One Chicken, One Day

Date: 04/19/2002 at 10:20:39
From: Erryn Bard
Subject: Fractions

It's the same old chicken and egg question, but with a twist ending:

If a chicken and a half lays an egg and a half in a day and a half, 
how many eggs can one chicken lay in one day?

Date: 04/19/2002 at 12:25:12
From: Doctor Peterson
Subject: Re: Fractions

Hi, Erryn.

Have you seen our answers to related questions?

   A Chicken and a Half?!
   http://mathforum.org/library/drmath/view/58675.html 

   A Hen and a Half
   http://mathforum.org/library/drmath/view/57464.html 

   Laying Eggs Better by Half
   http://mathforum.org/library/drmath/view/56844.html 

(I found these by searching our site for the phrase "and a half.")

The second of these is the simplest. I like to just transform the 
problem one step at a time into something simpler:

    How many eggs will a chicken and a half lay in three days?

    In one day?

    How many will three chickens lay in one day?

    How about one chicken?

There are many ways to think about this, suited for different levels 
of experience. I would like to see what ideas you came up with, so I 
can help you do it in your own way.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/19/2002 at 14:31:32
From: Erryn Bard
Subject: Fractions

We found this via the URL's you gave:

For a quick solution to your version, we can start by finding the 
laying rate in eggs per hen-day:

         1.5 eggs
    ------------------- = 2/3 eggs/hen-day
    1.5 hens * 1.5 days

so the answer to our question is 2/3 egg.

However, after drawing it out on paper we surmise that if the half 
chicken lays 1/2 egg in 1 1/2 days, the whole chicken will lay 1 egg 
in 1 1/2 days. Therefore, how much will that one chicken lay in one 
day?

We could split the one and half days into 3rd's and thus get 2/3, but 
how do you get 2/3 egg per chicken?

Are we supposed to think of 1 1/2 chickens as a whole, and the 1 1/2 
eggs as a whole, rather than splitting the eggs up between the 1 1/2 
chickens?

Date: 04/19/2002 at 15:19:15
From: Doctor Peterson
Subject: Re: Fractions

Hi, Erryn.

You have the right answer. Using my approach,

    1 1/2 chickens lay 3 eggs in 3 days
    1 1/2 chickens lay 1 egg in 1 day
    3 chickens lay 2 eggs in 1 day
    1 chicken lays 2/3 egg in 1 day

So what does that mean? How can there be a fraction of an egg?

When we talk about a fractional rate, such as 2/3 egg per day, we have 
to visualize it in whatever way is appropriate to the problem. 
Obviously if each hen lays once a day at this rate, there will be a 
lot of broken eggs lying around. Instead, the rate has to be thought 
of as an average. It may be that every day 2/3 of the chickens lay an 
egg, so that if you have 3 chickens you get 2 eggs every day. It may 
be that each chicken lays 2 eggs every 3 days, either by taking every 
third day off, or by laying one egg every 36 hours. So it might look 
like
                day 1           day 2           day 3
          +---------------+---------------+---------------+
    hen 1   o               o
    hen 2   o                               o
    hen 3                   o               o
    -----
    total   oo          =2  oo          =2  oo          =2

or
                day 1           day 2           day 3
          +---------------+---------------+---------------+
    hen 1   o                       o
    hen 2           o                       o
    hen 3                   o                       o
    -----
    total   o       o   =2  o       o   =2  o       o   =2

or even
                day 1           day 2           day 3
          +---------------+---------------+---------------+
    hen 1   o               o
    hen 2   o               o
    hen 3   o               o
    -----
    total   ooo         =3  ooo         =3              =0

where the number differs from day to day, but in any 3 days you get 
6 eggs. All of these fit.

Or maybe 1/3 of the chickens are roosters and never lay an egg, but 
for every 3 chickens, 2 eggs are laid every day!
 
                day 1           day 2           day 3
          +---------------+---------------+---------------+
    hen 1   o               o               o
    hen 2   o               o               o
    rooster
    -----
    total   oo          =2  oo          =2  oo          =2

Talking about averages always involves pretending things are more 
uniform than they really are; rather than talking about the number of 
whole eggs each individual hen lays, and when, we just spread around 
the eggs that are laid, as if each hen laid 2/3 of an egg. The total 
number of eggs would be the same, and that's all the farmer cares 
about (unless he'd like to stop feeding some of those roosters 
and increase his profits).

The same thing happens if we say that the average family has 2.5 
children. That doesn't mean there are a lot of pieces of children 
running around. Rather, one family may have 2 and the next has 3, so 
that there are 5 children for every 2 families. To average it out, we 
divide them equally and pretend there are 5/2, or 2.5, children in 
each family.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/