Current Affairs 6th Class

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
crores Ten Lakes Lakes Ten Thousand Thousands Hundred Tens Ones
    2 9 8 7 3 5
  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...

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.
  Simplest form of a Fraction A fraction is said to be in the simplest or lowest form if its numerator and denominator have no common factor except 1.   Mixed Fractions Combination of a proper fraction and a whole number is called mixed fraction. Every mixed fraction has a whole and a fractional part.   Like and unlike Fraction When two or more fractions have same denominator then they are called like fractions whereas unlike fractions do not have equal denominators.   Equivalent Fractions Two fractions are said to be equivalent if they are equal to each other. Two equivalent fractions may have a different numerator and a different denominator  
  •             Example:
Convert  into a mixed fraction. (a)                                                              (b)              (c)                                                                 (d) All the above   (e) None of these Answer (a)  
  •              Example:
are:   (a) like fractions                                                  (b) unlike fractions (c) equivalent fractions                                  (d) Mixed fractions (e) None of these Answer (b)   OPERATIONS ON FRACTIONS   Addition of Like Fractions Addition of like fractions is the addition of their numerators and common denominator is the denominator of the resulting fraction. Hence, the sum of like fractions   Subtraction of Like Fractions Subtraction of like fractions is same as its addition except that addition is converted into subtraction. Let two like more...

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.  
  •             Example:
Find the least number which is exactly divisible by each of 28 and 42. (a) 64                                                                (b) 84            (c) 52                                                                (d) All of these (e) None of these Answer (b) Explanation: \[28\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }7,\text{ }42\text{ }=\text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }7\] LCM \[=2\times 2\times 3\times 7=\text{ }84\]   HCF (Highest Common Factor) Highest Common Factor is also called Greatest Common Measure (GCM) or Greatest Common Divisor (GCD). H.C.F of two or more numbers is the greatest number which exactly divides each of the numbers.   HCF by Prime Factorization Method The HCF of two or more numbers is obtained by the following steps: Step 1: Find the prime factors of each of the given number. Step 2: Find the common prime factors from prime factors of all the given numbers. Step 3: The product of the common prime factors is their HCF.   HCF by Continued Division Method The HCF of two or more numbers can also be obtained by continued division method. The greatest number is considered as dividend and smallest number as divisor. Follow the following steps to perform the HCF of the given numbers: Step 1: Divide the greatest number by smallest. Step 2: If remainder is zero, then divisor is the HCF of the given number. Step 3: If remainder is not zero then, divide again by considering divisor as new dividend and remainder as new divisor till remainder becomes zero. Step 4: The HCF of the numbers is last divisor which gives zero remainder.   HCF of more than two Numbers The HCF of more than two numbers is the HCF of resulting HCF of two numbers with third number. Therefore, HCF of more than two numbers is obtained by finding more...

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,  
  •              Example:
Mapped distance between two points on a map is 9 cm. Find the ratio of actual as well as mapped distance if 1 cm = 100 m. (a) 10000 : 1                                                      (b) 375 : 1       (c) 23 : 56                                                          (d) 200 : 1 (e) None of these Answer (a) Explanation: Required ratio \[=\text{ }900\text{ }\times \text{ }100:\text{ }9\text{ }=\text{ }90000:\text{ }9\text{ }=-\text{ }10000:\text{ }1\]  
  •              Example:
Consumption of milk in more...

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\[-\frac{5}{4}\].  
  •         Example:
Sign of resulting addition of two like terms depends on which one of the following? (a) Sign of biggest term                (b) Sign of smallest term (c) Sign of positive term            (d) Sign of negative term (e) None of these Answer (a)   Operations on Algebraic Expressions When constant and variables are linked with any of the following fundamental arithmetic operations i.e. addition, subtraction, multiplication and division, then the solution of the expression is obtained by simplification of the expression.   Addition and Subtraction of Terms The addition of two unlike terms is not possible and their addition is obtained in the same form. Addition of 2x + 3x is 5x but the addition of 2x + 3y is 2x + 3y. Subtraction of two like terms is same as the subtraction of whole numbers. For example, 4x - 2x = 2x  
  •        Example:
Simplify: \[\left( \mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}{{\mathbf{x}}^{\mathbf{2}}} \right)\mathbf{ - }\left( \mathbf{5}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+11}{{\mathbf{x}}^{\mathbf{2}}} \right)\] (a) \[15{{x}^{3}}\]                                                                   (b) \[15{{x}^{2}}\]          (c) \[-3{{x}^{2}}\]                                                                    (d) \[13{{x}^{2}}\] (e) None more...

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 \[=\frac{\sqrt{3}}{4}\times {{\left( Side \right)}^{2}}\]
  Perimeter and Area of a Parallelogram Parallelogram is a quadrilateral whose opposite sides are equal and parallel to each other. In the given figure ABCD is a Parallelogram in which \[AB||\,\,CD,\,\,BC||\,\,\,AD,\text{ }AB\text{ }=\text{ }CD\text{ }and\text{ }AD\text{ }=\text{ }BC\] Perimeter of a Parallelogram = 2(sum of two adjacent sides) Hence, perimeter of a parallelogram ABCD = 2(AB + BC) Area of a Parallelogram \[=\text{ }Base\text{ }\times \text{ }Height\] Therefore, the area of a parallelogram \[ABCD=AB\times CE\]   Perimeter and Area of a Rectangle A rectangle has four right angles and its opposite sides are equal Longer side of a rectangle is called length and shorter side is called width. Perimeter of rectangle ABCD = AB + BC + CD + DA = length + width + length + width \[=\text{ }2\left( length\text{ }+\text{ }width \right)\] Hence, perimeter of a rectangle \[=\text{ }2\left( length\text{ }+\text{ }width \right)\] Area of a rectangle \[=\text{ }length\text{ }\times \text{ }width\] Perimeter and Area of a Rhombus A rhombus is a parallelogram with four equal sides. Therefore, perimeter of rhombus\[=\text{ }4\text{ }\times \text{ }side\]. In the figure given below ABCD is a rhombus. Perimeter of a rhombus, \[=\text{ }4\text{ }\times \text{ }side\] Area of a rhombus \[=\text{ }base\text{ }\times \text{ }height\] Also area of a rhombus =\[\frac{1}{2}\] product of length of diagonals.   Perimeter and Area of a Square A square has four equal sides and each angle of\[90{}^\circ \]. In the picture given below, ABCD is a square because its all sides are equal and each angle is a right angle. Perimeter of square \[=\text{ }side\text{ }+\text{ }side\text{ }+\text{ }side\text{ }+\text{ }side\text{ }=\text{ }4\text{ }\times \text{ }side\] more...

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.  
  •               Example:
The following pictography shows the number of cakes sold at a bakery over five days. more...
 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.  
  •              Example:
Find cardinal number of a set A of the composite numbers between 10 and 25. (a) 4                                                                  (b) 6            (c) 8                                                                  (d) 9 (e) None of these Answer (d) Explanation: Here, A = {12, 14, 15, 16, 18, 20, 21, 22, 24} \[\Rightarrow \text{ }n\left( A \right)\text{ }=\text{ }9\text{ }\Rightarrow \]Cardinal number of set A = 9   Average Average is a calculated central value of a set of given number. The average is the sum of the numbers divided by the count. So, Average = \[\frac{sum\,of\,the\,number}{count}\] For example. Average of first ten natural numbers is \[\frac{1+2+3+4+5+6+7+8+9+10}{10}=\frac{55}{10}=5.5\]   Percentage The word 'percent', more...

 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.  
  •       Example:
Complete the series given below: 4, 10, 26, 72, 208, 614? (a) 1815                                                            (b) 1820         (c) 1830                                                             (d) 1836 (e) None of these Answer (c) Explanation: Here the pattern is given below: \[4\times 3-2\times 1=\text{ }10;\text{ }10\times 3\text{ }-\text{ }2\times 2\text{ }=\text{ }26\] \[26\times 3\text{ }-\text{ }2\times 3\text{ }=\,\,\,72;\text{ }72\times 3\text{ }-\text{ }2\times 4\text{ }=\text{ }208\] \[208\times 3\text{ }-\text{ }2\times 5\text{ }=614;\text{ }614\times 3\text{ }-\text{ }2\times 6\text{ }=1830\] Problems Based on Coding-decoding In these types of problems, you will learn to code a word by using a certain pattern or rule.  
  •       Example:
If 'SAMPLE' is coded as 'FMNOBT' then how would you code 'CLIMAX'? (a) YBJLMD                                                      (b) YJBMLD      (c) XJMNLD                                                       (d) DBJLMY (e) None of these Answer (a) Explanation: Here we have the following pattern.                               Problems Based on Puzzle In these types of problems, the given information is summarized by a table.  
  •       Example:
Joseph and John are good in hockey and cricket. Ketan and Yash are good in cricket and football. Arjun and Tuffey are good in baseball and volleybalL Ketan and Tapan are good in hockey and baseball. Based on above information answer the given question. Who is good in Hockey, Cricket, Football and baseball? (a) Joseph                                                         (b) John (c) Ketan                                                           (d) Yash (e) None of these Answer (c) Explanation:
Days Number of cake
Monday
Tuesday
Wednesday
Thursday
Friday
  Joseph John Ketan Yash Arjun more...



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