The Hundred Fowls

Date: 09/29/2001 at 16:14:28
From: Yana
Subject: Word Problem

A Chinese puzzle found in the sixth-century work of mathematician 
Chang Chiu-chen called the "hundred fowls" problem asks:

If a rooster is worth five coins, a hen three coins, and three 
chickens together are worth one coin, how many roosters, hens, and 
chickens totaling 100 can be bought for 100 coins?

I tried to figure this problem out algebraically, but what would equal 
the variable, or how would the equation be set up?

Date: 09/29/2001 at 16:26:59
From: Doctor Jodi
Subject: Re: Word Problem

Hi Yana,

You can solve this problem by trial and error, but it can take a 
while. I used it in a history of math class for gifted middle 
schoolers this summer. Since they didn't know algebra, they used a LOT 
of tables, and the fact that you can't have more than 20 roosters. It 
turns out that there are three different solutions.

If you want to set it up algebraically, first call the number of 
roosters R, the number of hens H, and the number of chickens C.

Then we have two constraints. First of all, the total number of fowl 
must be 100. So R + H + C = 100.

Second, the total COST of the fowl must be 100. The cost of roosters 
is 5R, the cost of hens is 3H, and the cost of chickens is (1/3)C, 
right?

So what's the total cost of the fowl? 5R + 3H + (1/3)C. So our second 
equation is that  5R + 3H + (1/3)C = 100.

Next you can solve for one of the variables in terms of another. I 
expect that you'll still need to use a bit of "guess and check" since 
there is more than one solution.

Do write back if you want more help.

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

Date: 09/29/2001 at 21:13:17
From: Yana
Subject: Re: Word Problem

Thank you for your help but I still have a few questions.

From the equations I got the following:

5R + 3H + (1/3)C = 100
   divide everything by 5
R + 3H + (1/3) = 20
   divide everything by 3
R + H + (1/3) = 6 2/3
   divide everything by 1/3
R + H + C = 20

I get stuck at this point and don't know where to go. Please help.

Date: 09/30/2001 at 13:48:30
From: Doctor Jodi
Subject: Re: Word Problem

Hello again!

First of all, remember that if you divide ONE part of the equation by 
5 (or 3 or 1/3), you need to divide the rest of the equation by the 
same number.

So we have the equations 

R + H + C = 100 and 
5R + 3H + (1/3)C = 100

Let's solve for C in the second equation.

First multiply both sides by 3:

3 * (5R + 3H + (1/3)C) = 3 *100

We get 15 R + 9 H + C = 300.

Next we want to subtract 15 R from both sides.

 15R + 9 H + C = 300
-15R             -15R
--------------------------
       9H + C = 300 - 15R

Next we want to subtract 9H from both sides:

 9H + C = 300 -1 R
-9H       -9H
-------------------
C = 300 - 15R - 9H

Now we have solved for the number of chickens in terms of the 
number of hens and roosters.

So we can plug 300 - 15R - 9H in for C in the other equation, 
R + H + C = 100.

This gives us:  R + H + (300 - 15R - 9H) = 100.

Add the R's and the H's together. This gives

-14R -8H + 300 = 100.

Now let's subtract 300 from both sides:

-14R -8H +300  =  100.
          -300    -300
-----------------------
-14R -8H = -200

Next we can divide by -2:

7R + 4H = 100

At this point, algebra has done as much for us as it can. Now we need 
to find solutions to this equation. When we DO find a solution, we can 
find the number of chickens C using R + H + C = 100. Then the numbers 
R, H, and C that we have found will satisfy the above equation.

How can we find solutions to this equation?

We'll have to use guess and check. We can have no more than 14 
roosters (since 7*15 = 105). We can pick various numbers of roosters 
and see how many hens we will need.

For instance, if R = 14, then 7*14 + 4H = 100, 4 H would have to be 2, 
which doesn't make sense.

If R = 13, then 7*13 + 4H = 100, so 4H = 9. Again, we don't get an 
integral answer.

You can continue the guess and check for R = 12, 11, etc. to find the 
solution.

Does this make sense? Write back if you have more questions.

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

Date: 09/30/2001 at 16:40:49
From: Yana
Subject: Re: Word Problem

Thank you so mush for your explanations and help. Through the guess 
and check method with the use of the equations. So the final result 
would be either the roosters could equal 12 or 8.

Yana

Date: 09/30/2001 at 18:56:24
From: Doctor Jodi
Subject: Re: Word Problem

Hi Yana,

The possible solutions you've found are 

12 Roosters, 4 Hens, and 84 Chickens.
8 Roosters, 11 Hens, and 81 Chickens.

There's actually ONE MORE. Can you find it?

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