Simplifying Fractions

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Simplifying Fractions

Writing a Number as a Product of Primes

We call a whole number greater than one prime if it cannot be divided evenly except by itself and one. For example the number 7 is prime but the number 6 is not, because

        6 = 2 x 3

A number that is not prime is called composite.  The first primes are

        2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53  These are all prime

One of the most common uses of primes is to write a number as a product of primes.

Example

Write the number 140 as a product of prime numbers.

Solution

We write

        140 = 10 x 14 = (2 x 5) x (2 x 7) = 2 x 2 x 5 x 7

We can also use a factor tree to write a number as a product of primes

Example

Write the number 882 as a product of primes using a factor tree.

             $A factor tree with 196 going to 4 and 49, 4 going to 2 and 2, and 49 going to 7 and 7$
So that

        196 = 2 x 2 x 7 x 7

Exercise

Write each number as a product of prime numbers

Remark: As we have seen from the following examples the task of writing a number as a product of primes is always possible. In fact, the result will always be the same. This remark is so important that it is called

The Fundamental Theorem of Arithmetic

Every composite number can be written in exactly one way as a product of prime numbers

Reducing Fractions

Consider a pizza that has been cut into four slices such that two of the slices are left. There is more than one way of writing this as a fraction. One way is

        2

        4
since that are two slices left out of four total. The other way is to notice that exactly one-half of the pizza is left. So we write

        1

        2

Notice that we can write the numerator and denominator of the first fraction as

        2            2 x 1
               =
        4            2 x 2

              2         1
      =           x
              2         2


                     1
      =    1 x                Any number divided by itself is 1.
                     2

               1
      =
               2

We define a common factor of two number to be a number that is a divisor of both. As was shown in the example, we can always divide out a common factor.

Example

Simplify

           18

           24

Solution

We see that 6 is a common factor of 18 and 24, so

           18            6 x 3            3
                   =                   =
           24            6 x 4            4

Exercises

Simplify the following
1.            14
  
             35
2.             24
  
              33
If we do not immediately see the common factor, we can factor the numerator and the denominator into their prime factorizations and then cancel all common factors

Example

Simplify

           126

           350

Solution

We write

        126 = 2 x 63 = 2 x 3 x 21 = 2 x 3 x 3 x 7

and

        350 = 10 x 35 = 2 x 5 x 5 x 7

so that

           126             2 x 3 x 3 x 7
                     =
           350             2 x 5 x 5 x 7

               3 x 3              9
         =                =
               5 x 5             25

Exercises
1.             90
  
             165
2.             225
  
              441
Testing for Equality

How can we tell if two fractions are equal? One way is to simplify and see if they simplify to the same fraction. An easier way is to take the cross products and test for equality.

Example

       24    ?    8
               =
       27          9

We check

        24 x 9 = 27 x 8

Since these are both equal to 216. We can conclude that the two fractions are equal.

Example

Show that the following two fractions are not equal

       3             6

       4             7

Solution

We have

        3 x 7 = 21     and     6 x 4 = 24

Since

        21 24

We can conclude that the two fractions are not equal

Exercises

Determine which pairs of fraction are equal
1.         3     ?     5
                 =
         11          14
2.        16    ?    18
                 =
         40          45
Applications

Example

Gasoline costs 144 cents per gallon at the pump. 54 cents of this goes to taxes. What fractional part of the cost goes to taxes?

Solution

We write

        54             9 x 6                  2 x 3 x 3 x 3
                =                     =
       144         12 x 12            2 x 2 x 2 x 2 x 3 x 3

             3                 3
                         =
       2 x 2 x 2            8

We can conclude that 3/8 of the total cost goes to taxes.

Exercise

You have found that at your restaurant out of the 96 patrons, 8 complained that food took too long to come. What fractional part of the patrons made this complaint?

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