Number System and Its Operations
Numbers are the symbolic representation of counted objects. There are infinite counting numbers from 1. Some arc-divisible by another whereas some are not divisible. Numbers are differentiated according to their divisibility and factors. A numeral system is a writing system for expressing numbers. The most commonly used system of numerals is Hindu-Arabic numeral system. In this chapter, we will learn about various numeral systems, types of numbers and operation on numbers.
Indian or Hindu-Arabic Number System
This number system was introduced by Indians, and is therefore, called Indian Number System. In this number system 10 is considered as the base.
10 ones = 10, 10 tens = 1 hundred, 10 hundreds = 1 thousand
Hindu - Arabic number system is based on the place value of digits in number
Indian Place Value Chart

The number two lakh ninety-eight thousand seven hundred and thirty-five is written by placing 2 at the place of "lakhs', 9 at the place of "Ten thousands', 8 at "Thousands', 7 at "Hundreds', 3 at "Tens' and 5 at "Ones',
Place Value
If a number contains more than one digit then the place more...

crores | Ten Lakes | Lakes | Ten Thousand | Thousands | Hundred | Tens | Ones |

2 | 9 | 8 | 7 | 3 | 5 |

Fractions and Decimals
Fraction
Fraction is a method for representing the parts of a whole number. An orange is divided into two equal parts and so the first part of orange is half of the whole orange and represented by of the orange.
TYPES OF FRACTIONS
Proper Fractions
A fraction whose numerator is less than denominator is called a proper fraction. are proper fractions.
Improper Fractions
A fraction is called improper fraction even if:

- It has smaller denominator than numerator
- It has equal numerator and denominator are improper fractions.

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LCM and HCF
LCM (Least Common Multiple)
LCM of two or more numbers is their least common multiple, LCM of 4 and 6 is 12, It means, 12 is the least common multiple of 4 and 6, therefore, 12 is exactly divisible by each of 4 and 6.
LCM by Prime Factorization Method
The following steps are used to determine the LCIVI of two or more numbers by prime factorisation method:
Step 1: Find the prime factors of each number
Step 2: Product of highest power of prime factors is their LCM.
LCM by Division Method
The following steps are used to determine the LCM of two or more numbers by division method:
Step 1: Numbers are arranged or separated in a row by commas.
Step 2: Find the number which divides exactly atleast two of the given numbers.
Step 3: Follow step 2 till there are no numbers (atleast two) divisible by any number
Step 4: LCM is the product of all divisors and indivisible numbers.

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Ratio and Proportion
Ratio
Ratio of two quantities is the comparison of the given quantities. Ratio is widely used for comparison of two quantities in such a way that one quantity is how much increased or decreased by the other quantity.
For example, Peter has 20 litres of milk but John has 5 litres, the comparison of the quantities is said to be, Peter has 15 litres more milk than John, but by division of both the quantity, it is said that Peter has, \[\frac{20}{5}\text{ }=\text{ }4\]times of milk than John. It can be expressed in the ratio form as 4: 1.
Note: In the ratio\[a:\text{ }b\text{ }\left( b~\ne 0 \right)\], the quantities a and b are called the terms of the ratio and the first term (ie. a) is called antecedent and the second term (i.e., b) is called consequent.
Simplest form of a Ratio
If the common factor of antecedent and consequent in a ratio is 1 then it is called in its simplest form.
Comparison of Ratio
Comparison of the given ratios are compared by first converting them into like fractions, for example to compare 5: 6, 8: 13 and 9: 16 first convert them into the fractional form i.e.\[\frac{5}{6},\frac{8}{13},\frac{9}{16}\]
The LCM of denominators of the fractions \[=2\times 3\times 13\times 8=\text{ }624\]
Now, make denominators of every fraction to 624 by multiplying with the same number to both numerator and denominator of each fraction.
Hence,\[\frac{5}{6}\times \frac{104}{104}=\frac{520}{624},\frac{8}{13}\times \frac{48}{48}=\frac{384}{624}\] and\[\frac{9}{16}\times \frac{39}{39}=\frac{351}{624}\]. Equivalent Tractions of the given fractions are,\[\frac{520}{624},\frac{384}{624},\frac{351}{624}\]We know that the greater fraction has greater numerator, therefore the ascending order of the fractions are, \[\frac{351}{624}<\frac{384}{624}<\frac{520}{624}\] or \[\frac{9}{16}<\frac{8}{13}<\frac{5}{6}\] or 9 : 6 < 8 : 13 < 5 : 6 thus, the smallest ratio among the given ratio is 9 : 16 and greatest ratio is 5 : 6.
Equivalent Ratio
The equivalent ratio of a given ratio is obtained by multiplying or dividing the antecedent and consequent of the ratio by the same number. The equivalent ratio of \[a\,\,\times \,\,b\]is \[a\,\,\times \text{ q }:\text{ }b\text{ }\times \text{ }q\]whereas, a, b, q are natural numbers and q is greater than 1,
Hence, the equivalent ratios of 5 : 8 are,

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Algebraic Expressions
In an algebraic expression constant and variables are linked with arithmetic operations. The value of unknown variable is obtained by simplification of the given expression.
TERMS OF AN ALGEBRAIC EXPRESSION
Literals or Variables
Alphabetical symbols used in algebraic expressions are called variables or literals. a, b, c, d, m, n, x, y, z ........... etc. are some common letters which are used for variables.
Constant Terms
The symbol which itself indicate a permanent value is called constant. All numbers are constant. \[6,\text{ }10,\,\,\frac{10}{11},\text{ }15,\text{ }-6,\text{ }\sqrt{3}\text{ }....\] etc. are constants because, their values are fixed.
Variable Terms
A term which contains various numerical values is called variable term. For example,
Product of 4 and \[X\text{ }=\text{ }4\text{ }\times \text{ }X\text{ }=\text{ }4X\]
Product of \[2,\text{ }X,\text{ }{{Y}^{2}}\]and \[Z\text{ }=\text{ }2\text{ }\times \text{ }X\text{ }\times \text{ }{{Y}^{2}}\times \text{ }Z\text{ }=\text{ }2X{{Y}^{2}}Z\]
Thus, 4X and \[2X{{Y}^{2}}Z\] are variable terms
Types of Terms
There are two types of terms, like and unlike. Terms are classified by similarity of their variables.
Like and unlike Terms.
The terms having same variables are called like terms and the terms having different variables are called unlike terms. For example,\[6x,x,,-2x,\frac{4}{9}x,\], are like terms and\[6x,2{{y}^{2}},-9{{x}^{2}}yz,4xy,\], 4xy, are unlike terms.
Coefficient
A number or a symbol multiplied with a variable in an algebraic expression is called its coefficient. In \[-\text{ }6{{m}^{2}}\]np, coefficient of\[n{{m}^{2}}\]p is -6 because \[{{m}^{2}}\]np is multiplied with -6 to form \[-\text{ }6{{m}^{2}}\]np.
The variable part of the term is called its variable or literal coefficient. In\[-\frac{5}{4}\] abc, variable coefficients are a, b and c.
The constant part of the term is called constant coefficient. In term\[-\frac{5}{4}\], abc, constant coefficient is\[-\frac{5}{4}\].

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Geometry and Symmetry
Basic Geometrical Shapes
Lines and angles are the main geometrical concept and every geometrical figure is made up of lines and angles. Triangles are also constructed by using lines and angles.
Point
A geometrical figure which indicates position but not the dimension is called a point. A point does not have length, breadth and height. A point is a fine dot. P is a point on a plane of paper as shown below.
Line
A set of points which can be extended infinitely in both directions is called a line.
Line Segment
A line of fix length is called a line segment.
In the above figure RS is a line segment and the length of RS is fixed.
Ray
A ray is defined as the line that can be extended infinitely in one direction.
In the above figure AB can be extended towards the direction of B. Hence, called a ray.
Note: A line segment has two end points, a ray has only one end point and a line has no end points.
Angle
Angle is formed between two rays which have a common point.
Vertex or common end point is 0.
OA and OB are the arms of AOB
The name of the above angle can be given as AOB or BOA
The unit of measurement of an angle is degree (°)
TYPES OF ANGLES
Acute Angle
The angle between is called an acute angle.
For example, are acute angles.
Right Angle
An angle of measure is called a right angle.
Obtuse Angle
An angle whose measure is between is called an obtuse angle.
Straight Angle
An angle whose measure is is called a straight angle.
Reflex Angle
An angle whose measure is more than and less than is called a reflex angle.
Complementary Angle
Two angles whose sum is is called the complimentary angle.
Complementary angle of any angle is more...

Mensuration
Perimeter and Area of Plane Figures
Perimeter of geometrical figure is the sum of its sides. There are different types of geometrical figures. Figures are classified by their shapes and sizes. Area of a geometrical figure is its total surface area.
Perimeter and Area of a Triangle

- Perimeter of a triangle = Sum of the length of all sides.
- Area of a right triangle \[=\frac{1}{2}\,\,\,\times \text{ }Base\text{ }\times \text{ }Height\]
- Perimeter of an equilateral triangle \[=\text{ }3\text{ }\times \text{ }Side\]
- Area of an equilateral triangle \[=\frac{\sqrt{3}}{4}\times {{\left( Side \right)}^{2}}\]

Data Handling
In this chapter we will learn about pictograph and bar graph.
Data
Data is a collection of facts, such as numbers, observations, words or even description of things.
Observation
Each numerical figure in a data is called observation.
Frequency
The number of times a particular observation occurs is called its frequency.
Statistical Graph
The information provided by a numerical frequency distribution is easy to understand when we represent it in terms of diagrams or graphs.
To represent statistical data, we use different types of diagrams or graphs. Some of them are:
(i) Pictograph
(ii) Bar graph
Pictograph
A pictograph represents the given data through pictures of objects. It helps to answer the questions on the data at a glance.
more...

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Applied Mathematics
Set
Set is a collection of well-defined objects which are distinct from each other. The objects in the set are called its elements. Sets are usually denoted by capital Letters A, B, C, ?.. and elements are usually denoted by small letters a, b, c, ........
For example/ the set of all even natural numbers less than 10 can be represented by
N = {2, 4, 6, 8}.
Methods for Describing a Set
(i) Roster Method: In this method, a set is described by listing elements, separated by commas, within braces.
e.g. A = {a, e, i, o, u}
Note: This method is also called listing method or tabular form method.
(ii) Set Builder Method: In this method, we write down a rule which gives us all the elements of the set by that rule.
e.g. A = {x : x is a vowel of English alphabet}
Finite Set
A set containing finite number of elements or no element, is called a finite set, eg. The set of all persons in India is a finite set.
Infinite Set
A set containing infinite number of elements is called an infinite set.
Cardinality of a Finite Set
The number of elements in a given finite set is called cardinal number of finite set, denote by n (a), where A is the given set.
e.g. \[P\text{ }=\text{ }\left\{ 5,\text{ }15,\text{ }25,\text{ }35,\,\,45 \right\}\Rightarrow ~n\left( P \right)=5\]
Universal Set (U)
A set consisting of all possible elements which occurs under consideration is called a universal set.
e.g. Let the set U defines the set of all natural numbers then set of all odd natural numbers is another subset of U and the set of all even natural numbers is another subset of U.
Equal Sets
Two sets A and B are called equal, if every element of A is a member of B and every element of B is a member of A, Thus we write A = B.
e.g. A = {2, 4, 6, 8, 10,} and {all the even natural numbers less than or equal to 10} i.e., A and B are equal sets.

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Reasoning and Aptitude
Reasoning and logic skills are an integral part of subjects like Mathematics. In this chapter, we will learn various problems related to reasoning and aptitude.
Problems Based on Missing Numbers
In these types of problems, we find out a missing number from a given set of numbers, which is appropriate and follow a certain pattern.

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Days | Number of cake | |

Monday | ||

Tuesday | ||

Wednesday | ||

Thursday | ||

Friday |

Joseph | John | Ketan | Yash | Arjun | more...
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