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