Introduction
In the previous chapter we have studied about the fractions. In this chapter we will study some operations on the fractions like addition, subtraction, multiplication, I and division on fractions. 
Comparison of Unit Fractions
Comparison of Unit Fractions
\[\frac{1}{P}\]and\[\frac{1}{Q}\]are unit fractions where P and Q are natural numbers. Lf P > Q then\[\frac{1}{P}<\frac{1}{Q}.\]
Compare between\[\frac{1}{5}\] and \[\frac{1}{7}.\] Which is greater?
Solution:
Compare between their denominators 5 < 7 ( 5 is smaller than 7) Therefore, \[\frac{1}{5}>\frac{1}{7}.\]
Comparison of Like Fractions
Like fractions have same denominator. The fraction which has greater numerator is greater. For example,\[\frac{5}{7}\]and\[\frac{3}{7}\]are like fractions and 5 > 3, therefore, \[\frac{5}{7}>\frac{3}{7}.\]
Is \[\frac{24}{71}\] greater than \[\frac{12}{71}?\]
Solution:
Both the fractions have same denominator and 24 > 12Therefore, \[\frac{24}{71}>\frac{12}{71}.\]
Comparison of Unlike Fractions
To compare unlike fractions, unlike fractions are converted into like fractions, Numerator and denominator of one fraction is multiplied with the denominator of other fraction and vice-versa. For example, \[\frac{P}{Q}\]and \[\frac{R}{Q}\]are unlike fractions thus\[\frac{P\times S}{Q\times S}\]and\[\frac{R\times Q}{S\times Q}\]are like fractions. Now compare the numerators.
Compare between \[\frac{3}{4}\] and\[\frac{5}{7}.\] which is greater?
Solution:
First convert\[\frac{3}{4}\]and\[\frac{5}{7}\] into like fractions \[\frac{3\times 7}{4\times 7}=\frac{21}{28}\]and\[\frac{5\times 4}{7\times 4}=\frac{20}{28}\] Now compare numerators 21 > 20, therefore\[\frac{21}{28}>\frac{20}{28}\]or\[\frac{3}{4}>\frac{5}{7}.\]
In which one of the following figures does the unshaded part represent?\[\frac{2}{5}?\]
(a)
(b)
(c)
(d)
(e) None of these
Answer: (c)
Explanation
In the figure, which has been shown in the option C, 3 part out of the 5 part is shaded. Therefore, fractional representation often shaded part is \[\frac{2}{5}.\]
Which one of more... 
Reciprocal of a Fraction
Reciprocal of a fraction is the fraction by which if the fraction is multiplied the product is 1. The reciprocal of a fraction has reversed numerator and denominator. For example,\[\frac{Q}{P}\]is the reciprocal of the fraction\[\frac{P}{Q}.\]
Find the reciprocal of \[\frac{11}{13}.\]
Solution:
Reciprocal of \[\frac{11}{13}=\frac{13}{11}.\] 
Equivalent Fractions
Two or more fractions are said to be equivalent fractions if they have the same value. In other word s when equivalent fractions are reduced into their simplest form, they give the same fraction. For example, \[\frac{10}{15},\frac{20}{30},\frac{30}{45},\frac{40}{60}...\] etc. are equivalent fractions.
Are the fractions\[\frac{24}{27}\]and\[\frac{8}{9}\]equivalent fractions?
Solution:
The simplest form of \[\frac{24}{27}=\frac{8}{9}.\]Therefore, \[\frac{24}{27}\] and \[\frac{8}{9}\]are equivalent fractions.
How to Find Equivalent Fractions of a Given Fraction
Multiply both the numerator and denominator of the given fraction by a common number.
Find three equivalent fractions of\[\frac{6}{7}.\]
Solution:
(a)\[\frac{6}{7}=\frac{6\times 2}{7\times 2}=\frac{12}{14}\]
(b)\[\frac{6}{7}=\frac{6\times 3}{7\times 3}=\frac{18}{21}\]
(c)\[\frac{6}{7}=\frac{6\times 4}{7\times 4}=\frac{24}{28}\]
Thus \[\frac{12}{14},\frac{18}{21}\]and \[\frac{24}{28}\]are equivalent fractions of \[\frac{6}{7}.\] 
Lowest or Simplest Form of a Fraction
When HCF of numerator and denominator of a fraction is 1, the fraction is in its simplest or lowest form.
For example: The fraction \[\frac{5}{7}\]is in its simplest form as HCF of 5 and 7 is 1.
Is the fraction \[\frac{27}{72}\] in its lowest form?
Solution:
HCF of 27 and 72 is 9. Therefore, the fraction \[\frac{27}{72}\] is not in its simplest form.
How to Reduce a Fraction into Lowest Form
To reduce a fraction into its lowest form, numerator and denominator of the fraction is divided by their HCF. The resulting fraction is the reduced form of the given fraction.
Reduce the fraction \[\frac{16}{38}\]into its lowest form.
Solution:
HCF of 16 and 38 = 2
Now divide both numerator and the denominator by \[2\frac{16\div 2}{38\div 2}=\frac{8}{19}\]
Thus \[\frac{8}{19}\]is the reduced form of\[\frac{36}{38}.\] 
Coversion of Fractions
Conversion of an Improper Fraction into a Mixed Fraction Divide the numerator by the denominator. The quotient represents the whole number, the remainder represents the numerator and the divisor represents the denominator for the fractional part in the mixed fraction.
Change \[\frac{189}{18}\]into a mixed fraction.
Solution:
Divide 189 by 18
Thus remainder = 9, Quotient = 10, and divisor = 18 Therefore, the mixed fraction for \[\frac{189}{18}=10\frac{9}{18}.\]
Conversion of a Mined Fraction into an Improper Fraction
The whole number is multiplied with the denominator of the fractional part and the product is added by numerator. The sum represents numerator for the improperfraction and denominator of the improper fraction is same as the denominator offractional part of the mixed fraction.
Convert \[4\frac{11}{17}\]into a mixed fraction.
Solution:
Multiply 4 with 17 and add the product by 11 = 4 x 17 = 68 + 11 = 79.
Thus 79 is numerator and 17 is denominator for the improper fraction.
Thus \[\frac{79}{17}\]is the improper fraction for the mixed fraction\[4\frac{11}{17}.\] 
Fraction
Like Fraction
The fractions which have the same denominator are called like fractions. For example, \[\frac{5}{7},\frac{9}{7},\frac{5}{7}\]are like fractions as they have the same denominator.
Write the like fraction of \[\frac{8}{21}\]whose numerator is 4.
Answer:
According to the question numerator should be 4 and like fractions have same denominator thus the required fraction will be \[\frac{4}{21}.\]
Unlike Fraction
The fractions which have different denominators are called unlike fractions. For example, \[\frac{58}{87},\frac{52}{75},\frac{45}{88}\]are unlike fractions as they have different denominators.
Represent the shaded part in the following figures into fractional form and check are they unlike fractions?
Fractional representation for the shaded part in first figure is \[\frac{2}{4}\]and for the second figure is \[\frac{4}{6}.\] The fractions have different denominators. Therefore, they are unlike fractions.
Unit Fraction
The fractions which have the numerator 1 are called unit fractions. For example,\[\frac{1}{5},\frac{1}{8},\frac{1}{12}\]are the unit fractions as each of them has the numerator 1.
The following picture has been divided into 5 equal parts. How many parts of the figure should be shaded so that fractional representation of the shaded part of the figure is a unit fraction.
Solution:
A unit fraction has the numerator 1, therefore, only 1 part should be shaded.
Proper Fraction
If the numerator of a fraction is smaller than denominator, the fraction is called proper fraction. For example, \[\frac{5}{7}\] is a proper fraction as 5 is smaller than 7.
A flowering plant contains 19 flowers. 11 flowers fall down. What fraction of the total flower falls down? Is it a proper fraction?
Answer:
Total number of flower = 19
Number of flower which falls down = 11
Fractional representation of the flowers which fall down out of total flower \[=\frac{11}{19}\]
11 is smaller than 19 thus \[\frac{11}{19}\]is a proper fraction.
Improper Fraction
If the numerator of a fraction is greater than denominator, the fraction is called improper fraction. For example,\[\frac{19}{17}\]is an improper fraction as 19 is greater than17.
11 kg of sweet is distributed among 5 persons. Represent the amount each ofthe person would get as a fraction and check it is a proper or improper fraction?
Solution:
Total amount of sweet = 11 kg
Number of person = 5
Total amount which each of the persons would get \[=\frac{11}{5}kg\]
\[\frac{11}{5}\] is an improper fraction as numerator more... 
Introduction
Fraction is a number which is used to represent a part of a whole. It is in the form P of \[\frac{P}{Q}\]. Where P and Q are natural numbers. P is called numerator of the fractionand Q is called denominator. For example, \[\frac{5}{9}\] is a fraction, where 5 is numeratorand 9 is denominator of the fraction.
Represent the shaded part of the following figure as a fraction and write numerator and denominator of the fraction.
Explanation
The above figure has been divided into 5 equal parts. Out of 5 parts 1 part is shaded. Therefore, fractional representation of the shaded part \[\frac{1}{5}\] and numerator = 1, and denominator = 5. 
Multiples
When two or more than two numbers are multiplied with each other, the resulting number is the multiple of all that numbers. For example, if A x B = C, C is multiple of both A and B.
Multiples of 5 = 5, 10, 15, 20, 25, ____
Multiples of 2 = 2, 4, 6, 8, 10, 12,____
Multiples of 10 = 10, 20, 30, 40, ____
Common Multiples
The same multiples of two or more than two different numbers are called common multiples.
Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40,__
Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45,___
Common multiples of 4 and 5 = 20, 40, 60, etc.
Least Common Multiple (L.C.M.) The least common multiple among the common multiples of two or more than two numbers is called least common multiple.
Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ___
Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, ___
Common multiples of 5 and 7 = 35, 70,___
Least common multiple of 5 and 7 = 35.
LCM by Prime Factorization Method
In the prime factorization method numbers are written in the form of product of prime factors. The common and non-common factors are separated out and multiplied with each other. Their product are the LCM of the given numbers.
Prime factorization of \[28=2\times 2\times 7\]
Prime factorization of \[36=2\times 2\times 3\times 3\]
Common prime factors \[=2\times 2\]
Non-common factors \[=7\times 3\times 3\]
Product of the common factors \[=2\times 2\times 3\times 3\times 7=252\]
Thus LCM of 28 and 36 = 252.
LCM by Division Method
Division method of LCM involves the following steps:
Step 1: Write the numbers whose LCM is to be found in a row.
Step 2: Choose the least prime number by which at least two numbers are divisible. Then divide the numbers by the prime number.
Step 3: Write the quotient and undivided number in the next row just below the respected number.
Step 4: Repeat the process of division by the prime numbers unless only 1 or co-prime numbers remain in the last row.
Step 5: Now multiply the divisor and co-prime numbers of the last row. Product is the LCM.
Find the LCM of 12, 20, 32 and 28.
more...
 Common Factors
The same factors of two or more than two different numbers are called common factors.
Factors of 24 = 1, 2, 3,4, 6, 8, 12, 24
Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30.
Here 1, 2, 3, and 6 are the common factors of 24 and 30.
 Co-prime or Relatively Prime Numbers
If two numbers have only one common factor, the numbers are called co-prime or relatively prime numbers.
Factors of 2 = 1, 2
Factors of 5 = 1, 5
Thus 2 and 5 have only one common factor 1, therefore, 2 and 5 are co- prime numbers.
Factors of 15= 1, 3, 5, 15
Factors of 8 = 1, 2, 4, 8
15 and 8 have only one common factor which is 1. Therefore, 15 and 8 are the co- prime numbers.
Highest Common Factor (H.C.F) or Greatest Common Factor (G.C.F)
The common factor which is highest among the common factors of two or more than two numbers is called highest common factor of that numbers.
Find the highest common factors of 12 and 32
Explanation
Factors of 12 = 1, 2 , 3, 4, 6, 12
Factors of 32 = 1, 2, 4, 8, 16, 32
Common factors of 12 and 32 = 1, 2, 4,
The highest common factor of 12 and 32 = 4
HCF by Division Method
In case of greater numbers it is difficult to find HCF by writing factors. With the help of division method we can get the HCF easily even of greater numbers. Long division method involves the following steps:
Step 1: The greater number is divided by the smaller number.
Step 2 : The remainder left after subtraction is taken as divisor and divisor as dividend for the next step of division.
Step 3: The process is repeated unless remainder becomes 0.
Step 4: The last divisor is the HCF.
Find the HCF of 132 and 64.
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