Unit 7 - Objective 1 - Equivalent Fractions


Multiplying or dividing both the numerator and denominator of a fraction by the same numbers, except zero, results in a fraction that is equivalent to the original fraction.  Equivalent means they represent the same number.  We want to reduce the fraction to lowest terms or simplest terms.

Note:

u7obj1-01.gif (388 bytes)

because u7obj1-02.gif (513 bytes)  and

if we have -x/y the negative can be with the numerator as in -x/y or in the denominator as in x/-y but the negative cannot be with both. General procedure for reducing a fraction is to factor numerator and denominator then divide out factors they have in common.

 

Examples:

1.

Reduce  u7obj1-03.gif (485 bytes)  to lowest terms

The numerator and denominator have a common   factor of 8xyz3.  We can rewrite the fraction as

u7obj1-04.gif (1266 bytes)

Since 8xyz3 cancel we are left with 3x/y2 which is equivalent to

u7obj1-03.gif (485 bytes) which is equal to (3x)/y^2
 


2.

Reduce  u7obj1-06.gif (462 bytes)

Notice that x2 is a factor in the numerator but it is not a factor for both terms in the denominator.   We need to factor the denominator which is the difference of two squares.  The numerator cannot be factored farther.

u7obj1-07.gif (586 bytes)

We can now cancel the x - 4 term and we are left with -2x2/(x+4) as an equivalent fraction. (answer: (-2x^2)/(x+y)

 

Note:

u7obj1-08.gif (2543 bytes)
(factor out a (-1) changes the signs)

 

Examples:

1.

Reduce  u7obj1-09.gif (383 bytes)  to lowest terms

We do not have a common factor for the numerator and denominator so we have to factor them separately.

u7obj1-10.gif (448 bytes)

Notice that (5 - b) and (b - 5) differ by a negative 1 so we can rewrite either one.

u7obj1-11.gif (462 bytes)

We can now cancel the quantity (b - 5).

= -a/3
 


2.

Reduce  u7obj1-12.gif (685 bytes)  to lowest terms.

Notice that (2x - 1) and (1 - 2x) differ by a negative 1 so we can rewrite either one..

u7obj1-13.gif (729 bytes)

We can now cancel the quantity (1 - 2x).

u7obj1-14.gif (393 bytes)  or u7obj1-15.gif (267 bytes)


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