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Greg wants to knew how many cows and ducks are in the meadow. After counting 56 legs and 17 heads, the farmer knows. How many cows and ducks are there?

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    Any thoughts? Hint: if there are $C$ cows and $D$ ducks....how many legs are there? How many heads? – 2017-02-16
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    In Animal Farm by George Orwell, the animals defined bird's wings to count as legs because their main function is locomotion rather than manipulation. This allowed them keep chanting "Four legs good, two legs bad" without excluding the ducks and geese and chickens. I just assume that this is not the case here. – 2017-02-16

2 Answers 2

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Let $C$ be the number of cows, and $D$ the number of ducks. Each animal has just one head (I hope), so $C + D = 17$. On the other hand, each cow has four legs, and each duck has two legs, so $4C + 2D = 56$. Now you have two linear equations in two unknowns. Can you solve for $C,D$?

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There are $17$ animals (unless there might be some two-headed cows or something).

If they were all ducks, there would be $34$ legs. Each cow adds 2 extra legs. We need $56-34=22$ extra legs, so there are $11$ cows and $6$ ducks.

Or you can set this up as a simple algebra problem:

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    I always prefer non-algebra solutions to these problems. Setting up the two equations with two unknowns, while guaranteed to find an answer if there is one, takes time to do, and it takes time to solve, and the method of solution has little to do with cows and ducks. – 2017-02-16
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    Having polled three cows, I find that *they* do not consider two of their legs "extra." – 2017-02-16