Worksheet on Algebraic Fractions (multiplication and division)

8/21/2019 Worksheet on Algebraic Fractions (multiplication and division)

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Worksheet on ALGEBRAIC FRACTIONS (II)

(Multipliction n! "i#ision)

1 What Are Algebraic Fractions?

Algebraic fractions are fractions that include pronumerals (i.e. letters) as

well as numbers.

2 Multiplying and Dividing Algebraic Fractions

The rules used in multiplying and dividing algebraic fractions are exactly

the same as those used for purely numerical fractions. So lets review these

rules with some purely numerical examples.

Well begin with multiplication. ur goal is to find a rulethat will allow us

to multiply fractions.

3 O! "eans "ultiply!#

!athematical problems begin with words" ne of the #obs of amathematician is to translatethe words into mathematical symbols.

$n %nglish the word &of is often used to indicate multiplication.

' When we as what is * lots of+,- we mean &what is * multipliedby +,-.

' When we as what is a uarter of /,- we mean what is 0 multiplied

by /,-

$ Multiplying a raction and a %hole nu"ber

$n maths it is nearly always a good strategy to start with the simplest case.

So lets begin by multiplying a fraction and a whole number. We will then

move on to multiplying two fractions.

We will first wor out the answers to some simple uestions by using

&commonsense 1 that is2 by using our nowledge of what the uestions

mean. $n this way2 we hope to discover apattern.

Gary Pocock 12/06/14 1


2/26

3emember that a whole number can be thought of as a fraction with a

denominator of 4.4

55= 2

4

**= 2

4

66= 2 etc.

Examples

4. What is a half of 7, (r8 What is 7 lots of a half,)

Solution

Without any calculations2 we can use our nowledge of what the uestion

meansto wor out the answer.

$nterpretation $ (eual parts)

&What is a half of 7, means&$f 7 is divided into two eual parts2 what is the

si9e of each part, :learly2 the answer is *.

$nterpretation $$( repeated addition)&What is 7 lots of a half, means&What is the total if 7 lots of one'half added

together, Again2 it is clear that the answer is *.

;ets state this conclusion in words8 one-half of 6 equals 3.

.

$n words8 one-half of 7 equals 3 .

$n symbols85

4*=

5

4=

& Finding a pattern

At first glance2 the above multiplications dont seem to display much of a

pattern. ?ut if we re'write the answers as improperfractions a pattern

emerges.

*5

7

4

7

5

4==

5

4*

5

=

4

=

5

4==



3/26

These simple examplessuggesta rule for multiplying fractions.

'ule 1( 'ule or "ultiplying t%o ractions

)o "ultiply t%o ractions* "ultiply the nu"erators together +to obtainthe nu"erator o the ans%er,* and "ultiply the deno"inators together

+to obtain the deno"inator o the ans%er,-

.n sy"bols(bd

ac

d

c

b

a=

/ )esting 'ule 1

We proposed this rule on the basis of only two examples. We need to test

the rule by seeing if it wors for a wider range of cases.

3ule 4 applies to both proper and improper fractions. ?ut it cannot be

applied to mixed numbers. Therefore2 it is important to remember8

When using 'ule 1al%ays convert "i0ed nu"bers into

i"proper ractions-

Examples

4. What is > of 0 ,

Solution

:ommonsense reasoning tells us that the answer is 4@/. (raw a number line if

you need to convince yourself.)

oes our rule give the correct answer,

/4

6544

64

54 =

= Bes"

5. What is*

4of

5

44 ,

Solution

Again2 commonsense reasoning tells us that the answer is > . (raw a number

line and divide 4 > into three eual parts.)

oes our rule give the correct answer, (3emember we must convert 4 > into



4/26

an improper fraction.)

5

4

7

*

5*

*4

5

*

*

4

5

44

*

4*

*

==

==

Bes"

*. What is +

*

of 5

4

,

Solution:ommonsense reasoning tells us one'fifth of5

4euals

4C

4. Dence

three!fifths of5

4must eual three lots of

4C

4' that is2

4C

*.


4C

*

5+

4*

5

4

+

*=

= Bes"

$n fact2 3ule 4 al"a#sgives the correct answer for multiplying fractions 1

that is2 it always gives the same answer as that obtained by commonsense

reasoning.

This is very useful to now because it means that we no longer have to

always resort to &commonsense reasoning (which can often tae uite a

long time to wor out) when multiplying fractions. $nstead2 all we have to

do is apply the rule.

0tending 'ule 1 to the "ultiplication o three +or "ore, ractions

3ule 4 is easily extended. Dere is the rule for multiplying threefractions

together. (The extension to four or more fractions follows the same pattern.)

'ule 1( 'ule or "ultiplying three ractions

)o "ultiply three ractions* "ultiply the nu"erators together +to

obtain the nu"erator o the ans%er,* and "ultiply the deno"inators

together +to obtain the deno"inator o the ans%er,-

.n sy"bols(bdf

ace

f

e

d

c

b

a=



5/26

Examples

4. :alculate=

4

+

*

5

4

Solution

=C

*

=+5

4*4

=

4

+

*

5

4=

=

5. :alculate+

6

/

*

E

/

Solution4+

6

*7C

E7

+/E

6*/

+

6

/

*

E

/56

56

==

=

4 5ancelling do%n! ractions

3ecall from $or%sheet on &lgebraic 'ractions ( that a single fraction can

be written in an infinite number of different ways. %.g. .....7*

65

54 ===

These are called equi)alent fractions.

3ecall also that we often want to write a fraction using the smallest possible

integers. This is called "riting a fraction in simplest form.

To convert a fraction into its simplest form2 we simply divide both the

numerator and the denominator by the highest common factor. %.g.

+

*

4+

E

4+

E*

*

==

ften the phrase &cancelling do"n (or #ust &cancelling) is used to describe

this process of converting a fraction to its simplest form.

$ will try to avoid this phrase because it can be misleading. $t tends to mae

people thin that there is some special mathematical operation called

&cancelling2 whereas in reality it is #ust a process involving ordinary

division.



6/26

?ut the phrase is widely used (and2 $ admit2 it can be useful")2 so you need

to be aware of it. ?ut remember8 cancelling down! 6ust "eans converting

to simplest form!-

7 8sing cancellation! to si"pliy the process o "ultiplying ractions

;oo again at the last example of F=.


7/26

5. :alculate6C

44

**

57

4*

4C

Solution

7

4

45

5

*46

454

***4C6

4475C4

6C**4*

44574C

6C

44

**

57

4*

4C*46

454

===

=

=

19 Multiplying Algebraic Fractions With :ronu"erals

The method for multiplying fractions with pronumerals is exactl#the same as

that for numerical fractions.

Examples

4. Simplify+

5

*

xx

Solution

4+

)5(

+*

)5(

+

5

*

=

=

xxxxxx

5. Simplify45

=

6 +

x

x

Solution

)45(6

=

45

=

6 +=

+

x

x

x

x

The multiplication of algebraic fractions gets a bit more complicated (but not

that much") when common factors are involved. ?efore doing some examples2

lets revise the rules for multiplying and dividing with powers 1 i.e. the index

rules.


"here is no need to e'#and thebrackets. $actorised form is nearlyalways best.


8/26

11Revision: The Index Rules (aka Power Rules or Exponent Rules)

;For "ore detail see "y Wor


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Examples

4. Kse index rules to simplify (where possible) the following expressions.

(a) 55H 5+

(b) =6H =*H =5

(c) +5H 5*

(d) x7H x6

(e) x*H y+

Solutions

(a) 55H 5+I 5=

(b) =6H =*H =5I =E

(c) +5H 5* cannot be simplified +base numbers aren,t the same

(d) x7H x6I x4C

(e) x*H y+I x*y+ (cannot be simplified further as the bases aren,t the same

When applying the index rule for division2 the resulting power will sometimes

be 9ero2 or a negative number. There are two more index rules that tell us what

such powers mean.

)%o More .nde0 'ules

'ule .3 +>ero po%er rule,( A nu"ber raised to po%er o >ero is eual to 1-

)hat is* a9@ 1- +0ception( 99is undeined-,

'ule .$ +negative po%er rule,( an@ 1Ban +a C 9,



10/26

Examples

5. Kse index rules to simplify (where possible) the following expressions.

(a) 5=56

(b) +6H +*+7

(c) 757+

(d) x5H x6x7

(e) x*y+

Solutions

(a) 5=56I 5*

(b) +6H +*+7I +4I +

(c) 757+I 7'*I 4@7*

(d) x5H x6x7I xCI 4

(e) x*y+I x*@y+ (cant be simplified further as the bases arent the same)

12 A Mathe"atical 5onvention For Writing Fractions With :ronu"erals

$n the next section we will apply the index rules to the multiplication of

fractions. ?ut first its important to explain a certain mathematical convention

for writing fractions with pronumerals.

A Mathe"atical 5onvention

When %riting a raction such as t%othirds o 0!* the 0! "ay be placed

eitherin the nu"erator

*

5x* oronthe side

x*

5- )hese are 6ust t%o

dierent %ays o e0pressing the sa"e thing-



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This convention can cause confusion when the number in the numerator is 4.

Gor example2 the convention implies that

6

4

6

4 xx =

Dowever2 when the coefficient ofxis 4 we normally #ust write &x rather than

&4x. Thus we usually write

66

4 xx=

Examples

4. Where possible2 write the following fractions in a different way using the

above convention.

(a) x=

4

(b)+

*x

(c)E

x

(d)x

+

Solutions

(a)==

4 xx=

(b) xx

+

*

+

* =

(c) xx

E

4

E=

(d)x

+ (Cannot be written

differently as ' is in the

denominator.)

13 Applying .nde0 'ules to the Multiplication o Fractions



12/26

Examples

4. Simplify4+

+5 *xxx

Solution

xorx

x

xx

xx *

5

*

5

4+

4C

4+

+55

**

==

5. Simplify.

x#

x

#.

#

x 4C

+

5

+ *

5

5

Solution

.xorx

.

.#x

.#x

.

x#

x

#.

#

x 45*

555

*

5

5 +

6

+

6

5+

5C4C

+

5

+

==

*. Simplify8 55

*

7

5

)5(

6

5

*

x

x

x

x

x

x

Solution

5

5

*5

6

55

*

)5(

5

)5(7

)5(45

7

5

)5(

6

5

*

=

=

x

x

xx

xx

x

x

x

x

x

x

1$ )he :artial 5ancellation! rror



13/26

ne very common error made when woring with fractions is what $ call the

&partial cancellation error. Dere is an example.

x#

#x5

5=

+ TD$S $S W3


14/26

?ut now suppose we tried to simplify the expression using the &partial

cancellation procedure. We would get8

65

56=

+,,,

$t is clear that the procedure gives the wrong answer2 and so cannot be

valid.

/uestion 38 Why do people mae this error,

$n algebra2 there are rules for addition and rules for multiplication. The two

sets of rules are uite similar2 but not identical. !any mistaes2 including this

one2 are made through getting the rules for addition and multiplication

confused.

:onsider the following8

x#

#x5

5= TD$S $S :33%:T"

$n this example there are only two terms2 and &y is common to bothterms. So

dividing by &y gives &full cancellation rather than &partial cancellation.

Algebra always reuires &fullcancellation.

Examples

4. Simplify (if possible)8+5

+6

x

x

Solution

There is no common factor. The expression cannot be simplified any further.

5. Simplify (if possible)85

/6 x

SolutionThere is a common factor of 5. The expression can be simplified by dividing

every term by 5.

654

65

5

/6

5

/65

55

=

=

=

xxxx



15/26

Examples

*. Simplify (if possible)8#

x

x

x# 4*

46

6 +

+

Solution

46

)4*(6

)46(

)4*(64*

46

6

++=++=++ xxx

x#

x#x

#

x

x

x#

1& Division .nvolving :urely u"erical Fractions


16/26

This means that the following two uestions are equi)alent(i.e. they

are #ust two different ways of asing the same thing).

ivision uestion8 What does 4C divided by + eual,-

!ultiplication uestion8 What number2 when multiplied by +2 gives 4C,-

o .nterpretation .. +eual parts,

Deres another way of thining about 4C +8

$f 4C is divided into + equal parts2 what is the si9e of each part,-

o .nterpretation ... +repeated subtraction,

Ginally2 4C

+ can viewed as asing the following8

Dow many lots of + can be subtracted from 4C,-

We will now loo at some examples of division involving fractions. We

will wor out the answer using one of more of the above interpretations.

%ventually2 we hope to find a pattern"

As always2 we start with the simplest examples 1 i.e. cases involving

fractions divided by whole numbers.



17/26

Examples

4. What is a half divided by *,

Solution

$nterpretation $8The euivalent multiplication uestion is8

What number2 when multiplied by *2 gives one'half.-

$nterpretation $$8

$f one'half is divided into * eual parts2 what is the si9e of each part,-

($magine half a cae divided into * eually si9ed pieces.)

$nterpretation $$$8

(oesnt mae sense in this case because the dividend (>) is smaller than the

divisor (*).)

n the basis of interpretations $ and $$ it is clear that the answer is one'sixth.

$n words8 one-half di)ided b# three equals one-sixth.

$n symbols87

4*

5

4=

5. What is two'thirds divided by 5,

Solution

$nterpretation $8The euivalent multiplication uestion is8

What number2 when multiplied by 52 gives two'thirds.-

$nterpretation $$8

$f two'thirds is divided into 5 eual parts2 what is the si9e of each part,-

($magine two'thirds a cae divided into 5 eually si9ed pieces.)

$nterpretation $$$8

(oesnt mae sense in this case because the dividend (>) is smaller than the

divisor (*).)

n the basis of interpretations $ and $$ it is clear that the answer is one'third.

$n words8 t"o-thirds di)ided b# t"o equals one-third.

$n symbols8*

45

*

5 =


18/26

Examples

*. What is * divided by 4@5,

Solution

$nterpretation $8

The euivalent multiplication uestion is8

What number2 when multiplied by >2 gives *.-

$nterpretation $$8

(oesnt mae sense in this case because the divisor is a fraction.)

$nterpretation $$$8

Dow many lots of one'half can be subtracted from *,

n the basis of interpretations $ and $$$ it is clear that the answer is 7.

$n words8 1hree di)ided b# one-half equals six.

$n symbols8 75

4* =

6. What is 4@* divided by >,

Solution

$nterpretation $8


What number2 when multiplied by >2 gives 4@*,-

$nterpretation $$8


$nterpretation $$$8

(oesnt mae sense because the dividend is smaller than the divisor.)

n the basis of interpretation $ it is clear that the answer is 5@*.(o the multiplication yourself")

$n words8 2ne-third di)ided b# one-half equals t"o-thirds.

$n symbols8*

5

5

4

*

4=

1/ Finding a pattern



19/26

:an you see a pattern, ;ets write out the above divisions again2 but this

time well write the whole numbers as fractions with a denominator of 4. $f

you focus on the diagonalsyou should notice a pattern.

7

4

4

*

5

4

= 54

4

*

7

4

*

4

7

5

4

5

*

5

==

74

7

5

4

4

* ==

*

5

5

4

*

4=

These examples suggest that weve found a rule for dividing fractions. All

one has to do is &multiply diagonally.

Dowever2 most maths boos express the rule in a slightly different way.

($ts still the same ruleN its #ust put differently.) Deres how the rule is

usually expressed.

'ule 2( 'ule or dividing ractions +invert and "ultiply! rule,

)o divide one raction by another raction do the ollo%ing(

Etep 1( .nvert the divisor +that is* turn the 2ndraction upside do%n,

Etep 2( multiply#

.n sy"bols(bc

ad

c

d

b

a

d

c

b

a==

1 )esting 'ule 2

We proposed this rule on the basis of only four examples. We need to testthe rule by seeing if it wors for a wider range of cases.



20/26

3ule 5 applies to both proper and improper fractions. ?ut it cannot be

applied to mixed numbers. Therefore2 it is important to remember8

When using 'ule 2al%ays convert "i0ed nu"bers into

i"proper ractions-

Examples

4. What is 5@+ divided by 4@4C ,

Solution

$nterpretation $8

The euivalent multiplication uestion is8What number2 when multiplied by 4@4C2 gives 5@+,-

$nterpretation $$8


$nterpretation $$$8

Dow many lots of 4@4C can be subtracted from 5@+,

$f you thin about these two uestions carefully you will see that the answer

is 6. (Dint8 5@+ I 6@4C)


6+

5C

4+

4C5

4

4C

+

5

4C

4

+

5==

== Bes"

5. What is*

4divided

5

44 ,

Solution

$nterpretation $8


What number2 when multiplied by5

44 2 gives

*

4,-

(The other two interpretations dont mae sense.)

This is a harder uestion to answer. ?ut after some trial and error you should

find that the answer isE

5.

oes our rule give the correct answer, (3emember we must convert 4 > into

an improper fraction.)

E5

**54

*5

*4

5*

*4

544

*4 ==== Bes"



21/26

$n fact2 3ule 5 al"a#sgives the correct answer for dividing fractions 1 that

is2 it always gives the same answer as that obtained by commonsensereasoning.

This is very useful to now because it means that we no longer have to

always resort to &commonsense reasoning (which can often tae uite a

long time to wor out) when dividing fractions. $nstead2 all we have to do is

apply the rule.

14 Dividing Algebraic Fractions With :ronu"erals

The method for dividing fractions with pronumerals is exactl#the same as that

for numerical fractions.

As was the case with multiplying fractions2 it is important to remember the

index rules and factorisation rules.

Examples

4. Simplify+

5

*

xx

Solution

)5(*

+

5

+

*+

5

* =

=

x

x

x

xxx

5. Simplify 45

=

6 + xx

Solution

5/

)45(

=

45

645

=

6

+=

+=

+

xxxx

x

x

17 Applying .nde0 'ules to the Division o Fractions


"here is no need to e'#and thebrackets. $actorised form is nearlyalways best.


22/26

Examples

4. Simplify4++

5 *x

x

Solution

6

66*

*

77

+

*C4+

+

5

4++

5

=== xorxxxx

x

x

5. Simplify

.

x#

.

#x

x#

. 6

+

/

+ *

5

5

5

Solution

(3emember8 always do the operation inside the bracets first")

5

55

5

55

5

*

5

5

5

*

5

5

5

*

5

5

5

5+

5

5+

5

+

5

+

5C

/

+

6+

/

+

6

+

/

+

#

.x#

x.

.

x

x#

.

x#.

#.x

x#

.

x#

.

.

#x

x#

.

.

x#

.

#x

x#

.

=

=

=

=

=

29 )he Elash! Ey"bol or division

Sometimes you will find the slash symbol (@)2 or a fraction line2 4being

used to indicate division. The safest way to deal with this is to first

convert the slash or fraction line to the familiar division symbol ()

and than apply 3ule 5.4Deres a bit of trivia for you. The correct term for the fraction line is the )inculum.



23/26

$f there is more than one fraction line2 it is the longerline that indicates

division.

Examples

4. %valuate

*+

5

Solution

4+

5

*

4

+

5

4

*

+

5

*

+

5

===

5. %valuate

*

57

Solution

E4

E

54

*7

5

*

4

7

*

5

4

7

*

5

74

*

==

===

*. %valuate

5

44

6

*

Solution

5

4

45

7

*

5

6

*

5

*

6

*

5

446

*

5

44

6

*

=====

21 uestions

4. Simplify the following fractions.

(a) 6

*4=

7

*

5



24/26

(b)

7

4

4C

E

+

*

(c)

=

**

*

45

(d)/

+

*

+

(e) 5

5

/

*

6 #

x.

#

x

(f) xx

+

5

*

(g)

5=

6#

(h)

#

x+

+

(i)x#

#x6

54

(#)

x

x

*6

4

()6+

55 5x

xx

x

+

(l)a

bccab

*5

55

(m)

/

4

5

45##

(n)*

+75

+

+xx

x

(o)ba

ba

55

4

*

55

+

22 Ans%ers

1-

(a) 45

5

445

454

=*6

=75

6=*

=75

6

=

=

7

*

5

6

*4

=

7

*

5445

454

==

=

=

==



25/26

(b) 6*

45

+E

C7*

E

7C

+

*

7C

E

+

*

74C

4E

+

*

7

4

4C

E

+

*4*

454

==

===

=

(c) /*=

65=

=

56

*

=

=

**

*

45

44

/4

=

==

(d)*

55

*

/

*+

/+

+

/

*

+

/

+

*

+==

==

(e).

x#

x#.

#x

x.

#

#

x

#

x.

#

x

*

5

45

/

*

/

6/

*

6

5555

5

5

===

(f) 4+

5

+

5

*+

5

*

5xxx

x

x

==

(g)

=

5

46

6

5

4

=

65

=

6

5=

6

####

#

====

(h)x

#

x

##

x#x

#

x ====+

+

+

+++

+

+

(i)5555 5

4

6

5

6

456

56

54

6

54

#x#xx#x#x#

x#x#

#xx##x ====

=

(#)

45

4

45*

4

6*

6*6

4

====

x

x

x

xx

x

x

x

())+(5

)5(

)+(6

)5(5

6+

55 55

x

xx

xx

xxx

xx

x

+

=+

=

+

(l)c

ba

bc

cba

bc

acab

a

bccab

5

5

55

5

555 77*

4

5

*5 ===

(m) ##

#

##

#######6

4

6

5

/

4

/

5

4

/

4

5

4

/

44

5

4/

4

5

4 55

555 =====

=



26/26

(n)xxx

x

xx

x

xx

x 4C

)*(

)*(4C

)*(

+)*(5

*

+75=

++

=+

+=

+

+

(o)7)(7

))((

)(5*

))((

55

4

*

55 ba

ba

baba

ba

baba

ba

ba =

+

+=

+

+=

+

Documents

Worksheet on Algebraic Fractions (multiplication and division)