Mixing Peanuts and Cashews

Date: 11/19/1999 at 23:28:15
From: Kelley Remington
Subject: Peanuts and Cashews Mixture Problem 

My daughter has a problem that I have tried to figure out for hours: 
Peanuts sell for $3.00 per pound. Cashews sell for $6.00 per pound. 
How many pounds of cashews should be mixed with 12 pounds of peanuts 
to obtain a mixture that sells for $4.20 per pound?

I sure hope you answer my question or I will never be forgiven for not 
knowing how to solve this problem. I will go out and buy a book if you 
are unable to answer my question.

Thank you, 
Kelley Remington

Date: 11/20/1999 at 10:54:08
From: Doctor Anthony
Subject: Re: Peanuts and Cashews Mixture Problem

Let C be the weight of cashews to buy.

Total cost of mixture =

     (12 x 3) + (C x 6) = (12+C) x 4.2

                36 + 6C = 50.4 + 4.2C

                   1.8C = 14.4

                      C = 8

So you need to buy 8 pounds of cashews.

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

Date: 06/23/2000 at 17:02:57
From: Doctor Johnny
Subject: Re: Peanuts and Cashews Mixture Problem

Kelly,

Hello there. My name is Dr. Johnny. I am a high school algebra teacher 
who really understands your frustrations. When trying to teach this 
same type of problem to high school students, I find that they 
experience a variety of concerns. I hope I can help you to better 
understand this situation.

The best way to set up this problem is to construct a table that shows 
the relations among the 3 items (peanuts, cashews, and mixture).

     Type        Cost per pound     Pounds     Total Cost
     ----        --------------     ------     ----------
     Peanuts         $3.00            12       3(12) = 36
     Cashews         $6.00             x         6x
     Mixture         $4.20           12+x      4.20(12+x)

Here are some explanations for why I set the chart up this way. The 
peanuts line is pretty self-explanatory. The amount for the cashews is 
the unknown, because you know the least about that part of the 
problem. The mixture is made up of cashews and peanuts, so the amount 
of the mixture is the same as the amount of peanuts + the amount of 
cashews. From here, we know that the cost of the peanuts and cashews 
must be the same as the mixture: 36 + 6x = 4.20(12+x). Now we must 
simplify the equation to obtain the amount of cashews needed to 
satisfy the problem.

       36 + 6x = 4.2(12+x)
       36 + 6x = 50.4 + 4.2x   Distributive property
     6x - 4.2x = 50.4 - 36     Get the variables on one side and the
                                constants on the other
          1.8x = 14.4          Combine like terms
             x = 8             Division property of equality

We now have a value that we anticipate is the answer. Let's check to 
see if it makes sense:

     12 pounds of peanuts @ $3.00 per pound costs $36
      8 pounds of cashews @ $6.00 per pound costs $48
     20 pounds of mixture @ $4.20 per pound costs $84

$36 + $48 does equal $84; therefore 8 pounds of cashews does make this 
situation hold true. 

I always tell my students to look at the smaller pieces of the picture 
instead of trying to look at the whole picture, because all of that 
information can run together and really get a person confused. Good 
luck in the future and if you ever need any additional help, feel free 
to write.

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

Date: 11/20/1999 at 19:44:33
From: Kelley Remington
Subject: Re: Peanuts and Cashews Mixture Problem

Thank you for your answer. I finally figured it out but went about it 
a different way. Thank you so much for your answer though.