How Many Chickens and Ducks?

Date: 8/16/96 at 10:22:45
From: Leong Theng Kwong
Subject: How Many Chickens and Ducks?

A man bought 20 chickens and ducks altogether, with a $2 discount per 
chicken and 50 cent discount per duck. He saved $22 in all. How many 
chickens and how many ducks did he buy?

Please explain how to do this problem. Thank you.

LTK

Date: 8/16/96 at 11:33:53
From: Doctor Patrick
Subject: Re: How Many Chickens and Ducks?

Hi!

To solve this let's start by making C stand for the number of chickens 
and D stand for the number of ducks.

From the first part of the problem we know that the number of Chickens 
plus the number of Ducks is equal to 20.  

Or in other words, C + D = 20.

Now we turn to the other half of the problem.  Since we know how much 
the man saved per chicken and per duck, and what the total savings 
was, we can write another equation to express that $2 per chicken and 
$0.50 per duck gave a total of $22.  How do you think we can write 
this?

How about 2C + 0.5D = 22?
  Do you understand how I got that?

Now to solve these equations you will have to use both equations in 
your favorite way for solving for two variables.  

One way is like this:

Going back to the first equation (since its simpler) we can rewrite it 
as C = 20 - D, right?  Then, we go to the second equation and 
substitute in 20 - D for the C.  This way we only have one variable 
left, which is easy to solve for.  

Once we substitute for the C we get:

2(20-D) +.5D =22

Let's simplify the equation and get 40 - 2D + 0.5D = 22.  
  Do you see what I did?

Now we add like terms and get 40 - 1.5D = 22.  

We can further simplify by subtracting 22 from both sides:
 
  40 - 22 - 1.5D = 22-22, or 18 -1.5D=0

Now if we add 1.5D to both sides we get 18 = 1.5D 

Now we solve for D by dividing each side by 1.5 to get D = 12.

Now that we know D we can substitute it back into one of the 
equations to solve for C.  Let's use C+D = 20 again since it is much 
easier.  When we substitute in the D = 12, we get C + 12 = 20.  
Subtracting 12 from each side we find that C = 8.

Together we have C (chickens) = 8 and D(ducks) = 12.  You should go 
back and check this by putting these numbers into the problems and 
making sure that they work. You may also want to try to solve it using 
a different method, if you know one, for practice.

Good luck, let us know if we can be of more help.

-Doctor Patrick,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/