Blending Seed

Date: 09/27/98 at 00:26:11
From: Anonymous
Subject: Grass seed problem

Hi!  Here is a problem that my father and I have been working on for 
half an hour:  

A lawn-and-garden dealer wants to make a new blend of grass seed by 
using 200 pounds of $0.45 per pound seed and some $0.65 per pound 
seed. How much of the $0.65 seed does the dealer need to make a $0.55 
per pound blend?

Thank you so very much for your help. Hope to hear from you soon.

Date: 09/27/98 at 01:22:14
From: Doctor Ken
Subject: Re: Grass seed problem

Hi,

First I'll give you a sort of intuition-based solution, and then I'll 
go back and be a little more formal, writing things out in algebraic 
notation.

The first thing I noticed about this problem was that a pound of one 
kind of seed costs 45 cents, a pound of the other costs 65 cents, and 
we want to make a mix that costs 55 cents. Well, 55 is halfway between 
45 and 65, so it seems like we should combine equal parts of the two 
kinds of seed. Since we need to use 200 pounds of the first kind, we 
should use 200 pounds of the other kind, for a total of 400 pounds.  

To verify this solution, let's see how much the pieces would cost.  
200 pounds of the first kind of seed would cost 200 * $0.45 = $90, and 
200 pounds of the second kind of seed would cost 200 * $0.65 = $130.  
We bought 400 pounds total and paid $220, so we paid on average 
$220/400 = $0.55. Seems to check out.

Now, how can we use algebra to find this same solution? The first 
thing I usually try to do is to find a sentence (an English sentence, 
or whatever language you like best) that says something true and 
useful about the problem. Then I translate that sentence into an 
algebraic equation.  

In this problem, I think I'd make this sentence: "The cost per pound 
of the combined seed mixture is 55 cents." Let's work on translating 
that into an equation. First it becomes:

   cost per pound = $0.55

What is the cost per pound? It's the number you get when you divide 
the total price by the number of pounds of seed:

     total price
   ---------------- = $0.55
    pounds of seed

The total price is the cost of the 45c part and the 65c part. The 
number of pounds of seed is the number of pounds of the first kind 
plus the number of pounds of the second kind. The only unknown thing 
here is the number of pounds of the second kind, so let's call that x.

    200*$0.45 + x*$0.65
   --------------------- = $0.55
         200 + x

Now we've done the translation. I'll leave it to you to work out the 
solution from here, and verify that the answer really is 200.

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