Cha pter 2

Fractions

Worksheet

As a result of this lesson thestudent will be able to understand

how fractions work by applyingmathematical operations to

fractions and using the associatedvocabulary.

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WorksheetMath091Chapter2

1. A is anynumberthat can be writtenin the forma , where b is notb

O.Thenumber a is calledthe , and the number b is calledthe

2. are fiactiODSwith different terms but have the same size.

3. In a if a < b then the ftactionisb

, if a ~ b then the fraction is

4. The sum of a whole number and a ftaction is called

5. Any two nwnberswhoseproductis I are called

6. Every whole number can be written as a ftaction with a denominator of

7. What does "reduce to lowest terms" means?

8. Beforewe_change them into

mixed numbers, we mustorftactions

9. What operation does a ftaction indicate?

]o. When we "cancel" in multiplication of 1iactiODSwe are really tolowest teDns. .

II. Whatoperationswith ftactionsIequirea LCD? and

12. To find the reciprocal of a mixed number we must first change the mixed numberto

13. A ftaction that contains a ftaction in its numerator or denominator or both iscalled

a14. If a representsany number,then - =]

anda-= (a is not 0)a

15. A number with 1 and itself as factors is a number.

16. The formula for the area of a triangle is,triangle requires the of all sides.

. The perimeter of a

property can be used to simplify 6( ! + 2 ).2 317. The

18. Find the missing number:651 %-=-1674 18

19. Arrange the following in order ftom largest to smallest.

251733'12' 4'24'8

20. How are multiplication and division of ftactions related?

21. When you are asked to take ~of12 is this an addition or multiplication problem?7

22. When adding ftactions with the same denominator, we always addtheir but we never add their

23. To subtract 1 2fT ftom 10, we must rewrite ten as what mixednumber?

Info Sheet

Instructions for applyingmathematical operations tofractions and applying the

associated vocabulary.

C211

Subject Name

Multiply and DividingFractions

Remember: Integers (-3,3,O,1,-2,etc) are tractions. Just put it over 1.3 129165

Ex) - 3=-- Ex.)129165=1 1

Remember: CAN NOT MULTIPLYIDIVIDE USING A MIXED NUMBER.(Whole # with a traction. Ex. 4.!. ). 5

without changing it to an IMPROPER FRACTION (Top # is bigger than or the same as the bottom. Ex. 11 )3Chanp Mixed Number to Improper Fractions

Improper fi'action numerator = Denominator times the whole number plus the numeratorImproper fi'actiondenominator = Denominator ftom the mixed number

4 1 . _-= 21Ex.) Change - to anunproperUMiUon Answer:-5 . 51

4...:L Numerator=5 x 4 + 1=21.X5

215

Denominator = S

Change Im,properFractions to Mixed NumberDivide. The number of even times it goes into it is the whole number. Numerator=remainder Same denominator

Ex.) Change 11 to a mixednumber.3

5

3Ji715

02

Answer:5~3

Reci,procal:Just a fimcywordfor flippinsthe top and bottomof a traction. The signstaysthe same.

Example: Find the reciprocal of S. S =S which has a reciprocal of .!.1 S

Example: Ymdthe reciprocal of - S. The reciprocal of _ 5 is _ 3335

SimpIiO'VI.Reduce: Always be mindful ofwbat the directions say with tractions.Reduce: Cancel what you can cancel. It doesn't matter if the ftaction is proper or not.Simplify:Cancel what you can cancel (reduce) then change improper tractions to mixed numbers.If you are in doubt. always simplify..

The easiest operations for tractions are multiplication and division.Multiplication is straight forward (top times top and bottom times bottom). For division. flip and foUowmultiplication.

MUL11PLY:Change integers &.mixed tractions to improper tractions.Multiply top times top and bottom times bottom.Simplifyand/or reduce ifnecessuy.

Page1on

Subject Name_Multiplyand DividingFractions

IAnswer: -

33 4

Ex.) -8-4 9

3 4 384-8-= -4 9 48912 1-=-36 3

5 3Ex.)-- 82-

6 43 11

Sidework: 2- =-4 4

5 11--8-6 45811--684

55-- Reducefinalanswer.24

Top times top &.bottom times bottom

Multiply &. reduce.

55 7 .Answer: -- or -2- (depends on the directions)

24 24

4 times 2 is 8. 8 plus 3 is 11. Denominator is the same, 4.

Replace the mixed number with the improper ftaction. .

Top tbnes.top &.bottom tbnes bottom. (-) x (+) =(-) So our answer is goins to be negative.

7- 2 - Simplifyfinal answer.24

DIVIDE:Change integers &.mixed ftactions to improper ftactions.Take the reciprocal (flip) the second # and change + to x.Now multiply.NOm: Alwaysflipthe 2- number

3 4Ex.)-+-4 9

3 9-8-4 4389 27-=-484 16

27ReducedAnswer:-

16

Flip the ,... and change multiply to divide.

S' lified Answer: 1Y-unp 16

Multiply top tbnes top and bottom times bottom.

To simplifyto a mixed number: 16 goes into 27, 1 tbne. Remainder is 7. 1Y-16

SimplifiedAnswer:2.!.'4Ex.)-2+(- :)

- : 8(- :)28918818 9 1289 189 9-=- Multi~&.~. ~ ---=--=-8 4 181. 184 4

To simplifythe answer to a mixed number: 4 goes into 9, 2 times. Remainder is 1. 2.!.4

9ReducedAnswer: -

4

Change -2 to an improper ftaction. Flip the 2- number and ~hange multiply to divide.

Multiply top times top &. bottom times bottom. Remember (-) x (- ) =(+ )

Cancel then multiply.

NOTICE: In both multiplication&.division you are multiplying. So the operation that I am goins to change is the one that Ichange one of the numbers.

PaF2of2

Subject Name

LeastCommonDenominator(LCD)FINDING THE LCD

CHART ME11IOD:

1.) Break up your denominators into prime factors (break up the factors 88 small 88 possible).

2.) Createthischart. I I IFactor DenolD.1 DfJnom.2 I Biebest3.) Chartit.

L List each factor in the 1- denominator (Factor) and the number of times it is there (DenolD.l).b. Do the same for the 2- denominator, except list the number of times under DenolD. 2. If a factor in the

~ denominator is already there, do not re-write it. Instead, just list the number of times it is in the ~denominator under DeBOlD.2.

c. For each factor in the chart, write the higher of DeBom. 1 &:Denom. 2 in Bigbest.4.) To get the LCD, the write the factors raised to the number in Bigbest and multiply.NOTE: If there are more than 2 denominators, just add co1umns. .

NOTE: This may seem long, but after doing it repeatedly you will get faster. Eventually, you will figure out step 4.)without using the chart. Some books teach this method without using the chart.WARNING: This method is used when the denominator has variables. Learn it now &:save yourself time later on.

7 11 5Ex.) Findthe leastcommondenominatorfor -, -, - - . Answer: 360

9 30 24Denominators are 9, 30 and 24.

9= 3.3, 30=3.2.5 (not 6.5 because6 canbe brokenup to 3.2)24=3.2.2.2Chartit.

LCD =32 .23 .51 (Each factor in the chart raised to its number in ffigbest)LCD=9.S.5=36O .

LIST ALLMULl1PLESME11IOD:

1.) For each denominator, make a list ofits multiples.2.) Stop when you have a match.NOTE: Sometimes, this way can take awhile.

.711 5Ex.) FindtheleastcommondenomlDatorfor -, -, -- .9 30 24

9: 9, 18,27,36,45,54,63, 72, 81, 90, 99, 108, 117, 126, 135,.", 351,~30: 30,60,90,120, 150, 180,210,240,270,300, 330, ~ 39024: 24,48, 72,96,120,144, 168, 192,216,240,264,288, ..., 336,~

NOTICE: The 1- waywillbe fasterfor thesedenominators.In the nextexample,try it both ways&:noticethe~ waymaybefaster.

7 5Ex.) FindtheLCDfor -, --. ANSWEll:36. 12 IS

CHART: 12= 2.3.2 18= 2.3.3Factor DenolD.1 DenolD.2 Bi:2 2 1 23 1 2 2

LCD= 22.32= 4.9=36

MULTIPLES:

12: 12,24,36LCD=36

18: 18,36

,... alZ

Factor Denom. 1 Denom.2 Denom.3 Biebest tExDonent)3 2 1 1 22 1 3 35 1 1

Subject Name

Least Common Denominator (LCD)CONVERTING FRACTIONS TO EQUIVALENT ONES WITH LCD

Equivalent means equal. We will use the tact that a number divided by itseIfis 1.Whenever the new denominator is bigger than the old. we need to know what is missing ftom the old denominator to get the new.Multiply the top and bottom by what is missing. Multiplying top and bottom by the same number is multiplyingby 1.We are focusins on the LCD as the new denominator. The LCD is going to be the same as the denominator or bigger.WARNING: Do not reduce the ftactions once you have changed it.

. . . 7 SEx) Convert the ftactions to an eqwvalent one usmg the LCD: -, --

12 187 21 S _ _ 10

Answer: 12 =36 ' - 18 - 36We-already know the LCD is 36.

7 1What . .- =- tunes 12IS361

12 367 7- 3 21 .- =- =- 3 times12is 36. Multiplythe top andbottomby 3.12 12-3 36

NOTE: Thefollowingsayingcanbe useful: 12goes into36, 3 timesand 3 times7 is21. 2. =~ .12 36

You are doing the same thing, but you are just looking at it using the definition of division.S 1

What . . 6--=-- tunes181S3.18 365 5- 2 10

- - = -- = -- 2times18is36.Multiplythetopandbottomby2.18 18.2 365 10. . . d . .

- - = - - Saymg:18goesinto36,2 tunesan 2 times5IS10.. 18 36

. . . 7 11 5 7 280 11 132 5 75Ex.) Convertthe ftactionsto aneqwvalentoneusmgthe LCD:- - -- ANS.:- =- - =- - - =--

9 ' 30' 24 9 360' 30 360 '24 360LCD :=360

7 1-=-9 3607 7.40 280-=-=-9 9-40 3607 280-=-9 360

11 1-=-30 36011 11-12 132-= =-30 30-12 36011 132-=-30 360

5 1--=-24 3605 5-15--=

24 24.155 75--=--

24 360

What times 9 is 3601

40 times 9 is 360. Multiply top and bottom by 40.

Saying; 9 goes into 360 40 times and 40 times 7 is 280.

What times 30 is 3601

30 times 12 is 360. Multiply top and bottom by 12.

Saying; 30 goes into 360 12 times and 12 times 11 is 132.

What times 24 is 3601

75=--360

IS times 24 is 360. Multiply top &.bottom by 15. The sign does not change.

Saying: 24 goes into 360, 15 times. 15 times 5 is 75.

"'2"'2

Name

Adding and Subtracting Fractions

Subject

AddinglSubtracting Just FractioDlThe key to adding &:subtracting &actions is to have like/common denominators.We use the LCD for the common denominator. See LCD handout to refi'eshyour memory, ifnecessary.Think about eating a medium and a large pizza. When you put the left over pieces together, it is hard to say what portion

(~on) of the pizza you have left. Medium and large pizza slices are cut different. Ifwe re-cut the slices to be the same size,then we can say whether or not we have ~ or Y., ete, pizza left.

Steas to add/subtract:1.) Convert the &actions to equivalent &actions usiDsthe LCD as the denominator.2.) Add/subtract the tops over the common denominator.3.) Reduce or SimplifYyou answer, as per the directions.

53Answer:-120

Sidework: LCD= 120

2. = 25 24 goes into 120, 5 times. 5 x 5=2524 120

2- = 28 30 goes into 120,4 times.4 x 7=2830 12025 28 25+28 53

-+- = =- addthetopsoverthedenom.120 120 120 120

53 is already reduced120

5+2-Ex.) 24 30 Ex.) .! +2 _2. Reducedans: 23 Simplifiedans.: 12-4 8 16 16 . 16

Side work: LCD "" 16

.! = 12 4 goes into 16, 4 times. 4 x 3 = 124 16

2 = 14 8 goes into 16, 2 times. 2 x 7 = 148 16

1.. is alreadyoverthe LCD. No work needed.1612 14 3 12+14-3.23- +- -- = - addtopsoverthedenom.16 16 16 16 16

23 is reduced.To simplifY,23+16= 1remainder7. 12-16 16

AddingfSubtracting with Mixed NumbenAdd/subtract the whole number parts with the whole and the ftactionaJ parts with the ftaction. That is the same as saying

we have to add/subtract orangC;Swith oranges and apples with apples.

METiiOD 1: Apples &:Oranges.1.) Convert the ftactions to equivalent ftactions using the LCD as the denominator.2.) Add/Subtract the ftactionaJ part as usual.

If you have to borrow, then follow these steps.a.) borrow a 1 &om the .wholenumberb.) add that 1 to the ftaction that needs more.

ftaction' s denominatori.) That is. take the filet that 1- &:add it to the ftaction.

ftaction' s denominator

ii) Short cut: add the denominator to the numerator to get the new numerator.3.) Add/Subtract the whole number parts.4.) Reduce or simplifYthe ftactionNOTE: If you have to borrow because you do not have a ftaction, then once you borrow your ftaction is going

LCD.to be-

LCD3 4 53

Example)2-+1- Answer:3-4 15 60

Side work: LCD= 60. .! =45 .4 goes into 60,15 times. 15x3=45.1. =-.!. .15 goes into 60,4 times. 4x4=164 60 15 60

Add the &actions: 45+16=61 over the denominator 60. Now add the whole numbers, 2+1=3.

is not simplified.(60 goes in 61, 1 time with remainder 1.)

We know have a 3 and a I...! We put these together by adding to get 4...!....60 60

Subject Name

Adding and Subtracting Fractions

1 5 7Example) 7- - 5- Answer:1-3 9 9

Sidework: LCD=9. 1.=.! 3goesinto9, 3times.3 x 1= 3.3 9Cannot take 3 from 5, so we must borrow.

We borrow 1 from 7, which is the same as ! & adding that9is the same as adding numer. with denom.

5- alreadyhas the LCD(No work)9

METiiOD 2: No Mixed Numbers.Whenever you determine that you will need to borrow, 'you can avoid borrowing by CoDoq these steps.1.) Change aDthe mixed numbers to improper ftactiODS(Including changing integers by puttins them over 1)2.) Convert the ftactiODSto equivalei1tftactiODSusing the LCD as the denominator.3.) Add/subtractftactioDS. (Borroq is not needed)4.) Reduceand/orchangeftactiODSbackto mixednumbers.

1 5 7Example)7--5- Answer:1-3 9 '9

Side work: LCD =9. 1.=.! 3 goes into9,3 times. 3 x 1= 3.3 9Cannot take 3 'from 5, so we must borrow. Instead, let'sChange aDmixed numbers in to improper ftactiODS.

3 66 '

7-=- (9x 7 =63 +3 =66)9 95 50

S-=- (9xS=4S +5=509 9

~ already has the LCD (No work)9

(9 goes in to 16, 1 time with remainder 7)

**REMEMBER: A mixed number is nothing but a whole number and a ftaction added together.That is why we add whole with whole and part with part.Thatisalsowhywe cansimplifYour answerthe waywe did intho~ example.

Activity

This is an innovative methodfor adding and subtracting

fractions, which may bemore effective for visual learners

than the standard procedure

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IActivity 3: 1a - If I

7=1 2 3 4 5 6 7 8 9 10 11 12

1

2

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5

6

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12

If) 4/5 + 1/3 =1 2 3 4 5 6 7 8 9 10 11 12

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A3-5

lQ)4/5 - t/3 = .1 2 3 4 5 6 7 8 9 10 11 12

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12.-

J.g)8/11 + 3/8 =1 2 3 4 5 6 7 8 9 1011 12

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ftlJ(3uJ c r<. s+t ee+"'

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1b)2/3 tf/7 =i\ 123457tJr=- (cJ 6 7 8 9 10 11 12

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1

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19 4/5 + 1/3 =1 , ~ 4 ~ &

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IActivity3: 1a - If I

lQ) 4/5 - 1/3" 1 2 3 4

l.{(3)-=- I'd 1 .. . . . .' · 7 II 1011"5C I) / S :

9

10

11

12

If)8/11 - 3/8 =

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1\)(3 ~ 33 3

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3 I 66

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11

12

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1 , 3 4 6 6 7 8 9 10 11 12

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7 18

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1

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- - - - - -- .. --1

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7 6

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5 7

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J&)8/11 + 3/8 =1 2 4 5 6 7 8 9 10 "11 12

- - .. .-

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