Section 2.3 Multiplication and Division of Fractions. Simplification.

 

Multiplication

 

How can we multiply fractions? What would be the result of the following multiplications?

,   , 

 

Multiplication by an Integer

Let’s start with multiplication of a fraction by integers. The best example of a fractions used in real life are the coins:

Half a dollar or

1 quarter or

1 dime or

1 nickel or

1 penny or

 

Example:

What would be ?

Thinking about the money 1 quarter taken 3 times is equal to 3 quarters.

So,

 

To Multiply a Fraction by an Integer

1.      Multiply the numerator of a fraction by an integer.

2.      Keep the same denominator.

 

Example:

Multiply fraction by integer

  4.  Recall, any number multiplied by 0 is equal to 0

 

 

Multiplication of Fractions

 

Example: Money

  1. What would be?

Thinking about the money it is the same as asking what is a half of one dime (one tenth of a dollar or 10 cents)?

Half of one dime is one nickel (one twentieth of a dollar or 5 cents)

So,

  1. What would be?

Thinking about the money it is the same as asking what is a half of eight dimes (eight tenth of a dollar or 80 cents)?

Half of eight dimes is eight nickels (eight twentieth of a dollar or 40 cents)

So,

            Also we can think about 40 cents as 4 dimes or

 

Example: Graphical

What would be?

            1/6                 1/6              1/6                1/6                1/6               1/6

 

 

 

 


    1/12   1/12    1/12   1/12    1/12   1/12   1/12    1/12   1/12    1/12  1/12    1/12                  

If we take half of we will get

 

To Multiply a Fraction by a Fraction

1.      Multiply the numerators of the fractions to get the new numerator.

2.      Multiply the denominators of the fractions to get the new denominator.

 

Example:

Multiply fractions

Simplifying

Example:

Look at the following shaded parts of an object

 

 

 

 


Both of them represent the same quantity but are represented by different fractions, the first one as , while the second as .

So

Multiplying by 1

Recall

Now if we look back at the equation

If we divide both numerator and denominator of a fraction by 2 the resulting fraction is equal to the original one. In general multiplication or division of numerator and denominator of the fraction by the same non-zero number changes the form of the fraction by keeps its value unchanged.

 

Multiplicative Identity for Fractions

When we multiply any number by 1, we get the same number

, for any integer n, such

 

Since we know that and are two names for the same number. We say that and  are equivalent.

 

 

Example:

Find a fraction with denominator 24, equivalent to the fraction

Since ,

 

Simplifying Fractions

A fraction in the simplest form is a fraction that has the smallest numerator and denominator, the fraction that has no common factors, other than 1, in numerator and denominator.

 

Simplifying Fractions

1.      Write both numerator and denominator as products of their prime factors.

2.      Divide both numerator and denominator by all the common factors.

 

 

Example:

Simplify a fraction

 

Caution with Cancelling

You can cancel only common factors!!!

Can cancel:

Cannot cancel:

 

If cannot factor, cannot cancel!

 

 

To compare two fractions with the same denominator, compare their numerators.

 

To compare two fractions with different denominators, bring both fractions to the same denominator and compare the numerators.

 

Example:

Compare and

 

Find the common denominator. It is the same as the Least Common Multiple of the numbers

3 and 8

LCM is 24

Bring both fractions to the same denominator 24 by multiplying each of them by 1 using the missing factors:

            

Since ,                                              

Division

Reciprocals

 

Example:

Multiply

  3.  

Reciprocals

If the product of two numbers is 1, these numbers are called reciprocals of each other or multiplicative inverses. To find a reciprocal of a fraction, interchange the numerator and denominator

Number ->  <- Reciprocal

 

Example:

Find the reciprocal of a number

  1.                     Reciprocal is
  2.                      Reciprocal is , since               
  3.                  Reciprocal is

 

Number 0 has no Reciprocal

The number 0, or , has no reciprocal (Since is undefined)

 

Division

 

Example:

How can we divide a fraction by a fraction? The question turns into how many times does go into?

   1/6                 1/6              1/6                1/6                1/6               1/6

 

 


                                    1/3                                      1/3

 

        2/3          

goes into:             4 times

 If we multiply dividend  by the reciprocal of the divisorthan we will get the same result:

 

To Divide Fractions

Multiply the dividend by the reciprocal of the divisor

1.      Multiply the numerator of a dividend by the denominator of the divisor

2.      Multiply the denominator of a dividend by the numerator of the divisor

3.      Simplify the result if possible

 

Example:

Divide fractions and simplify:

 

Solving Equations with Fractions

Example:

Solve an equation:

  1.  Divide both sides by                    b.   Divide both sides by

                                                          

Multiply by the reciprocal                    Multiply by the reciprocal

 

                                                             

Check:                                                               Check: