Lemonade
(page 90)

That sixth-grade class from Hermann, MO sent in this solution (April 2005):

Dear Tim,  

We are sixth-grade students at Hermann Middle School in Hermann, MO.  We had two different groups working on this problem.  Each group decided to do different things.  Group one decided to sell 4 oz.(1/2 cup) Dixie cups of lemonade.  Group 2 decided to sell 8 oz. (1 cup) plastic cups of lemonade.   

Each group then decided how many people to plan for, what to buy, and what would be free.  Group 1 decided to plan for 60 people.  Group 2 planned for 64 people.  Each group decided that the lemons, pitchers, tables and chairs, water, and markers would be free.  That left sugar, napkins, cups, ice, and poster board to be purchased.  

Before we could do too much planning, we needed a lemonade recipe.  We found the following recipe on the Internet:

6 lemons
3 cups of water
3/4 cups of sugar
This recipe makes 4-8 oz. servings.

We then had to figure out how many gallons of lemonade to make.   Since this recipe makes 4 cups of liquid, we had to convert the recipe to 1 gallon.  We knew that there are 4 cups to a quart, so this recipe makes 1 quart.  We also know that there are 4 quarts in a gallon. So, we would need to multiply everything by 4.   

New recipe:
24 lemons
12 cups of water
3 cups of sugar

Group 1 could get 8-4 oz dixie cups to one recipe. This group multiplied their servings by 4.  They can get 32 servings per gallon.  This group would need 2 gallons to serve at least 60 people.  With two gallons they could actually serve 64 people.   

Group 2 had bigger servings.  They could only get 4 servings per quart, so they multiplied their 4 servings by 4 to get 16 servings per gallon.  Since this group needed 64-8 oz servings, they did repeated addition until they got to 64 servings.  They found out they would need to make 4 gallons.  

Each group got prices of products from the Internet.  They found out they could purchase a 4 pound bag of sugar for $1.49.  Napkins could be purchased for $.02 a piece.  Four ounce cups could be purchased for 3 cents each, and 8 ounce cups could be purchased for 6 cents per cup.  Group one estimated that they could buy ice for $1.50. Group 2 figured they could buy ice for $1.25.  Poster board could be puchased for $2.50.  

Group 1 Expenses
  One bag of ice   1.50
  60 cups          1.80
  60 napkins       1.20
  sugar; 1bag      1.49
  poster           2.50
For a total of  $8.49
 
Group 2 Expenses
  2 bags of ice    2.50
  75 cups          4.50
  75 napkins       1.50
  sugar; 2 bags    2.78
  poster           2.50
For a total of $13.78

Group 1 sold 58 servings at 25 cents a cup for $14.50.  This left them with a profit of $6.01.  Group 2 sold 58 servings at 50 cents a cup for $29.00.  This left them with a profit of $15.22.  

Sincerely,
The Sixth Grade at Hermann Middle School

This is the first solution to this problem. Thanks!

I'm really grateful that you described what you did so thoroughly.

Readers, as usual, think: did they do it right? Are their calculations correct? Is their reasoning sound? (I'm not saying what I think -- you really need to look carefully and decide!)

Then think about a big-picture question: in real life, would Group 2 have a profit so much larger than Group 1? Why or why not? Are the numbers realistic? And for that much profit, is it worth the effort to sell lemonade?

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