Math CentralQuandaries & Queries


Subject: quadratic equations
Name: Norman
Who are you: Other

My grandson is testing me on a question he recently solved.

The question is "A farmer has cows and chickens - there are 252 heads and 81 legs - how many of each animal does he have?"

I immediately thought of the quadratic equations I learned 50 years ago at school, but cannot remember how to do it.
It is further complicated by the fact that 1 animal must have 3 legs as the total number of legs is 81 (or perhaps he his trying to catch me out)
Any helpful advice would be welcome.

Many thanks

Norman West

We have two responses for you.

Hi Norman.

There are far more heads than legs! Could you perhaps have the numbers reversed?

If there are 252 legs and 81 heads then consider it this way:

Let B = the number of bovines and F = the number of fowl

Then B + F = 81 (counting the heads) and 4B + 2F = 252 (counting the legs).

Now you have two equations with two unknowns. Can you solve it from here?
When I solve it, I get two sensible whole numbers.

Stephen La Rocque.>

Hi Norman,

I like these "head and feet" problems but I try to do them without using equations at all.

Imagine that you are a child constructing make-believe chickens and cows from marshmallows and toothpicks. A marshmallow with two toothpicks in it is a chicken and with four toothpicks in it is a cow. Take your 81 marshmallows and stick 2 toothpicks in each to make 81 chickens. That took 2 x 81 = 162 toothpicks leaving 252 - 162 = 90 toothpicks, that is 90/2 = 45 pairs of toothpicks. With each of these 45 pairs of toothpicks I can turn a chicken into a cow giving me 45 cows and the rest are left as chickens.


About Math Central


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS