Four Methods of Solving Classic Chicken and Egg Problems

Date: 02/24/2009 at 18:41:05
From: Mikayla
Subject: Solve the word problem.

If 6 cats can kill 6 rats in 6 minutes, how many cats will it take to
kill 100 rats in 50 minutes?  I think the answer is 50 cats but I 
really can't tell.  I just don't understand how to solve it.

Date: 02/24/2009 at 20:09:53
From: Doctor Greenie
Subject: Re: Solve the word problem.

Hi, Mikayla --

A more common problem of this type involves chickens laying eggs 
instead of cats killing rats.  But the idea of the problem is the 
same.  You can find links to several pages in the Dr. Math archives 
where this type of problem is discussed by searching the archives 
using the keywords "chicken egg".

Following is my quick summary of some possible approaches to the 
problem.

(1) Start with the given information and vary two of the three 
numbers at a time, keeping the third one unchanged.  For example, 
twice as many rats and the same amount of time required twice as many 
cats; or twice as many cats and the same number of rats requires half 
as much time.

The following table shows one of many different paths to the answer 
to your problem using this method:

  cats   rats   minutes
  ---------------------
    6      6      6   [given]
    6      1      1   [same number of cats; 1/6 as much time
                         means 1/6 as many rats]
    6    100    100   [same number of cats; 100 times as many rats
                         (to give us the number of rats we want)
                         means 100 times as many minutes]
   12    100     50   [same number of rats; half as many minutes
                         (to give us the number of minutes we want)
                         means twice as many cats]

It takes 12 cats to kill 100 rats in 50 minutes.

(2) Here is a variation of the above method, getting directly to the 
answer by using ratios.

  It takes 6 cats to kill 6 rats in 6 minutes.

  We want to kill 100 rats instead of 6; that will require
  (100/6) times as many cats.

  We have 50 minutes instead of 6; that will require (6/50)
  times as many cats.

  So the number of cats we need is

    (6)(100/6)(6/50) = 12

(3) Another approach is to find the number of "cat-minutes" it takes 
to kill each rat.  6 cats take 6 minutes to kill 6 rats, so the 
number of cat-minutes it takes to kill a rat is

  (6)(6) cat-minutes
  ------------------ = "6 cat-minutes per rat"
    (6)  rats

We are supposed to find the number of cats it takes to kill 100 rats 
in 50 minutes.  100 rats, at 6 cat-minutes per rat, means 600 cat-
minutes.  If we have 50 minutes, then the number of cats we need is

  600/50 = 12

(4) A formal mathematical approach that is very much like the 
preceding method uses direct variation (actually, if you have 
studied this topic, "joint" variation).  The idea is that the number 
of rats killed increases with either increased numbers of cats or 
increased numbers of minutes.  So we can write a joint variation 
equation

  r = kcm

where r is the number of rats, k is a constant of variation to be 
determined (it is related to the "6 cat-minutes per rat from the 
previous method), c is the number of cats, and m is the number of 
minutes.

We can use the given information to find the value of our constant 
of variation.  r=6 when c=6 and m=6; so

  6 = k(6)(6)  -->  k = 1/6

Now we use this constant of variation and the new numbers of rats 
(100) and minutes (50) in our equation to find the new number of 
cats required:

  r = kcm
  100 = (1/6)(c)(50)   -->  c = 12

Try understanding each of the above methods and pick the one that you 
find easiest to understand and use.

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