Fractions

IndexIndex1. What are Fractions

2. Examples

3. Need of Fractions

4. Parts of Fractions

5. Example

6. Types of Fractions- Proper fractions & Improper Fr

actions Mixed Fractions Like Fractions & Unlike Fracti

ons Equivalent Fractions

7. Operations of Fractions Addition Subtraction Multiplication DivisionDivision

8.8. ComparisonComparison Greater than and Smaller Greater than and Smaller

thanthan How does the Denominator How does the Denominator

controls the Fractioncontrols the Fraction How does the Denominator How does the Denominator

controls the Fractioncontrols the Fraction

What are Fractions?What are Fractions?Fractions are for counting part of something.Fractions are for counting part of something.

Loosely speaking, a fraction is a quantity that cannot be Loosely speaking, a fraction is a quantity that cannot be represented by a whole number.represented by a whole number.

A fraction (from the Latin fractus, broken) is a number that A fraction (from the Latin fractus, broken) is a number that can represent part of a whole. The earliest fractions were can represent part of a whole. The earliest fractions were reciprocals of integers: ancient symbols representing one part reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much of two, one part of three, one part of four, and so on. A much later development were the common or "vulgar" fractions later development were the common or "vulgar" fractions which are still used today (½, ⅝, ¾, etc.)which are still used today (½, ⅝, ¾, etc.)

1 1 22//10101 1 22//101011//1212

11//1212

11//8811//88

1 1 ½½1 1 ½½

1111//12121111//1212

5555//60605555//6060

ExamplesExamples

Need Of Fractions?Need Of Fractions?Consider the following scenario.Consider the following scenario.

Can you finish the whole cake?Can you finish the whole cake? If not, how many cakes did you eat?If not, how many cakes did you eat?1 is not the answer, neither is 0.1 is not the answer, neither is 0.

This suggest that we need a new This suggest that we need a new

kind of number i.e. kind of number i.e. FractionFraction..

ba

Parts Of Fractions?Parts Of Fractions?

I’m the NUMERATOR.I’m the NUMERATOR. I tell you the number of partsI tell you the number of parts

I’m the DENOMINATOR.I’m the DENOMINATOR. I tell you the name of partI tell you the name of part

The The denominatordenominator tells us how tells us how many congruent pieces the whole is many congruent pieces the whole is divided into, thus this number divided into, thus this number cannot be 0. (b)cannot be 0. (b)

The The numeratornumerator tells us how many tells us how many such pieces are being considered. (a)such pieces are being considered. (a)

ExampleExample

8

7

How much of a pizza do we have below?How much of a pizza do we have below?The blue circle is our whole. The blue circle is our whole. if we divide the whole into 8 congruent pieces,if we divide the whole into 8 congruent pieces,- the denominator would be - the denominator would be 88..We can see that we have 7 of these pieces.We can see that we have 7 of these pieces.Therefore the numerator is Therefore the numerator is 77, and we , and we have have

of a pizza.of a pizza.

Proper fractionsProper fractions &&

Improper FractionsImproper Fractions

Proper fractionsProper fractions &&

Improper FractionsImproper FractionsEquivalent Equivalent FractionsFractions

Equivalent Equivalent FractionsFractions

Like Fractions Like Fractions & &

Unlike FractionsUnlike Fractions

Like Fractions Like Fractions & &

Unlike FractionsUnlike Fractions

Mixed FractionsMixed FractionsMixed FractionsMixed FractionsUnit FractionsUnit Fractions

&&Whole FractionsWhole Fractions

Unit FractionsUnit Fractions&&

Whole FractionsWhole Fractions

Types Of FractionsTypes Of Fractions

In Proper Fractions the In Proper Fractions the numerator is less than the numerator is less than the

denominator.denominator.E.g.. – 1/2 ,3/4 ,2/7 etc.E.g.. – 1/2 ,3/4 ,2/7 etc.

  

In Proper Fractions the In Proper Fractions the numerator is less than the numerator is less than the

denominator.denominator.E.g.. – 1/2 ,3/4 ,2/7 etc.E.g.. – 1/2 ,3/4 ,2/7 etc.

  

11441144

In Improper Fractions the In Improper Fractions the numerator is greater than (or numerator is greater than (or equal to)equal to) the denominator.the denominator.

E.g. – 4/2 ,9/5 ,6/6 etc.E.g. – 4/2 ,9/5 ,6/6 etc.Every whole no is Improper Every whole no is Improper

FractionFractionE.g. – 24 = 24/1 E.g. – 24 = 24/1

In Improper Fractions the In Improper Fractions the numerator is greater than (or numerator is greater than (or equal to)equal to) the denominator.the denominator.

E.g. – 4/2 ,9/5 ,6/6 etc.E.g. – 4/2 ,9/5 ,6/6 etc.Every whole no is Improper Every whole no is Improper

FractionFractionE.g. – 24 = 24/1 E.g. – 24 = 24/1

77447744

Proper Proper && Improper Improper FractionsFractionsI’m SmallerI’m Smaller

I’m BiggerI’m Bigger

1 ¾1 ¾1 ¾1 ¾ 7/87/87/87/8

Mixed FractionsMixed Fractions

In Mixed Fractions a whole number In Mixed Fractions a whole number and a proper fraction are together.and a proper fraction are together. E.g.. –2 1/4, 16 2/5 etc.E.g.. –2 1/4, 16 2/5 etc.

Mixed Fractions and Improper Mixed Fractions and Improper Fractions are same.Fractions are same.

We can use any to show the same We can use any to show the same amount. amount.

==

ConversionConversion

81981

9 1 = 1 8 8

7

17

7

372

7

32

ImproperImproper FractionFraction toto MixedMixed FractionFractiondivide the numerator by the denominator the divide the numerator by the denominator the quotient is the leading number, the remainder as quotient is the leading number, the remainder as the new numerator.the new numerator.

MixedMixed Fraction to Improper FractionFraction to Improper Fractionmultiply the whole number with the denominator multiply the whole number with the denominator and add the numerator to it. The answer is the and add the numerator to it. The answer is the numerator and the denominator is same numerator and the denominator is same

    

    

Like Like && Unlike Unlike FractionsFractions

In Like Fractions the denominators of In Like Fractions the denominators of the Fractions are samethe Fractions are same

In Unlike Fractions the denominators In Unlike Fractions the denominators of the Fractions are different.of the Fractions are different.

    

    

Unit FractionsUnit Fractions &&

Whole FractionsWhole FractionsIn Unit Fractions the In Unit Fractions the numerator of the numerator of the Fraction is 1.Fraction is 1.

In Whole Fractions the In Whole Fractions the denominators of the Fraction is denominators of the Fraction is 1.1.

Convert Unlike Fractions to Like Convert Unlike Fractions to Like FractionsFractions

Convert Unlike Fractions to Like Convert Unlike Fractions to Like FractionsFractions

Simplify all the Fractions.Simplify all the Fractions. Find LCM of all the Denominators.Find LCM of all the Denominators. Multiply all the fractions with a special form of 1 Multiply all the fractions with a special form of 1

to get 84 (here). Now these are Like Fractions.to get 84 (here). Now these are Like Fractions.

3 5 43 5 44 3 74 3 7

, , 4,3,74,3,72,3,72,3,71,3,71,3,71,1,71,1,71,1,11,1,1

22237

2 2 3 7 = 842 2 3 7 = 84

63 113 4863 113 4884 84 8484 84 84

, ,==

Equivalent FractionsEquivalent Fractions

They are the fractions that may have many They are the fractions that may have many different appearances, but are same.different appearances, but are same.

In the following picture we have ½ of a cake as the In the following picture we have ½ of a cake as the cake is divided into two congruent parts and we have cake is divided into two congruent parts and we have only one of those parts.only one of those parts.

But if we cut the cake into smaller congruent But if we cut the cake into smaller congruent pieces, we can see thatpieces, we can see that½ = 2/4 = 4/8 = 3/6½ = 2/4 = 4/8 = 3/6

Equivalent FractionsEquivalent Fractions

To know that two or more Fractions are To know that two or more Fractions are Equivalent we must simplify (change to its lowest Equivalent we must simplify (change to its lowest term) them.term) them.

Simplify: A fraction is in its lowest terms (or is Simplify: A fraction is in its lowest terms (or is reduced) if we cannot find a whole number (other reduced) if we cannot find a whole number (other than 1) that can divide into both its numerator and than 1) that can divide into both its numerator and denominator.denominator.E.g.- 5 : 5 & 10 can be divided by 5. 5 1E.g.- 5 : 5 & 10 can be divided by 5. 5 1 10 10 210 10 2

ExampleExampleExampleExampleAre 14 & 30 equal ?

21 45

2 = 2 14 = 30

3 3 21 45

reduce

32

721714

1421

reduce32

15451530

30

45

Making Equivalent FractionsMaking Equivalent Fractions

To make Equivalent Fractions we multiply the To make Equivalent Fractions we multiply the Fraction with a special form of 1 (same numerator & Fraction with a special form of 1 (same numerator & denominator- 4/4, 10/10 etc.)denominator- 4/4, 10/10 etc.)

E.g. : 4 = 4 5 = 20E.g. : 4 = 4 5 = 20 5 5 5 255 5 5 25

Operations Of FractionsOperations Of Fractions

SubtractionSubtractionSubtractionSubtraction MultiplicationMultiplicationMultiplicationMultiplicationAdditionAdditionAdditionAddition DivisionDivisionDivisionDivision

Addition Of FractionsAddition Of FractionsThings To Know !!!!Things To Know !!!!

SimplifyingSimplifying Like and Unlike FractionsLike and Unlike Fractions Like fractions are Compulsory to add.Like fractions are Compulsory to add. If there are Unlike Fractions then convert them to like If there are Unlike Fractions then convert them to like

fractions.fractions. The Denominator should not be added.The Denominator should not be added. Always change Improper fraction to a mixed Always change Improper fraction to a mixed

fraction.fraction.

Adding FractionsAdding Fractions

Addition means combining objects in two or more Addition means combining objects in two or more setssets

The objects must be of the same type, i.e. we The objects must be of the same type, i.e. we combine bundles with bundles and sticks with sticks.combine bundles with bundles and sticks with sticks.

In fractions, we can only combine pieces of the In fractions, we can only combine pieces of the same size. In other words, the denominators must be same size. In other words, the denominators must be the same.the same.

Adding Fractions WithAdding Fractions With Same DenominatorsSame Denominators

+ =

Add the numerator and

leave the denominator as it is.

1 2 3 11 2 3 1 + = = + = = 6 6 6 2 6 6 6 2

Adding Fractions with Adding Fractions with Different DenominatorsDifferent Denominators

If there are different denominators in the If there are different denominators in the fractions, then we change them to like fractions, then we change them to like fractions.fractions.

15

5

3

1

5

2

15

65

2

3

1

15

11

15

6

15

5

5

2

3

1

Adding Mixed FractionsAdding Mixed Fractions

Change the Mixed fractions to Change the Mixed fractions to Improper Fractions and then to Improper Fractions and then to Like Fractions.Like Fractions.At last add the Improper Like At last add the Improper Like Fractions.Fractions.Don’t forget to change the Don’t forget to change the answer to Mixed Fraction again.answer to Mixed Fraction again.

Subtraction Of FractionsSubtraction Of FractionsThings To Know !!!!Things To Know !!!!

SimplifyingSimplifying Like and Unlike FractionsLike and Unlike Fractions Like fractions are Compulsory to subtract.Like fractions are Compulsory to subtract. If there are Unlike Fractions then convert them to like If there are Unlike Fractions then convert them to like

fractions.fractions. The Denominator should not be subtracted.The Denominator should not be subtracted. Always change Improper fraction to a mixed Always change Improper fraction to a mixed

fraction.fraction. The numerator can be negative.The numerator can be negative.

Subtracting FractionsSubtracting Fractions

Subtraction means taking objects away.The objects must be of the same type, i.e. we can only take away apples from a group of apples.In fractions, we can only take away pieces of the same size. In other words, the denominators must be the same.

Subtracting Fractions WithSubtracting Fractions With Same DenominatorsSame Denominators

- =

Subtract the numerator and

leave the denominator as it is.

4 2 2 14 2 2 1 = = = = 6 6 6 3 6 6 6 3

Subtracting Fractions with Subtracting Fractions with Different DenominatorsDifferent Denominators

If there are different denominators in the If there are different denominators in the fractions, then we change them to like fractions, then we change them to like fractions.fractions.

15

10

3

2

5

2

15

65

2

3

2

15

4

15

6

15

10

5

2

3

2

Subtracting Mixed FractionsSubtracting Mixed Fractions

Change the Mixed fractions to Change the Mixed fractions to Improper Fractions and then to Improper Fractions and then to Like Fractions.Like Fractions.At last subtract the Improper At last subtract the Improper Like Fractions.Like Fractions.Don’t forget to change the Don’t forget to change the answer to Mixed Fraction again.answer to Mixed Fraction again.

Multiplication Of FractionsMultiplication Of Fractions

Things To Know !!!!Things To Know !!!! SimplifyingSimplifying The Denominator are always multiplied.The Denominator are always multiplied. “ “ Of ” : Of means multiply. E.g. : ½ of 2 = ½ Of ” : Of means multiply. E.g. : ½ of 2 = ½ × 2 = 1× 2 = 1 Product of two Proper Fractions is always less than both of them.Product of two Proper Fractions is always less than both of them. Product of two Improper Fractions is always Product of two Improper Fractions is always greatergreater than both of than both of

them.them. Product of one Improper Fraction and one Proper Product of one Improper Fraction and one Proper

Fraction is always less than the Improper Fraction Fraction is always less than the Improper Fraction and greater than the Proper Fraction.and greater than the Proper Fraction.

Multiplying FractionsMultiplying Fractions

To Multiply Fractions we Multiply both - The To Multiply Fractions we Multiply both - The numerators and the Denominators separately.numerators and the Denominators separately.

2 3 2 2 3 2 ×× 3 6 3 3 6 3 ×× = == = = =4 2 4 × 2 8 44 2 4 × 2 8 4

Multiplying Mixed FractionsMultiplying Mixed Fractions

Change the Mixed fractions to Change the Mixed fractions to Improper Fractions.Improper Fractions.Then multiply the Improper Then multiply the Improper Fraction.Fraction.Don’t forget to change the Don’t forget to change the answer to Mixed Fraction again.answer to Mixed Fraction again.

Addition Of FractionsAddition Of Fractions

Things To Know !!!!Things To Know !!!! SimplifyingSimplifying MultiplicationMultiplication Always change Improper fraction to a mixed fraction.Always change Improper fraction to a mixed fraction. Reciprocal: The inverse of fraction Reciprocal: The inverse of fraction

E.g. – 2/3 = 3/2 = 1 ½ , 2 ½ = 5/2 = 2/5E.g. – 2/3 = 3/2 = 1 ½ , 2 ½ = 5/2 = 2/5

Dividing FractionsDividing Fractions

To Divide Fractions we change the Second Fraction with its Reciprocal.Then Multiply the Reciprocal with the First Fraction.

2 4 2 × 5 10 5 ÷ = = =4 5 4 × 4 16 8

Dividing Mixed FractionsDividing Mixed Fractions

Change the Mixed fractions to Change the Mixed fractions to Improper Fractions.Improper Fractions.Then Multiply the Improper Then Multiply the Improper Fraction.Fraction.Don’t forget to change the Don’t forget to change the answer to Mixed Fraction again.answer to Mixed Fraction again.

ComparisonComparison

Comparison Comparison • Greater than and Smaller thanGreater than and Smaller than• How does the Denominator controls the FractionHow does the Denominator controls the Fraction• How does the Numerator controls the FractionHow does the Numerator controls the Fraction

Greater than and Smaller thanGreater than and Smaller than

Change the Fractions to Like Fractions.The fraction with greater numerator is bigger than the other one.If the Numerators are same, then the Fraction with smallest Denominator is the Biggest.

>2 3 10 9 & = 3 5 15 15

How does the Denominator How does the Denominator controls the Fractioncontrols the Fraction

Denominator represents the total number of pieces.Denominator represents the total number of pieces.If we share a Pizza with 2 people, we get ½ of pizza.If we share a Pizza with 2 people, we get ½ of pizza.If we share a Pizza with 4 people, we get ¼ of pizza.If we share a Pizza with 4 people, we get ¼ of pizza.If we share a Pizza with 8 people, we get 1/8 of pizza.If we share a Pizza with 8 people, we get 1/8 of pizza.

Conclusion: Conclusion: The larger the denominator the smaller the pieces, and The larger the denominator the smaller the pieces, and if the numerator is kept fixed, the larger the if the numerator is kept fixed, the larger the denominator the smaller the fraction,denominator the smaller the fraction,

How does the Numerator controls How does the Numerator controls the Fractionthe Fraction

Numerator represents the number of pieces.Numerator represents the number of pieces.If we share a Pizza with 16 people, we get 1/16 of pizza.If we share a Pizza with 16 people, we get 1/16 of pizza.If we share a Pizza with 13 people, we get 3/16 of pizza.If we share a Pizza with 13 people, we get 3/16 of pizza.If we share a Pizza with 5 people, we get 5/16 of pizza.If we share a Pizza with 5 people, we get 5/16 of pizza.

Conclusion :Conclusion :When the numerator gets larger we have more pieces. When the numerator gets larger we have more pieces. And if the denominator is kept fixed, the larger numerator And if the denominator is kept fixed, the larger numerator makes a bigger fraction.makes a bigger fraction.