Understanding Fractions

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Understanding Fractions

Definitions

In module A, all the numbers that we encountered were whole numbers. Although the whole numbers are important, they only tell part of the story. Module B is the study of fractions which are defined as parts of the whole.

Example

Consider the picture below which might represent what is left of a pizza. We see that the pizza was originally cut into eight slices and there are three slices left.
$A circle cut into 8 equal pieces with three of them shaded.$
We say that there is

        3

        8

of the pizza remaining. The number 3 is called the numerator of the fraction and the number 8 is called the denominator of the fraction.

Example

Write the fraction of red circles in the diagram below.
$9 circles with 5 of them shaded$

Solution

We see that there are 9 total circles, 5 of which are red. The total number of circles is the denominator and the number of reds is the numerator:

        5

        9

Exercise
1. Use a fraction to represent the shaded part of the whole.
  
  $6 rectangles with 5 of them shaded$
2. Draw a sketch to illustrate 4/5.
Some Properties of Fractions

Recall that there were several properties of division. Since fractions are really division problems, we arrive at the same properties:
1. If the numerator and denominator are the same then the fraction equals 1:
  
          4
                  = 1
          4
2. If the denominator is 1 then the fraction is just the numerator
  
          6
                  = 6
          1
3. If the numerator is 0 and the denominator is not 0, then the fraction equals 0
  
          0
                  = 0
          9
4. If the denominator is zero then the fraction is undefined
  
          2
                    is undefined
          0
Applications

Application of fractions are encountered everywhere.

Example Use a fraction to describe each situation
1. Out of ten problems on the test you scored nine correct
  
  Solution:
  
  You got 9/10 of the problems right
2. Of the 75 patrons on Ellie's run, 61 were boarders
  
  Solution:
  
  The fraction of boarders on Ellie's run was 61/75
3. Of the 51 bicycles at your rental shop, you discovered that 8 needed repair.
  
  Solution:
  
  8/51 of the bicycles needed repair.
Exercises Use a fraction to describe each situation
1. Of the seven students left in the class, three are women.
2. Seven days of January were snowy.
3. Fifteen of the ninety-one tourists are from Sacramento.

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