind...

If there are 14 cows and 10 chickens, then there will be (14 * 4 + 14) + (10 * 2 + 10) extremities, and that we can work quickly: (56 + 14) + (20 + 10) = (70) + (30) = 100. Can we tick all the boxes of the question stem?

Farm Animals:
chickens
cows
sheep

Chickens + cows = 3 times the number of sheep:
10 chickens + 14 cows = 3 * 8 sheep
24 = 24

More cows than chickens or sheep:
14 cows, 10 chickens, 8 sheep

Cows and chickens have a total of 100 feet and heads together:
(See above)

There is no room for doubt. The problem took me a little over a minute, even with the setback in the beginning of misinterpreting the extremity count. I had a lot of fun with this question, even if I felt a bit like a butcher. I hope maybe you enjoyed the problem, too (if you made it this far).

- Andrew
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Re: M06-12   [#permalink] 06 Oct 2019, 15:56

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