Hello, latinolp85!

Who assigned this problem?

A Venn diagram is the *worst* way to solve this problem!

In a group of cows and chickens i count 48 heads and 122 feet.

How many cows and fowl do i have? I spent several minutes trying to create a Venn diagram for this problem.

Let one circle be the set of cows.

Everything outside the circle is a fowl. Code:

* - - - - - - - - - - - - - - - *
| |
| |
| * - - - - - - * |
| | | |
| | | |
| | Cows | |
| | | |
| | | |
| * - - - - - - * |
| |
| |
| Fowl |
| |
| |
* - - - - - - - - - - - - - - - *

Let a second circle be the sets of Heads.

Everything outside the circle is a foot. Code:

* - - - - - - - - - - - - - - - *
| |
| |
| Feet |
| |
| |
| * - - - - - - * |
| | | |
| | | |
| | Heads | |
| | | |
| | | |
| * - - - - - - * |
| |
| |
* - - - - - - - - - - - - - - - *

Then our Venn diagram looks like this. Code:

* - - - - - - - - - - - - - - - *
| |
| |
| * - - - - - - * |
| | Cow | |
| | feet | |
| | * - - - + - - * |
| | | Cow | | |
| | | heads | | |
| * - - + - - - * | |
| | Fowl | |
| | heads | |
| * - - - - - - * |
| Fowl |
| feet |
* - - - - - - - - - - - - - - - *

Then I spent several more minutes trying to solve the problem

. . using this Venn diagram.

I failed abysmally. .I was forced to rely on Algebra.

. . This meant all the above work was **a total waste of time.**

Let $\displaystyle C$ = number of cows .and $\displaystyle F$ = number of fowl.

Since each cow has one head and four feet: .$\displaystyle \begin{array}{ccc}\text{Cow heads} & = & C \\ \text{Cow feet} & = & 4C\end{array}$

Since each fowl has one head and two feet: .$\displaystyle \begin{array}{ccc}\text{Fowl heads} & = & F \\ \text{Fowl feet} & = & 2F\end{array}$

We are told that: .$\displaystyle \begin{array}{ccc}\text{[Cow heads] + [Fowl heads]} & = & 48 \\

\text{[Cow feet] + [Fowl feet]} & = & 122 \end{array}$

This becomes the system: .$\displaystyle \begin{array}{ccc}C + F & = & 48 \\ 4C + 2F & = & 122\end{array}$

. . which has the solution: .$\displaystyle C\,+\,13,\;F\,=\,35$

Like I said, the *worst* way to solve it.