Mixture Problem

Date: 11/11/97 at 21:30:15
From: Erica Lower
Subject: Mixture problems

I am in ninth grade and in my first year of algebra. It was a breeze 
at first, but now it is getting very complicated. I know how to do the 
equation once it is set up, but the problem is that I can't figure out 
how to set them up.

Example:

Chris has 3 gallons of a solution that is 30 percent antifreeze, which 
he wants to use to winterize his car. How much pure antifreeze should 
he add to this solution so that the new solution will be 65 percent 
antifreeze?

Thank you so much for your time. I appreciate it and I'm sure my class 
will too!

Date: 11/12/97 at 05:42:13
From: Doctor Mitteldorf
Subject: Re: Mixture problems

Dear Erica,

Take heart! Everyone goes through a period of confusion about word 
problems. Translating words into useful equations is a skill that 
you can develop over a lifetime. The more you do, the better you'll 
get at it. Keep trying and don't be afraid to make mistakes. Follow 
the consequences of your reasoning, and try to learn from where it 
takes you.  Go back and try something else.  Whatever it takes.

About antifreeze... Let's see, you start out with x gallons of 
antifreeze that's been mixed with y gallons of water, and the total 
is 3 gallons, so I'd say x + y = 3 is a good equation to start with.  
You also know that the solution is 30 percent antifreeze, so 
x/(x+y) = .30. Since you have two equations and two unknowns, these 
equations are enough to tell you both x and y. 

Maybe you should find x and y before going any further, and then you 
can write the next equation more simply.

Now we want to add some more antifreeze until the total is 65 percent 
antifreeze. Say you're adding z gallons. The equation is 

    (x+z)/(x+y+z) = 0.65

The x+z is the total amount of antifreeze, and the x+y+z is the total
amount of solution, antifreeze+water. Since you already know x and y,
you can put them into this equation and you'll just have a single 
equation with z in it.  That should do the trick!

One more thing. I think the person who made up this question was 
thinking more about algebra than about chemistry. Antifreeze and water 
really don't mix this way! Try it and see: take one measuring cup full 
of antifreeze and mix it with one measuring cup full of water. Let me 
know if the mixture you get is two cups or if it comes out more or 
less than two cups!

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