# I have 12 animals (rabbits and ducks) loose in my barnyard. How many rabbits and ducks are there if I have counted 34 legs?

My answer to a question on Quora: I have 12 animals (rabbits and ducks) loose in my barnyard. How many rabbits and ducks are there if I have counted 34 legs?

There is a classical solution which uses only arithmetic.

• Observe that rabbits and ducks have different numbers of legs.
• Make all animals equal in a way that allows counting: at a pet shop, buy sufficient number of boots, $$1$$ pair for each duck and $$2$$ pairs for each rabbit, ask them put the boots on, and then ask each rabbit to return to you $$2$$ of its boots, so that each creature gets exactly $$2$$ boots.
• Ask a question: How many boots are left? It is easy: 12 animals with two boots each have $$12\times 2 = 24$$ boots.
• How many boots were removed? $$34 – 24 = 10$$ boots.
• From how many rabbits boots were removed? 2 boots from a rabbit means $$10 \div 2 = 5$$ rabbits.
• How many ducks are in the barnyard? $$12 – 5 = 7$$.
• As simple as that. But do not forget to return the boots to rabbits.

A comment for a teacher (if by any chance a teacher reads this reply): in this my solution, I am trying to demonstrate Igor Arnold’s characterisation of arithmetic:

The difference between the “arithmetic” approach to solving problems and the algebraic one is, primarily the need to make a concrete and sensible interpretation of all the values which are used and/or which appear at any stage of the discourse.

I was also using the classical old “questions method” for solving word problems; you many find its discussion in my paper A. V. Borovik, Economy of thought: a neglected principle of mathematics education, in Simplicity: Ideals of Practice in Mathematics and the Arts (R. Kossak and Ph. Ording, eds.). Springer, 2017, pp. 241 – 265. DOI 10.1007/978-3-319-53385-8_18. ISBN 978-3-319-53383-4. A pre-publication version (without editorial changes made by publishers): bit.ly/293orpk

After I published the answer, I found a simpler solution:

Give to each animal $$4$$ boots, and then ask ducks to return unnecessary (excessive) boots. The calculation now is

$$4 \times 12 = 48$$ boots all together,

$$48–34 = 14$$ excessive boots,

$$14 \div 2 =7$$ ducks,

$$12 – 7 = 5$$ rabbits.

The moral of this story: generosity pays.