Orange and Grape Sodas

Date: 08/24/2002 at 10:17:40
From: Dustin
Subject: Guess and check problem

Douglas bought a total of 50 cans of orange and grape sodas. He bought 
12 more cans of grape than orange. How many of each did he buy?

I have subtracted, added, and multiplied, and still can't get the 
answer. Would appreciate your help. Thanks.                          
                               

Dustin

Date: 08/24/2002 at 10:58:21
From: Doctor Sarah
Subject: Re: Guess and check problem

Hi Dustin - thanks for writing to Dr. Math.

Are you allowed to use a little algebra?  

You could let the cans of orange soda equal x. Then the cans of grape 
soda will be x+12, and the total will be 50:

   x + x + 12 = 50

Collect like terms (the x's):

      2x + 12 = 50

Subtract 12 from both sides of the equation:

           2x = 38

Divide both sides of the equation by 2:

            x = 19  (number of cans of orange soda)

       x + 12 = 31  (number of cans of grape soda)

Does 19 + 31 equal 50?

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

Date: 08/25/2002 at 15:33:46
From: Doctor Greenie
Subject: Re: Guess and check problem

Hi, Dustin -

Your question and the answer provided by Dr. Sarah interested me.

Her response was perfectly correct and appropriate - if you are 
allowed to use algebra. However, it appears likely to me that, if you 
are really 11 years old, then you aren't expected to use algebra in 
the solution of the problem. Here is another approach to the 
problem which you can use to get the answer without using algebra:

The problem says Douglas bought 12 more cans of grape soda than of 
orange, and that he bought a total of 50 cans of soda. Suppose, for 
the moment, we "take away" those 12 extra cans of grape soda from the 
problem - what do we have then?  Now, without those 12 "extra" cans 
of grape soda, he has bought the same number of cans of orange and 
grape sodas, and he has bought a total of 38 cans (50 - 12) of soda.  
That means he has bought 19 (half of 38) cans of each type of soda.  
Now we add those 12 extra cans of grape soda back into the problem 
and find that he bought 19 cans of orange soda and 31 cans (19 + 12) 
of grape soda.

If you haven't yet started to learn any algebra, it might be 
interesting to compare my response to the one provided earlier by 
Doctor Sarah. You might see that the path to the solution using the 
formal algebra of her approach and the logical reasoning of my 
approach are in fact virtually identical - hers just uses the formal 
algebraic notation, while mine is more informal. Comparing the two 
responses, you might be able to see how algebra is used to solve 
problems like this in a formal manner.

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

Date: 08/26/2002 at 15:13:34
From: Dustin
Subject: Thank you (Guess and check problem)

Dr. Greenie,

Thank you so much for your answer on the sodas. I really appreciate 
it. This makes more sense than the other way. I don't think I should 
be having algebra either in the 6th grade. Maybe this will help me.

Thanks again,
Dustin