Using Tables to Tame a Problem

Date: 09/19/2003 at 03:05:30
From: Tiffany
Subject: Math Problem

The Ride-A-Mile bicycle company presented their new models at a bike 
show. Racing models were either silver or blue. Every silver bike was 
a racing model. 1/2 of the blue bikes were racing models. 1/3 of the 
bikes were racing models. There were 40 blue bikes and 30 silver 
bikes. How many bicycles were neither blue nor silver?

I find the fractions especially confusing. 

Date: 09/19/2003 at 16:47:43
From: Doctor Peterson
Subject: Re: Math Problem

Hi, Tiffany.

The hard part, I think, is to make sense of all the information and 
find an orderly way to think about it. Maybe we can start by listing 
the different categories of bikes there are:

  bikes can be racing, or non-racing
  bikes can be blue, silver, or another color

Now maybe we can make a table:

          racing    non
         +--------------
  blue   |
  silver |
  other  |

Now we can start to fill in information. We know that ALL racing 
bikes are either silver or blue, so we can put a 0 in the space for 
other colors:

          racing    non
         +--------------
  blue   |
  silver |
  other  |  0

We know that ALL silver bikes are racing bikes:

          racing    non
         +--------------
  blue   |
  silver |           0
  other  |  0

We know only that half of the blue bikes are racing bikes, so that 
doesn't give us a number to fill in for blue, but we can note the 
fact that the two numbers beside blue will be equal.

Similarly, we know that 1/3 of ALL bikes are racing bikes. Keep note 
of that. We may want to make a row for total number of bikes of each 
type.

          racing    non
         +--------------
  blue   |
  silver |           0
  other  |  0
         +--------------
  total  |

Now we have totals of blue and silver bikes, so we can add a total 
column:

          racing    non  total
         +--------------+-----
  blue   |              | 40
  silver |           0  | 30
  other  |  0           |
         +--------------+-----
  total  |              |

Now, can you take over from here? We have to finish filling in 
numbers, using the facts we wrote down but haven't used yet.

If you need more help, please write back and show me how far you got.

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