Addition Subtraction, Multiplication and Division of Fractions
Addition of Like Fractions
Addition of like fractions is the addition of its numerator and denominator of the resulting fraction is same as the common denominator.
Like fractions are \[\frac{9}{61},\frac{7}{61},\frac{5}{61},\] therefore, the addition of the fractions \[\text{=}\frac{\text{Additionofnumerators}}{\text{Common denominators }\!\!~\!\!\text{ }}\text{=}\frac{\text{9+7+5}}{\text{61}}\text{=}\frac{\text{21}}{\text{61}}\text{.}\]
Find the like fractions from the given fractions and add them:
\[\frac{5}{2},\frac{7}{5},\frac{1}{6},\frac{121}{2},\frac{7}{2}.\]
(a) \[\frac{133}{2}\]
(b)\[\frac{137}{3}\]
(c) \[\frac{13}{2}\]
(d) All of these
(e) None of these
Answer: (a)
Explanation
Like fractions from the given fractions \[=\frac{5}{2},\frac{7}{2},\frac{121}{2}.\] .Their addition \[=\frac{5+7+121}{2}=\frac{133}{2}\]
Addition of Unlike Fractions
The following are the steps to perform the addition of unlike fractions:
Step 1: Find the LCM of denominators of the fractions.
Step 2: Multiply the numerators and denominators of all the given fractions to their LCM.
Step 3: Add the numerators and write down the addition over common denominator.
Step 4: Reduce the resulting fraction into its lowest term if necessary.
Let two unlike fractions are \[\frac{7}{8},\frac{9}{4},\] therefore, LCM of denominators is 8.
Now one fraction has same denominator as the LCM but other has different, so, to make it same as LCM, it should be multiplied by 2, thus the equivalent fraction is\[\frac{9}{4}\times \frac{2}{2}=\frac{18}{8}.\]
Now the addition \[\frac{7}{8},\frac{18}{8}=\frac{7}{8}+\frac{18}{8}=\frac{25}{8}.\]
Add the unlike fractions from the given fractions. Choose the correct option for their resulting addition?
\[\frac{3}{4},\frac{7}{2},\frac{5}{9},\frac{3}{2},\frac{1}{8}.\]
(a) \[\frac{72}{103}\]
(b) \[\frac{103}{72}\]
(c) \[\frac{7}{103}\]
(d) All of these
(e) None of these
Answer: (b)
Explanation
\[\frac{3}{4}+\frac{5}{9}+\frac{1}{8}=\frac{54+40+9}{72}=\frac{103}{72}\]
Subtraction of Like Fractions
The subtraction of like fractions is same as its addition except that addition isconverted into subtraction.
Let two like fractions are \[\frac{567}{456},\frac{4546}{456},\]
\[\text{Their subtraction}\,text{=}\frac{\text{Subtraction of its numerators}}{\text{ }\!\!~\!\!\text{ Common denominators}}\] \[\frac{4546}{456}-\frac{567}{456}\,text{=}\frac{4546-567}{\text{ }\!\!~\!\!\text{ 456}}=\frac{3979}{456}\]
Choose like fractions from the given fractions and find the difference between greatest and smallest fractions: \[\frac{5}{3},\frac{6}{7},\frac{5}{9},\frac{7}{3}.\]
(a) \[\frac{7}{9}\]
(b) \[\frac{2}{3}\]
(c) \[\frac{5}{7}\]
(d) All of these
(e) None of these
Answer: (b)
Explanation
Like fractions are \[\frac{7}{3},\frac{5}{3}\] and their difference \[=\frac{7}{3}-\frac{5}{3}=\frac{2}{3}\]
Subtraction of Unlike Fractions
Steps to perform the subtraction of unlike fractions:
Step 1: Find the LCM of denominators of the fractions.
Step 2: Convert the fractions into its equivalent fractions in such a way that the denominator of every fraction should be equal to their LCM.
Step 3: Subtract the numerators and write down the subtraction over common denominator. Let two unlike fractions are,\[\frac{9}{7},\frac{8}{5},\] therefore, LCM of their denominators is 35.
Now multiply the fraction by a number in such a way that denominator should be equal to their LCM. Therefore, the equivalent of \[\frac{9}{7}\] is\[\frac{45}{35}\] and equivalent more... 
Representation of Fractions on Number Line
Fractions are greater than 0 but less than 1 come between 0 and 1. Number offractions can be obtained between 0 and 1 by dividing distance among them intorequired number of times. The following steps are used/or representation of fractions on number line:
Step 1: Draw a line and mark 0 on it.
Step 2 : If numerator of fraction that is to be represented on number line is greaterthan denominator then the given fraction is represented beyond thedistance of 0 to 1.
Step 3: If numerator is smaller than denominator then fraction comes between 0 and 1.
Fraction, \[\frac{9}{10}\] comes between 0 and 1 because numerator is smaller thandenominator.
Therefore, the following steps are used for representation of fraction \[\frac{9}{10}\] on number line:
Step 1: Draw the line and mark 0 on it.
Step 2: Mark another point Ion the same line.
Step 3: Divide distance between 0 and 1 into 10 equal parts.
Step 3: Represent the fraction \[\frac{9}{10}\] at the point of 9 on the number line whichhas been drawn in step 3 as shown in the following figure.
Choose the correct option from the following for representation of fraction \[\frac{2}{3}\] on number line:
(a)
(b)
(c) 
(d) all of these
(e) None of these
Answer: (c)
Explanation
The number line which has been given in the option (c)correctly represents the given fraction. 
Operations on Fractions
In the ratio form of a fractional number, numerator is called dividend and denominator is divisor. Therefore, \[6\div 3\] is expressed in the form of fraction and written as \[\frac{6}{3}\].
\[\frac{6}{3}\]is not in full simplified or reduced form therefore, by division it can be simplified and the quotient of the division is the full simplified form of the fraction. Let us consider an example of fraction\[\frac{24}{36}.\] The numerator of the given fraction is not divisible by denominator but a common divisor is there between both the numerator and denominator of the fractions. Therefore, its simplified form \[\frac{2}{3}\]is the solution of the fraction. The simplified form of the fraction is called reduced form of the fraction.
In the following picture, some parts of picture are shaded but some are not. Find the part of the un shaded portion of the picture.
(a)\[\frac{2}{3}\]
(b)\[\frac{1}{5}\]
(c)\[\frac{3}{5}\]
(d) All of these
(e) None of these
Answer: (b)
Explanation
One part of the picture is not shaded but 4 parts are shaded.
Proper Fractions
A fraction which has greater denominator than numerator is called proper fraction. \[\frac{3}{5},\frac{1}{2},\frac{7}{9}\] are proper fractions.
Choose the proper fraction from the following options:
(a)\[\frac{7}{9}\]
(b)\[\frac{102}{34}\]
(c)\[\frac{11}{34}\]
(d) All of these
(e) None of these
Answer: (c)
Explanation
The fraction which has smaller numerator than denominator is called proper fraction.
Improper Fractions
A fraction is called improper fraction even if:
1. It has smaller denominator than numerator
2. It has equal numerator and denominator
\[\frac{6}{5},\frac{5}{2},\frac{109}{34},\frac{6}{6}\]are improper fractions.
Choose the improper fraction from the following options:
(a)\[\frac{56}{3}\]
(b)\[\frac{2}{3}\]
(c)\[\frac{25}{43}\]
(d) All of these
(e) None of these
Answer: (a)
Explanation
The fraction which has greater numerator than denominator is called improper fraction.
Mixed Fractions
The combination of a fraction and a whole number is called mixed fractions. Every mixed fraction can be simplified into an improper fraction, i.e. \[5\frac{2}{7}.\]
The simplified form of a mixed fraction is always an improper fraction.
The solution of an expression is \[4\frac{1}{2}.\] which one of the following is correct about the common name of the solution of the expression?
(a) Mixed fraction
(b) Proper fraction
(c) Improper fraction
(d) All of these
(e) None of these
Answer: (a)
Explanation
\[4\frac{1}{2}\]is a mixed fraction.
Conversion of Mixed Fraction
Let us consider a mixed fraction \[4\frac{1}{2}\]. it is converted into fraction by multiplying denominator more... 
Introduction
Fraction is a method for representing the parts of a whole number. An orange isdivided into two parts then the first part of orange is half of the whole orange and represented by \[\frac{1}{2}\] of the orange, \[\frac{1}{2}\] is in fraction and called fractional number. The upper part of the fraction is called numerator and lower part of the fraction iscalled denominator. In the fraction,\[\frac{a}{b},\]a is its numerator and b is denominator. Reciprocal of a fraction is obtained on reversing its order. Therefore, reciprocal of \[\frac{a}{b}\] is \[\frac{b}{a}\]and product of a fraction with its reciprocal \[=\frac{a}{b}\times \frac{b}{a}=1\] i.e. \[\frac{24}{16}\times \frac{16}{24}=1\] 
Introduction
We all are familiar with the numbers. Everything is counted by the numberstherefore, the numbers are the symbolic representation of counted objects. In thischapter, we will learn about number system, and its operations. 
Types of Numbers
Natural
Numbers Every counting number is called natural numbers. Zero is excluded from the natural numbers because it does not represent the number of counted objects.
1, 2, 3, 4, 5, etc. are natural numbers.
Therefore, the numbers that are used in counting are called natural numbers.
Highest natural number is infinite or cannot be defined but lowest natural number is 1.
Natural numbers are represented by N (First capital letter of its name).
Find the first five natural numbers from the following sets of numbers:
(a) 2, 3, 5, 7, 8
(b) 1, 2, 3, 4, 5
(c) 10, 11, 12, 13, 14
(d) None of these
(e) All of these
Answer: (b)
Explanation
1, 2, 3, 4, 5 are the first five natural numbers.
Whole Numbers
When 0 is included with counting numbers, it becomes whole number. Whole number is represented by W (First letter of its name). Whole numbers (W) = {0, 1, 2, 3, 4 ...infinite}.
Highest whole number cannot be defined but lowest is 0.
All natural numbers are whole number but all whole numbers are not natural numbers.
If N is the set of all natural numbers and W is the set of all whole numbers then overlapped part of N represent of which one of the following numbers?
(a) 1
(b) 0
(c) 2
(d) All of these
(e) None of these
Answer: (b)
Explanation
Zero is not the member of the set of natural numbers. Hence, overlapped part of N represent zero.
Prime Numbers
The numbers which have only two factors, 1 and the number itself are called prime numbers. The following numbers 2, 3, 5,7,11,13,17,19, 23 do not have factor other than 1 and the number itself, therefore, they are called prime numbers.
Consider the following statements:
Statement 1: The set of all natural numbers is prime number.
Statement 2: The set of numbers which does not have factor other than 1 and the number itself is called prime number.
Which one of the following options is correct?
(a) Statement 1 is true and 2 is false
(b) Statement 1 is false and 2 is true
(c) Both the statements are false
(d) None of these
(e) All of these
Answer:(b)
Explanation
Every natural number is not prime number and the number which has only two factors 1 and the number itself is called prime number.
Composite Numbers more... 
Representation of Numbers on Number Line
Every number on number line represents position from its greater and smaller numbers. Negative, positive, whole, natural, prime, etc. all are represented on number line. Numbers on number line are represented by the following:
Step 1: Draw a line and mark a point zero on it.
Step 2: Distance between the numbers is always equal. Therefore, the length of number line should be divided into required number of interval.
Step 3: The arrow mark at the far end point of number line, indicates infinitive.
Representation of whole numbers from 0 to 8 on number line.
Step 1: Draw a horizontal line and mark a point 0 on it.
Step 2: Mark another point and divide the distance among them into 8 equal parts.
Step 3: Write numbers 1 to 8 at each division.
Operation on whole Numbers
We are familiar with the basic operations of addition, multiplication, subtraction and division of the whole numbers. We will learn about these operations on whole numbers.
Properties of Addition
Consider the following two statements:
Statement 1: Subtraction of two whole numbers is always a whole number.
Statement 2: Subtraction of two whole numbers never be a whole number.
Which one of the following options is correct?
(a) Statement 1 is true and B is false
(b) Statement 1 is false and 2 is true
(c) Statement 1 and 2 are false
(d) All of these
(e) None of these
Answer: (a)
Explanation
Subtraction of two whole numbers is always a whole number.
If the sum of two numbers remains same on changing the order of the numbers then what will be the difference of the numbers if their order is changed?
(a) Difference of the numbers is not same on changing their order
(b) Difference of the numbers is same on changing their order
(c) Cannot be defined
(d) All of these
(e) None of these
Answer: (a)
Explanation
The result of subtraction of two numbers is changed on changing their order.
If the sum of \[x+(y+z)=m+n\] then the sum of \[(y+z)+x\] is?
(a) m - n
(b) n- m
(c) m+n more... 
Indian Number System or Hindu-Arabic Number System
This system was introduced by Indians therefore, called Indian number system. In this system 10 is considered as the base.
10 units = 10, 10 tens = 1 hundred, 10 hundreds = 1 thousand
Hindu -Arabic number system is based on the place value of the numbers. A single number is read as once and first digit from the right side of more than one digit number is also read as once while second, third, fourth, fifth, digits from right side are read as tens, hundreds, thousands, ten thousands respectively.
Indian Place Value Chart
| Crores | Ten lakhs | Lakhs | Ten Thousand | Thousand | Hundred | Ten | Unit |
| 9 | more...
 Bodmas Rule
When a single expression contains many mathematical operations then BODMAS rules are used for the simplification of the expression. The word BODMAS has been arranged according to the priority of the operations.
The letters of BODMAS express the following operations:
B Stands for Bracket
O Stands for of
D Stands for Division
M Stands for Multiplication
A Stands for Addition and
S Stands for Subtraction
 Brackets
There are three types of Brackets, small, middle or medium and big.
Small Bracket is denoted by ()
Middle Bracket is denoted by {}
Big Bracket is denoted by [ ]
Brackets are used according to the placement of operations. First we use small bracket then middle Bracket and Big bracket is used at last in an expression. i.e. \[\left[ \left\{ \left( 4+5 \right)\times 6 \right\}+7 \right].\]
Elimination of Brackets
The solution of the expression is obtained by eliminating the brackets. The first priority of eliminating the bracket is small bracket then middle bracket and finally big bracket.
The following steps are used to simplify the expression, \[\left[ \left\{ (4+5)\times 6 \right\}+7 \right]:\]
I st: Eliminate the small bracket by adding 4 and 5, in small bracket, \[[\{(4+5)\times 6\}+7][\{9\times 6\}+7].\]
\[\text{I}{{\text{I}}^{\text{nd}}}\] : Eliminate the middle bracket by multiplying 9 and 6, in middle bracket, \[[\{9\times 6\}+7]=[54+7].\]
\[\text{II}{{\text{I}}^{\text{rd}}}\]: Add 54 and 7 to eliminate big bracket [54 + 7] = 61. The solution of the above given expression is 61.
Simplify: \[[\{(34+17)-20\}+21]-20\]?
(a) 31
(b) 32
(c) 36
(d) Both (a) and (c)
(e) None of these
Answer: (b)
Explanation
\[[\{(34+17)-20\}+21]-20\] \[=[\{51-20\}+21]-20=[31+21]-20=52-20=32.\]
 How many different three digit numbers can be obtained by using the digits 0,1,3 without repeating any digit in the number?
(a) 4
(b) 5
(c) 3
(d) 2
(e) None of these
Answer: (a)
Which one of the following is the smallest seven digit number having four different digits?
(a) 1230000
(b) 0000123
(c) 1000023
(d) 1000032
(e) None of these
Answer: (c)
Simplify:\[~45+3\times 2\]of \[5-\text{(}16+4\text{)}-8\div 4.\]
(a) 55
(b) 43
(c) 53
(d) Both (a) and (c)
(e) None of these
Answer: (c)
Explanation
\[45+3\times 2\]of \[5-(6+4)-8\div 4\] \[=~45+3\times 2\]of\[5-20-8\div 4\] [Bracket removed] \[=~~45+3\times 2\]of 5 ?20 ? 2 [Operation of division \[8\div 4=2]\] = 45 + 30 - 20 - 2 [Operation of multiplication \[3\times 2\times 5=30]\] = 75 - 20 - 2 [Operation of addition 45 + 30 = 75] = 75 - 22 = 53 [Operation of subtraction]
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