worksheet on algebraic fractions

8/12/2019 Worksheet on Algebraic Fractions

1/21

Worksheet on ALGEBRAIC FRACTIONS (I)

(Addition and Subtraction)

1 What Are Algebraic Fractions?

Algebraic fractions are fractions that include pronumerals as well as

numbers.

2 Adding and Subtracting Algebraic Fractions

There are two basic principles involved in the adding and subtracting of

algebraic fractions. These are:

(I) The lowest common denominator principle

(II) The equivalent fraction principle

These are exactly the same principles that are used in adding and

subtracting purely numerical fractions. So lets review these principles with

some purely numerical examples.

3 The Lowest Common Denominator rinci!le" #umerical $%am!les

Examples

!. "hat is the lowest common denominator (#$%) for the sum:&

'

!+

Solution:The #$% of and & is &* because that is the smallest number that

both and & go into.

+. "hat is the lowest common denominator (#$%) for the sum:!+

'

,

+

Solution:The #$% of , and !+ is 2'* because that is the smallest number that

both , and !+ go into.

. "hat is the lowest common denominator (#$%) for the sum:-

+

.

!+

Solution:The #$% of and - is 2(* because that is the smallest number that

both and - go into.

Gary Pocock 06/06/14 1


2/21

/otice that in the first two examples the lowestcommon denominator is not

obtained by multiplying the two denominators together. This is because0 in

both of these examples0 the denominatorshave a common factor.

/otice that in the third example the lowest common denominator is

obtained by multiplying the two denominators together. This is because the

denominators and - do nothave a common factor.

' The $)ui*alent Fraction rinci!le" Con*erting +nto Sim!lest Form

A single fraction can be written in an infinite number of different ways. 1orexample0

+

!can also be written as

.

+or

&

(or

,

.and so on. All of these

fractions are e2ual to one*half because in each case the numerator is half of

the denominator.

Sometimes we want to write a fraction using the smallest possible integers.

This is called writing a fraction in simplest form.

The method for converting a fraction into its simplest form has two steps.

Step 1:1ind the highest number that divides into both the numerator and thedenominator. (This is called the highest common factor.)

Step 2:%ivide both the numerator and the denominator by the number

found in step !.

Examples

!. "rite the fraction!3

'in simplest form.

Solution:The highest number that divides into both ' and !3 is ,. So:

+!

3!'

+

!

=

+. "rite the fraction+3

!+in simplest form.

Solution:The highest number that divides into both !+ and +3 is '. So:

'

3+

+!'

=

, The $)ui*alent Fraction rinci!le" Con*erting FromSim!lest Form



3/21

"hen adding and subtracting fractions it is often necessary to convert a

fractionfrom simplest form. (Section - explains why.)

Method: To convert a fraction from simplest form0 simply multiplythe

numerator and denominator by the same number.

Examples

!. "hat is e2uivalent form of the fraction(

+obtained when the numerator and

denominator are both multiplied by &

Solution:

!,

!+

+&

&

=

+. "rite the fraction'

.in an e2uivalent form with !' in the denominator.

Solution:To convert the denominator to !' we as4: "hat number must ' be

multiplied by to give !' Answer: . So0 to find the re2uired

e2uivalent form0 we must multiply both the denominator andthe

numerator by .

!'

!+

'

.

=

52uivalent fraction conversions may be summed up by the following rule:

Whatever you do to the numerator of a fraction i!e! divide it "y a num"er# or

multiply it "y a num"er$# you must do exactly the same thing to the

denominator!

("hy is this a valid rule of algebra 6ecause in applying this rule we are

effectively multiplying or dividing the fraction by !. And as we 4now0

multiplying or dividing something by ! does not change its value.)

& Adding and Subtracting #umerical Fractions" Case + - Same Denominators



4/21

7roblems involving the addition and subtraction of fractions can be divided

into two cases.

%ase &: %enominators are all the same.

%ase &&:%enominators are notall the same.

$ase I problems are very easy since fractions with the same denominators

are all of the same type (they are all thirds0 or allfifths#etc.). 1or these

problems0 all you have to do is add the numerators0 4eeping the

denominators unchanged.

Example

!. 1ind,

+

,

!+

Solution:

,

,

+

,

!=+

(In words: 8ne*eighth of a pi99a plus two*eighths of a pi99a e2uals three*

eighths of a pi99a.)

. Adding and Subtracting #umerical Fractions" Case ++ - Di//erent

Denominators

If the denominators are different the fractions cannot be added or subtracted

directly because they are not the same type. To fix this problem we first use

e2uivalent fractions to ma'ethem the same type.

The method involves two steps.

Step 1: 1ind the lowest common denominator (#$%) of the two fractions.

Step 2:$onvert each fraction to its e2uivalent form with the #$% as the

denominator.



5/21

Examples

!. 1ind&

'

!+

Solution:

* The lowest common denominator is &.

* The second fraction already has & in the denominator0 so no conversion is

necessary.

* To convert the first fraction0 multiply both numerator and denominator by +.

&

!

!&

-

&

'

&

+

&

'

!

&

'

!+

+

or=

+=

+=+

+. 1ind-

+

.

!+

Solution:

* The lowest common denominator is +,.

* To convert the first fraction to a denominator of +,0 multiply both

numerator and denominator by -.

* To convert the second fraction to a denominator of +,0 multiply both

numerator and denominator by .

+,

!'

+,

,

+,

-

-

+

.

!

-

+

.

!.

.

-

-

=

+=

+=+

& AddingAlgebraic Fractions With ronumerals +n The #umerators Alone



6/21

If pronumerals occur only in the numerators0 the method for adding fractions

is exactlythe same as that for numerical fractions.

Examples

!. Simplify&

+

+

xx

Solution:

* The lowest common denominator is &.

* The second fraction already has & in the denominator0 so no conversion is

necessary.

* To convert the first fraction0 multiply both numerator and denominator by +.

&

+

&

)+(+

&

+

&

+

&+

&+

+

+

=

+=

+=

+=

+

x

xx

xx

xxxx

+. Simplify-

!+

.

++

xx

Solution:

* The lowest common denominator is +,.

* To convert the first fraction to a denominator of +,0 multiply both

numerator and denominator by -.

* To convert the second fraction to a denominator of +,0 multiply both

numerator and denominator by .

+,

.!'

+,.,

+,-

-

!+

.-

!+

. .

.

-

-

+=

++=

++=

++

x

xx

xxxx

(6oth the ! andthe +x are multiplied by )

. SubtractingAlgebraic Fractions With ronumerals +n The #umerators Alone



7/21

In algebra0 the fraction line is treated as a grouping symbol. This means that

the fraction line can be used in place of "rac'ets.

;owever0 when two fractions are combined into one fraction by addition or

subtraction it is important to write in the "rac'ets. This is especially important

forsu"tractionproblems.

(orgetting to write in the "rac'ets when com"ining fractions that are "eing

su"tracted is one the most common errors in alge"ra)

The next two examples illustrate this point in the simple case where the

denominators of the two fractions are the same.

Examples

!. Simplify&

&

'

xx

Solution:

&

.

&

'

&

)('

&

&

'

+=

+=

=

x

xx

xxxx

(


8/21

numerator is multiplied by that number. In other words0 it is as ifthe

numerator has brac4ets around it.

Examples

!. Simplify&

!

. +

xx

Solution:

&

>

&

!,+

&)!(,+

&

!

&

,+

&

!

.

&

!

.+

+

=

=

+

=

+

=

+

=

+

x

xx

xx

xx

xxxx

+. Simplify-

.

.

+ xx

+

Solution:

+,

+!>

+,

!+!&!.-

+,

)!+!&(!.-

+,

!+!&

+,

!.-

-

.

.

+

-

.

.

+.

.

-

-

=

++=

+=

+=

+=

+

x

xx

xx

xx

xxxx

( Algebraic Fractions With ronumerals +n The Denominators - Case +" Same

Denominators



9/21

1ractions with pronumerals in the denominators can also be divided into

two cases.

%ase &: %enominators are all the same.

%ase &&:%enominators are notall the same.

#ets loo4 at some easy $ase I examples first. ?emember the method: whendenominators are the same0 @ust add the numerators. /ote that the

denominators must be exactly the same.

Examples

!. Simplifyxx

'++

Solution:

xxxx

-'+'+=

+=+

+. Simplifyx

x

x

x

'

!

'

. ++

Solution:

x

x

x

xx

x

x

x

x

'

+

'

!.

'

!

'

.

=

++=

++

. Simplify:+

!

+

+

x

x

x

x

Solution:

+!+

+

)!(

+

!

+

=

+=

+

x

x

x

xx

x

x

x

x

0 Algebraic Fractions With ronumerals +n The Denominators - Case ++"

Di//erent Denominators



10/21

If the denominators are different then the method of e2uivalent fractions must

be used to ma4e them the same.

?ecall from the purely numerical examples that when the numbers in the

denominators have no common factor0 the #$% is found by simply

multiplying the two denominators together. 5xactly the same principle appliesto algebraic examples.

It is important to emphasise that factors such as x0 x=!0 x=+0 +x=!0

x=-0 etc. have no common factors. Therefore0 when such factors occur in a

denominator0 we simply multiply them together to obtain the #$%.

Example!. Simplify

+

-

'

.

+

+

xx

Solution:

* The denominators have no common factors. Therefore #$% (x * ')(+x = )

* To convert the first fraction to a denominator of (x * ')(+x = )0 multiply both

the numerator and denominator by +x = .

* To convert the second fraction to a denominator of (x * ')(+x = )0 multiply

both the numerator and the denominator by x * '.

)+)('(

+!'

)+)('(

'-!+,

)+)('(

)'(-)+(.

)+)('(

)'(-

)+)('(

)+(.

)+(

-

)'(

.

+

-

'

.)'(

)'(

)+(

)+(

+

=

+

++=

+

++=

+

+

+

+=

++

=

++

+

+

xx

x

xx

xx

xx

xx

xx

x

xx

x

xxxxx

x

x

x

"hat if the denominators have some numericalfactors in addition the

algebraic factors In that case we simply deal with the two 4inds of factors

(numerical and algebraic) separately. That is0 we do the following:

* Step 1: 1ind the #$% of the numerical factors alone.

* Step 2:1ind the #$% of the algebraic factors alone.

* Step *:Bultiply the above two #$%s to obtain the overall #$%.



11/21

Example

!. Simplify)'(&

!

.

+

+

xx

Solution:

* The numerical factors in the denominators are and &. Therefore the

numerical #$% &.

* The algebraic factors in the denominators are x and x = '. Therefore the

alge"raic#$% x(x = ')

* ;ence the overall#$% &x(x = ')

* To convert the first fraction to a denominator of &x(x = ')0 multiply both the

numerator and denominator by +(x=').* To convert the second fraction to a denominator of &x(x = ')0 multiply both

the numerator and the denominator by x.

)'(&

.3>

)'(&.3,

)'(&)'(&

)'(,

)'(&

!

.

)'(&

!

.)'(+

)'(+

+

+=

+

++=

+

+

+

+=

+

+=

+

+

+

+

xx

x

xx

xx

xx

x

xx

x

xxxx x

x

x

x

1 Algebraic Fractions With Powersn The Factors

$onsider the sum +'

.

'

+ . "hat is the #$% $learly it is '+since that is

the lowest number that ' and '+both divide into.



12/21

"riting '+ rather than +' in the last example was deliberate. It

emphasises the fact that the two numbers have a common factor

(namely0 ').

/ow consider the sum +)!+(

.

!+

+

+

+ xx . "hat is the #$% in this case It

is (+x=!)+since that is thesmallestfactor that both +x = ! and (+x = !)+

divide into.

This example shows that when the same factor occurs more than once0 "ut

with a different powers0 you do notmultiply the factors together to get the

#$%. ("hy not 6ecause +x = ! and (+x = !)+have a common factor*

namely0 +x = !.) ?ather0 you simply ta4e the factor with the highest power

attached to it.

Examples

!. Simplify

+.

xx

Solution:

* x and x have a common factor of x. Therefore the #$% x* i.e. the

common factor raised to the highestpower.

* To convert the first fraction to a denominator of x0 multiply both the

numerator and denominator by x

+

.* /o conversion is necessary for the second fraction.

+

+

+

+.

+.+.+

+

x

x

xx

x

xxxx x

x

=

=

=

Examples continued$

+. Simplify . )!(

+

)!(

'

x

x

x

Solution:



13/21

* 6oth denominators involve the factor x * !. Therefore the #$% (x * !)0

i.e. the common factor raised to the highestpower.

* To convert the first fraction to a denominator of (x * !) multiply both the

numerator and denominator by (x * !).

* /o conversion is necessary for the second fraction.

.

.

..

.)!(

)!(

.

)!(

'!

)!(

+'!'

)!(

+

)!(

)!('

)!(

+

)!(

'

)!(

+

)!(

'

=

=

=

=

x

x

x

xx

x

x

x

x

x

x

xx

x

xx

x

11 uestions

!. Simplify the following fractions. (7ronumerals in numerator only.)

(a).

+

'

! xx +

(d).

+

!3

)!(- +

xx



14/21

(b)>

+.

>

xx

(e)-

+

.+ xx+

+

(c)!+

!

,

-+ xx +

+

(f).

+

+

! ++

+ xxx

+. Simplify the following fractions. (7ronumerals in both numerator and

denominator.)

(a)

+!

.

++

x

x

x

x

(d),

!'

.

+ +

+ x

x

x



15/21

(b)+

!

'+

&

+ xx

(e)'

+

,!

x

xx

x

+

(c)!

-

-

.

+

xx

(f) +)!.(

)+('

!.

>

+

x

x

x

+. (continued)

(g)xx >

+

).(&

'

+(h) +)(,

'

)(.

!

+

x

x

x



16/21

. Simplify the following fractions. (;arder 2uestions)

(a)x

x

x .

+!

+!

+

(b))()'(

!

))('(>

+++

+

+ xxxx

(c)!+

)('+

. +

+

+

+

+

+ xx

x

x



17/21



18/21

12 Answers

!. (a)

+3

&-

+3

'!3.!+

+3

'!3

+3

.!+

.

+

'

!

.

+

'

!'

'

.

.

+=

+=

+

=

+

=

+

x

xx

xx

xxxx

(d)

+3

+.

+3

!3!'!.!.

+3

)+(')!(!.

+3

)+('

+3

)!(!.

.

+

!3

)!(-

.

+

!3

)!(-'

'

+

+

=

=

+=

+

=

+

=

+

x

xx

xx

xx

xxxx

(b)

>

+

>

+.

>

)+.(

>

+.

>

x

xx

xxxx

=

+=

=

(e)

!.+,&-

!.

&

!.

+,-

-

+

.

-

+

.

+

+

+

+

-

-++

++=

++

=

++

=++

xx

xx

xxxx

(c)

+.

+.

+.

+++!&

+.

++

+.

+!&

!+

!

,

-+

!+

!

,

-++

+

+=

++=

+

+=

+

+=

+

+

x

xx

xx

xxxx

(f)

!+

+'

!+

>),.(&&!+

>

!+

,.

!+

&&

.

+

+

!

.

+

+

!

.

.

&

&

+=

+++=

++

+=

++

+=

++

+

x

xxx

xxx

xxx

xxx

+. Simplify the following fractions. (7ronumerals in both numerator and

denominator.)



19/21

(a)

!&

+!.

+!

.

+=

++=

++

x

x

x

xx

x

x

x

x

(d)

x

xx

x

xxx

x

xx

x

x

x

x

xx

x

x

x

x

,

.'

,

'.+

,

)!'(

,

)+(+

,

!'

.

+

,

!'

.

+

+

+

+

+

++=

++

=

+

+=

+

+=

+

+

(b)

)+)('+(

!-.

)+)('+(

'+!+&

)+)('+(

)'+()+(&

)'+)(+(

'+

)+)('+(

)+(&

+

!

'+

&

+

!

'+

&

)'+(

)'+(

)+(

)+(

+

=

+

=

+

+=

+

+

+

=

+

=

+

+

+

xx

x

xx

xx

xx

xx

xx

x

xx

x

xx

xx

x

x

x

x

(e)

'

+

'

+

'

+

'

+

'

+

'

+

'

+

).-(+),!.(

,!

,!

,!,!+

+

x

x

x

x

x

xx

x

xx

x

x

x

xx

xx

xx

x x

x

=

=

+=

+=

+=

+

(c)

)!)(-(

'!>

)!)(-(

.>-.!+

)!)(-(

)-(-)!(.

)-)(!(

)-(-

)!)(-(

)!(.

!

-

-

.

!

-

-

.

)-(

)-(

)!(

)!(

=

+=

+=

+

=

+

=

+

xx

x

xx

xx

xx

xx

xx

x

xx

x

xx

xx

x

x

x

x

(f)

++

+

+

+

+)!.(

)!.(

+

)!.(

)!+!(+

)!.(

+.+&

)!.(

!'!3>&

)!.(

!'!3>&

)!.(

)+(')!.(>

)!.(

)+('

!.

>

)!.(

)+('

!.

>

=

=

=

=

+=

+

=

+

x

x

x

x

x

xx

x

xx

x

xx

x

x

x

x

x

x

x

x



20/21

(g)

).(!,

!&!!

).(!,

!&.!'

).(!,

).(.!'

).(!,

).(.

).(!,

!'

>

+

).(&

'

>

+

).(&

'

).(+

).(+

+

=

+

=

+

+=

+

+

+

=

+

=

+

+

+

xx

x

xx

xx

xx

xx

xx

x

xx

x

xx

xx

x

x

x

x

(h)

+

+

+

+)(+

)(+

+

)(,

&-

)(,

'&+

)(,

')(+

)(,

'

)(.

!

)(,

'

)(.

!

+=

+=

+

=

+

x

x

x

xx

x

xx

x

x

x

x

x

x

x

x

. Simplify the following fractions. (;arder 2uestions)

(a)

xx

x

xx

xx

xx

x

xx

x

x

x

xx

x

x x

x

x

x

+!.!!3

+!.

)+!(!+

+!.

)+!(

+!.

!+

.

+!

+!

.

+!

+!

+

+!

+!

.

.

+=

+=

+

=

+

=

+



21/21

(b)

++

++

++

++++

)(+

)(

)'(+

)'(

++

)()'(>

!>

)()'(>

>!3+

)()'(>)()'(+

)()'(>

)(

)()'(>

)'(+

)()'(

!

))('(>

+

)()'(

!

))('(>

+

+

+=

+

++=

+

+=

+

+

+=

+

+

=

+

+

+

+

xx

x

xx

xx

xx

xx

xx

x

xx

x

xxxxxxxxx

x

x

x

(c)

)!+()(

!3.&!'

)!+()(

)>&()',.()'+(.

)!+()(

)()!+)('+()!+)((.

)()(!+(

)(

)!+()(

)!+)('+(

)!+()(

)!+)((.

!+

)(

'+

.

!+

)(

'+

.

+

+

+

+++

+

+

+

+

++

)(

)(

)!+(+

)!+(

)!+)((

)!+)((

+

+

+

+

++=

+

++++++=

+

+++++=

+

++

+

++

+

+=

+

+

++

+

=

+

+

++

+

+

+

+

+

xx

xx

xx

xxxxxx

xx

xxxxx

xx

x

xx

xx

xx

xx

xx

x

x

xx

x

x

x

x

x

x

xx

xx

worksheet on algebraic fractions

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