Fractions

1. UNDERSTANDING A FRACTION
A fraction is a number representing part of a whole. The whole may be a single object or a group of objects and each to be equal.
Definition

The numbers of the form pq, where ‘p’ and ‘q’ are whole numbers and q 0 are called fractions.
Example:
12,23,14,35,812etc., are fractions, here 12 means the whole thing can be divided into two equal parts and we taken out 1 part similarly 33 means the whole thing can be divided into three equal parts and we take out two parts.

In the above figure the whole rectangle can be divided into 5 equal parts among which 2 parts are shaded. The shaded portion represents two-fifths and is denoted by 25. Here two-fifths is a fractional number and 25 is a fraction.
How to Read Fractions
12 read as one-half; 23 as two-thirds; 14 as one-quarter; 35 as three-fifth and 812 read as eight-twelths and so an.
Note:
For a given fractions pq, p is called the numerator and q is called denominator. Where p, q may have common factors
Example:
512,12,23,68,1012, etc, 5 is numerator and 8 is denominator
Fractions can also be written in its lowest terms.

Example:

(i) 512,12,23  fractions does not have any common factors in both numerator and denominator.
They are written in lowest terms.
(ii) Fractions 68,1012  have common factors in both numerator and denominator
We know that the numbers of the form pq  where p,q w (whole numbers), q 0 and p, q does not have any common factors are called rational numbers.
Example:
12,23,45,1011 , etc are rational numbers. Therefore ‘every rational number is a fraction’

2. SIMPLEST FORM OF A FRACTION
If the numerator and denominator of a fraction have no common factor except 1, then the fraction is said to be in its simplest form or in lowest terms. Or if a fraction is said to be in simplest form if the H.C.F of its numerator and denominator is 1.
I. Irreducible Fraction
If a fraction is in simplest form then it is called irreducible fraction.
II. Reducible Fraction
If a fraction is not irreducible fraction then it is called reducible fraction.
III. Method to find Simplest Form of a Fraction:
We can reduce given fraction into its simplest form using any of the following two methods.
Method 1
Divide numerator and denominator of given fraction by their H.C.F
Example:
To find simplest form of 915; the H.C.F of 9 and 15 is 3. Now divide both numerator and denominator of 915  by 3 to get reduce it into simplest form. Hence the simplest form of 915 is 35.
Method 2
In this method, we can divide both numerator and denominator of the given fraction by common factor till we are left with common 1 only.
Example:
We will find simplest form of 7290  using this method or follows, 7290=45. Therefore the simplest form of 7290  is 45.

3. REPRESENTING FRACTIONS ON NUMBER LINE
To represent given fraction 12 on a number line, we can divide the length between 0 and 1 into two equal parts and we can take 1 part as a fraction 12.

To represent 23 on a number line, we can divide the length between 0 and 1 into three equal parts and we can take into three equal parts and we can take 2 parts out of 3 as a fraction 23.

4. CLASSIFICATION OF FRACTIONS
I. Like Fractions
Fractions with the same denominator are called like fractions.
Example:
19,29,49,89,169 are like fractions.
II. Unlike Fractions
Fractions with different denominators are called unlike fractions.
Example:
12,23,35,58 are unlike fractions.
III Unit Fractions
Fractions with 1 as numerator are known as unit fractions.
Example:
12,13,14,112, etc, are all unit fractions.
IV. Proper Fractions
A fraction in which the numerator is less than its denominator is called proper fraction.
Example:
12,23,45,812, etc, are proper fractions lie between the 0 and 1 on a number line.
V. Improper Fractions
A fraction in which the numerator is greater than or equal to its denominator.
Note:
The numerical value of all the improper fraction does not lie between 0 and 1 on a number line.
VI. Mixed Fraction
A combination of a whole number and a proper fraction is called mixed fraction.
Example:
213,314,1623, etc, are all mixed fraction.
we can write mixed fraction = whole number part + fractional parts = whole number +  Numerator  Denominator  of fractional part
In a mixed fraction 213, 2 is whole number and 13 is a proper fraction. 213=2+13
VII. Equivalent Fraction
Two or more fractions representing the same point of a whole are called equivalent fractions.
Eg: 12=24=36=48=510=612…..are equivalent fractions

In above figures, the shaded regions of each figure are equal i.e., 12=24=36 are equivalent fractions.
Finding Equivalent Fractions
To find an equivalent fraction of a given fraction, we can multiply or divide both numerator and denominator of the given fraction by the same non-zero number.
Examples:
12=1×22×2=24; 2×34×3=612

12=1×22×2=2×34×3=1×32×3=1×52×5=…..      are equivalent fractions
1218=12÷218÷2=12÷318÷3=12÷618÷6              are also equivalent fractions
Note:
If two fractions are said to be equivalent fractions, the product of the numerator of the first and denominator of the second is equal to the product of denominator of the first and the numerator of the second. These products are called cross products. If pq,rs are two fractions then their cross product is denoted as

and if ps = rq then we say that pq,rs are equivalent fractions.
Example: 12,24 are equivalent fractions since 1 × 4 = 2 × 2 4 = 4

5. INTER-CONVERSION OF FRACTIONS
I. Conversion of Unlike Fractions into Like Fractions
To convert unlike fractions we can make denominators of all given fractions equal to their L.C.M (i.e., we convert each of the given fractions into an equivalent fractions having a denominator equal to the L.C.M of all the denominators of the given fraction.)
Example:
To convert the unlike fractions 34,56,49,18,712 into like fractions first we find L.C.M of denominator of given unlike fractions.
L.C.M of 4, 6, 9, 8, 12 = L.C.M of 8, 9,12 = 72
Now we are equating denominator of all fractions to 72 by multiplying numerators and denominators of all given fractions with a suitable number.

Therefore 343×184×185472; 565×126×126072; 494×89×83272; 181×98×9972 and 712=7×612×64272

Therefore 5472,6072,3272,972,4272 are the required like fractions.

I Conversion of mixed Fraction into Improper Fraction
We know that mixed fraction has two parts; one is whole number part and the other is fractional part. To convert mixed fraction into improper fraction, multiply the whole number part with the denominator of the fractional part and add the product to the numerator of the fraction part. This gives the numerator of the improper fraction and its denominator is same as denominator of fraction part.
i.e., if mixed fraction = whole number  Numerator  Denominator , then the improper fraction can be expressed as
Improper Fraction = (Whole number × Denominator ) Numerator  Denominator 
Example:
Convert mixed fraction 523 into improper fraction as follows:
523=523(5×3)23=15+23=173

III. Convert an Improper Fraction into a Mixed Fraction
We know that improper fraction has greater numerical value of numerator than denominator. To convert an improper fraction into a mixed fraction, divide the numerator by denominator then the quotient so obtained forms the whole number part and the remainder forms numerator of  fractional part.
Here denominator of fractional part is same as denominator of given improper fraction. So that mixed fraction can be written as follows

Mixed Fraction Quotient Remainder  Divsior 
Procedure to convert 237 into mixed fraction as follows divide 23 by 7 as follows

7)23(3    21      2

We have quotient 3 and remainder 2 when 23 divided by 7.

Therefore required fraction = Quotient  Re mainder  Divsior =327

Convert 587 into mixed fraction

7)58(8    56      2

Required mixed fraction = 827

6. COMPARISON OF FRACTIONS
To compare two are more fractions, their denominators should be equal. If the denominator of given fractions are not equal, then change each one of the given fractions into an equivalent fractions with denominator equal to the L.C.M of the denominators of the given fractions. Now the new fractions are like fractions so that the comparison between two or more fractions is possible when they are like fractions.
Conditions for comparison of fractions
To compare given fraction we can use following conditions.
(i) Fractions with same denominator
Among the fractions with same denominator, the fraction with greater numerator is greater than the other.
Example:
For the fractions 29,149;149>29 since both have same denominator and 14 > 2
(ii) Fractions with same numerator
Among two fractions with same numerator. The fraction with smaller denominator is greater than the other.
Eg: For the fractions 29 and 26;26>29 since both have same numerator and 6<9

(iii) Fractions with different numerator and denominator
(A) When compare fractions with different numerator and different denominators, we change them into fractions by equating their denominators to L.C.M of denominators of given fractions and then compare numerators of like fraction by applying first condition.
Example:

For 45,67; L.C.M of 5, 7 is 35

Now 45=4×75×7=2835; 67=6×57×5=3035

Now 2835, 3035 are like fractions (since they have equal denominators) and 30 > 28

3035>283567>45
(B) To compare fractions with different numerators and different denominators we can also use the following method for the fractions ab,cd find cross products ad and bc.
If ad > bc, then ab>cd; If ad < bc, then ab<cd; If ad = bc, then ab=cd

Example:
Compare the fractions 35 and 811 by considering cross multiplication

The products are 33, 40 and 40 > 33

5×8>3×11811>35

Therefore 811>35

7. ARRANGING FRACTIONS IN ASCENDING AND DESCENDING ORDER
Ascending order of fractions means arranging fractions from smaller value to greater value and descending order of fractions means arranging fractions from greater value to smaller value. This can be done by comparison of fractions.
Examples:
Rearrange the following fractions in ascending order and descending order 23,45,715,1120,2330

Here the given fractions have different numerators and different denominators.

L.C.M. of 3, 5, 15, 20, 30 = 2 × 3 × 10 = 60

Therefore 23=2×203×20=4060; 45=4×125×12=4860; 7157×415×42860; 112011×320×33360 and 2330=23×230×2=4660.

Since 28 < 33 < 40 < 46 < 48

2860<3360<4060<4660<4860   715<1120<23<2330<45

Therefore the ascending order of given fractions is 715,1120,23,2330,45 and the descending order of given fraction is 45,2330,23,1120,715.

8. FUNDAMENTAL OPERATIONS (+, -, ×, ÷) ON FRACTIONS
(i) Addition
Addition of two or more fractions is possible, when they are like fractions.
Addition of Like Fractions:
If the given set of fractions have same denominator (i.e., like fraction), then the numerator of sum of all fractions is the sum of numerators of given fractions and denominator is their common denominator.
Therefore sum of all like fractions = Sum of all Numerators of Fractions Common Denominator

Example: 56+86+136+226=2813226=486=8

Addition of Unlike Fractions:
Step1: We find L.C.M of denominators of all fractions.
Step2: Now convert each of the given fractions into equivalent like fractions by equating their denominators to L.C.M of denominators.
Step3: Now we can add all like fractions which are so obtained in Step2
Step4: Reduce the fraction obtained in Step3 into its lowest terms and convert it into mixed fractions if it is a improper fraction.
Examples:
Step1: Find the sum of 59,812,1516

Step2: Now 59=5×169×16=80144; 812=8×1212×12=96144; 1516=15×916×9=135144;

Step3: Now sum = 59+812+1516=80+96+135144=311144 is a improper fraction

Step4: Therefore 59812+1516=311144=223144

Addition of Mixed Fractions:
If given set of fractions are mixed fractions;
Step1: Convert each of the mixed fractions into an improper fraction.
Step2: Add all improper fractions using the procedure, either like fractions addition or unlike fractions addition.
Example:
Find the sum of 125+235+345
Method 1:
In above the addends are mixed fractions

12575; 235135; 345195

Now, 125+235345=75135+195=395=745
Method 2:
125+235345=(1+2+3)+2+3+45

=655+45

=6+1+45

=745

Example:
Find sum of 123+234+315

Method 1:
In above addends are mixed fractions

123=53;           234=114;       315=165

Now, 123+234+315=53+114+165

=(5×20)×(11×15)×(16×12)60 = 100+165+19260=45760=73760

Method 2:

123+234+315=(1+2+3)+2334+15

=6+(2×20)+(3×15)+(1×12)60

=6+(40+4512)60 = 6+9760=6+13760=73760

II. Subtraction
Subtraction of Like Fractions
Difference of like fractions is a fraction which is having numerator as difference of numerator and denominators is common denominator.
Therefore difference of like fraction =   Difference of Numerators   Common Denominator  

Examples:

913413=9413=513

2315

Subtraction of Unlike Fractions
First we find L.C.M of denominators of given fractions.
Convert each of the given fraction into equivalent like fraction by equating their denominators to L.C.M of denominators.
Now we can find difference between like fractions so obtained.

Method 1 :

2315

L.C.M of 3,5 = 15

23=2×53×5=1015

15=1×35×3=315

2315=1015315=10315=715

Method 2 :

2315=(2×5)(1×3)15=10315=715

Subtraction of Mixed Fractions
While subtracting two or more mixed fractions, convert each of mixed fractions in to an improper fraction and then find the difference of fractions so obtained using any of the above methods.
Examples:

Find 316214+413215

Since 316=196; 413=133  214=94; 215=115

Now 316214+413215=19694133115

(Therefore L.C.M of 6, 4, 3, 5 = 60)

=190135+26013260

(Since 60÷6 = 10; 60÷3 = 20; 60÷4 = 15; 60÷5 = 2)

=45026760=18660=3360 or  3120

Therefore  316214+413215=3360  or  3120

III. Multiplication
Multiplication of two fractions is a fraction whose numerator is products of numerators of given fractions and denominators is product of denominators of given fractions, we can define it as follows:

Product of Fractions =  Product of their Numerators  Product of their Denominator 

i.e., for any two fractions ab,cd,abz×cd=a×cb×d or acbd

Example:

Multiplication of 25 and 38 is 25×38=640=320

For the multiplication of mixed fractions convert them into improper fractions and then consider multiplication.
Example:

234×156=114×116=12124

Reciprocal or Multiplicative Inverse of a Fraction

If the product of any two fractions is 1, then each one of them is called reciprocal or multiplicative inverse of other.
For the fraction pq,qp,pq×qp=1

Therefore reciprocal of pq is qp and reciprocal of qp is pq

Examples:

The reciprocal of 56 is 65

The reciprocal of 123 is 35  Since 123=53

The reciprocal of 2 is 12  Therefore 2=251

IV. Division
To divide one fraction by another fraction, we multiply the dividend fraction by the reciprocal of the divisor.
Examples:

35÷23=35×32=910

123÷225=53÷125=53×512=2536

Note:
For any fraction ab,1×ab=ab×1=ab and 1÷ab=ba; ab÷1=ab; ab×0=0×ab=0

Every non zero fraction or rational number has multiplicative inverse.
Zero does not have multiplicative inverse.

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