Lemonade Proportions

Date: 04/02/98 at 20:50:01
From: Ria Evans
Subject: Algebra

Making lemonade requires water and lemon juice in the ratio of 3 cups 
of water to 2 cups of lemon juice. You need a total of 10 gallons for 
the fair. How many cups of lemon juice do you need? If you made 10 
gallons of lemonade for your class, how much would each person get? 
(Hint: 16 C = 1 gal).

Date: 04/04/98 at 15:23:09
From: Doctor Scott
Subject: Re: Algebra

Hi Ria!

We know we want to make 160 cups of lemonade for the fair.  We can use 
some algebra to solve the problem.

Every 5 cups of lemonade require 3 cups of water and 2 cups of lemon 
juice. We can use a proportion to help us:

     3 cups water           ? cups of water
   ------------------  =  --------------------
   5 cups of lemonade     160 cups of lemonade

Notice that the proportion allows us to compare the amounts of water 
and lemonade. We can compare the known ratio (3 to 5) to the one we 
want to make (? to 160), with the amount of water in the numerator 
and the total amount of lemonade in the denominator. Remember that we 
solve proportions by cross multiplying.  (I'm going to replace the
? with "x"):

   3 * 160 = 5x
       480 = 5x
        96 = x    <-- so we need 96 cups of water

Then, we know we are going to get 160 cups of lemonade, so we need: 

   160 - 96 = 64 cups of lemon juice.

We could have found the amount of lemon juice first, again by using a 
proportion to compare the quantities:

   2 cups lemon juice     ? cups lemon juice
   ------------------  =  -------------------
   5 cups of lemonade     160 cups lemonade

cross multiplying:

   320 = 5x
    64 = x   <--- amount of lemon juice

We can also check our answer.  We need a ratio of 3:2, and we have

      96 cups water
   -------------------
   64 cups lemon juice

Is this fraction equal to 3/2?  If so, then we're okay!

Proportions are a great way to solve a lot of problems. Whenever 
a problem involved comparing quantities, I usually try to use a 
proportion!

Hope this helps!

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