Class 6 Mathematics

Geometry Chapter 2 Points Lines Line Segment Ray And Their Concepts

Geometry Chapter 2 Ponts

1. If a piece of paper is folded twice in two ways, then we find, there are two long straight markings along the creases, and the place or the position where they intersect is called a Point.
2. Again the intersection of two adjacent edges of a page of a book or the upper surface of a square or rectangular table produces a point.
3. In the above examples, we can take two long straight markings and two adjacent edges of a page or the upper surface of a table as two straight lines (which we shall discuss in the next article) and they intersect at a point.
4. We know that a line has only length and its dimension is one.
5. If we gradually diminish the length of a line, then in the extreme case, the length of the line can not
6. be measured and the line is reduced to a point.
7. As the length of a point is not measurable, the point has no dimension.
8. So we can give the following definition of a point:
9. That, which has only position but no dimension is called a point.
10. A point has no length, breadth, or thickness.
11. Although a geometrical point can not be drawn, a point can be drawn by pressing lightly the sharp end of a pencil on a piece of paper.
12. A dot mark (.) which is produced on the paper is the required point.
13. In geometry, in order to distinguish the different points from one another, we denote them by the capital letters of the English Alphabet such as A, B, D, D, E, F, etc.

WBBSE Class 6 Points and Lines Notes

In this connection, the followings are to be kept in mind:

1. Two straight lines (which are not parallel) intersect at a point.
2. A line can be regarded as a continuous series of innumerable points.
3. By joining any two points, we can draw a line segment.
4. Collinear points: If a straight line is obtained by joining consecutively three or more points lying on the same plane i.e., if three or more points be situated on a straight line, then the points are called collinear points.
5. For example:
6. Here the points A, B, C, and D lie on the straight line AD===; so the points A, B, C, and D are collinear points.

Important Definitions Related to Geometry

Geometry Chapter 2 Lines

1. The geometrical figure which has only length, but no breadth or width is called a line:
2. For example, in the figure below both AB=== and PQ=== lines.
3. Lines are classified into two types, (i) Straight lines, and (ii) Curved lines.

 

Straight lines and curved lines :

1. In the previous chapter, we discussed lines only.
2. A line is a geometrical figure which has only length but no breadth or thickness.
3. It is a one-dimensional figure.
4. There are two types of lines: straight lines and curved lines.

Now we shall discuss some examples:

1. If we stretch both ends of a thread till it becomes straight, then the figure that the stretched thread forms are a straight line.
2. Two walls of a room meet at a line and also each of the walls of a room meets with the floor of the room at a line.
3. These lines are called straight lines.
4. If you walk along a circular track, then you will continuously change the direction of your movement.
5. Suppose you start your walking towards the south, after some time you will be moving towards the east and afterward you may stand facing north, etc.
6. The circular track along which you are walking is a curved line.
7. But if you walk along a straight line facing say, south, then all along you would have faced south.
8. Suppose, there are two stations located at A and B. There are innumerable ways of going from station A to station B as shown in the figure (among them, one is a straight line or way).
9. If you want to go from A to B through the shortest possible route, then obviously you will have to go through the straight line route and this is only the straight route.
10. All other routes are curved.
11. From this, we conclude that through two points only one curved route straight line can be drawn and innumerable curved lines can be drawn.
12. Definition: A-line, whose one end can be reached from the other end without changing direction, is called a straight line.
13. A straight line can also be defined in the following way:
14. A straight line is a line that can be extended on both sides uniformly without changing direction.
15. In the adjacent figure, AB is a straight line.
16. Definition: A line that is not a straight line or a line, whose one end can be reached from the other end by changing direction, is called a curved line.
17. A curved line can also be defined in the following way:
18. A curved line is a line that gradually deviated from
19. A B is straight for some or all of its length.
20. A curved line has many directions.
21. Curved lines if the sharp ends of a pencil along the side of a scale placed on the surface of a paper, a straight line is obtained.
22. The edges of a page, benches, tables, etc. are examples of straight lines.
23. If we draw the sharp end of a pencil along the side of a coin placed on the surface of the paper, a curved line is obtained.
24. The lines drawn on the surface of a sphere, cone, or cylinder are curved lines.

Properties of straight lines:

1. Innumerable straight lines can be drawn through a point.
2. Let O be a point on the plane of the paper.
3. A, B, C, D, E, F, G  be any number of points
4. on the plane of the paper at which point O lies.
5. We join these points with O, we get innumerable
6. straight lines OA, OB, OC, OD, OE, OF, OG…….
7. Thus through any point, we can draw as many straight lines as we, please.
8. One and only one straight line can be drawn through two given points.
9. Let A and B be two given points on a paper.
10. Then only one straight line AB can be drawn through A and B.
11. There is an infinite number of points on a line.
12. An infinite number of points lie on line AB.
13. The points which line on a line are called collinear
14. Here P, A, B, C, D, and Q are collinear points because they lie on the same line.
15. Three or more points may or may not lie on a line.
16. Here the points A, B, C, D, and E lie on a line.
17. Here the points P, Q, and R do not lie on the same line.
18. If two straight lines intersect, then they must intersect at a point only.
19. Here AB and CD be two straight lines and they intersect at a point P only.
20. A curved line and a straight line intersect at more than one point.
21. Here the curved line PQ intersects the straight line AB at four points X, Y, Z, and T.
22. A straight line is not always drawn through any three given points.
23. The maximum number of straight lines that can be drawn through three non-collinear points is three.
24. Here A, B, and C are given three non-collinear points.
25. Through them, only 3 lines AB, BC, and CA can be drawn.
26. Two straight lines lying on the same plane may or may not intersect each other.
27. If they intersect, then their point of intersection is only one.
28. If they do not intersect then the straight lines are parallel.
29. The opposite edges of a table, book, and brick are parallel to each other. even when they are extended to infinity on both sides, then they are said to be parallel straight lines.

Understanding Line Segments and Rays

Two Or More Straight lines may or may not lie on the same plane:

1. Here AB, CD, and EF are three straight lines and they do not intersect when they are extended in both ways. So AB, CD, and EF are parallel to each other.
2. If three or more straight lines lying on the same plane intersect at a point i.e. three or more straight lines pass through a single point, then they are said to be concurrent straight lines.
3. Here AB, CD, EF, and GH pass through the same point O.
4. Hence these lines are concurrent.
5. Point O is called the point of concurrence.
6. If the given straight lines do not pass through a single point i.e., if they do not meet at a point, then the straight lines are said to be not concurrent.
7. Here AB, CD, EF do not meet at a point and so they are not concurrent.
8. Two or more straight lines may or may not lie on the same plane.
9. If two or more straight lines lie on the same plane they are said to be coplanar lines.
10. If two or more straight lines do not lie on the same plane, then they are said to be non-coplanar lines.
11. Two straight lines which do not lie on the same plane and they neither intersect nor parallel are called skew lines.

 

Geometry Chapter 2 Line Segments

1. A line segment is a bounded segment or a portion of a straight line by two fixed points.
2. These two fixed points are called the endpoints of the line segment.
3. Let A and B be two points on the straight line AB as shown in the figure below.
4. The straight line AB can be extended on two sides (on the left and right sides) but AB is a bounded portion or segment of the straight line.
5. This portion is bounded by points A and B.
6. So, AB is a line segment. These two points A and B are the endpoints of the line segment AB.
7. As line segment AB is bounded by the two fixed points A and B, the length of line segment AB can be measured.
8. We generally denote the straight line AB by AB and the line segment AB by AB.
9. The arrowheads are placed at the two ends of the straight line AB in order to mean that the straight line can be extended on both sides indefinitely.
10. A and B are not the actual endpoints of the straight line.
11. But in general, it is understood from the context, by AB we mean the line segment AB.
12. We name the line segment according to the name of their endpoints.
13. Let A, B, C, and D be four fixed points on the same straight line as shown in the figure below.
14. The line segments are AB, BC, CD, AC, BD, and AD.
15. Each edge of the surface of a table, almirah, length, and breadth of a room, each side of a rectangle, square, parallelogram, each side of a book, etc. are examples of line segments

 

Geometry Chapter 2 Ray

1. In a straight line, both ends of a line segment are extended indefinitely.
2. Keeping one end of a line segment fixed, the other end is extended indefinitely, then it is called a Ray.
3. In (1), AB is a line segment; in (2) the line segment AB is extended on both sides and so it is a line.
4. In (3) A is fixed and the other end B is extended indefinitely.
5. It is a ray.
6. Again, in the end, B is fixed and the other end A is extended indefinitely.
7. It is also a ray.
8. In(3) and (4), arrowheads are given on the right-hand and left-hand sides only.
9. The rays are represented by X by AB and BA respectively.
10. The fixed end of a ray is called its vertex.
11. For the ray OX, O is its vertex and the end X is extended indefinitely.

 

Geometry Chapter 2 Distinguish Among Straight Lines Line Segments Ray

 

 

Geometry Chapter 2 Properties Regarding Points Segments And Rays

1. An infinite number of straight lines can be drawn through a fixed point.
2. One and only one straight line can be drawn through two given fixed points.
3. An infinite number of curved lines can be drawn through two given fixed points.
4.  There are an infinite number of points on a straight line or a curved line.
5. Two straight lines lying on the same plane either are parallel or intersect at a point.
6. Three or more points may lie on a straight line or may not lie on a straight line.
   If the points lie on a line then they are said to be Collinear Points.
7. The maximum number of straight lines that can be drawn through, three non-collinear points is three.
8. An Infinite number of rays can be drawn through a given point.
9. we extinct then indefinitely on both sides, then they are said to be parallel to each other and the straight lines are said to be parallel straight lines.

 

Concurrent straight lines:

If three or more straight lines lying on the same plane intersect at a point i.e., three or more straight lines pass through a single point then they are said to be concurrent straight lines.

 

 

 

Geometry Chapter 1 Geometrical Concept Regarding The Formation Of Regular Solid Bodies

Geometry Chapter 1 Introduction:

1. In our daily life, we encounter and also get in contact directly with different types of bodies within and outside our houses.
2. For example chair, table, book, pen, pencil, brick, die, ball, plate, glass, candle, electric bulb, box, pipe, drum, etc.
3. These are called solid bodies.
4. Some of them have length, breadth, and thickness; some of them have length and breadth but no thickness.
5. We also observe that among all the objects around us, there are some which have consistency in shapes and others that do not have consistency in shapes.
6. We take a straight line. We find that it has only length, it has no breadth and height. Again we take a brick.
7. It has all three lengths, breadth, and height.
8. A ludo die has consistency in shape but a piece of broken glass has no consistency in shape.

WBBSE Class 6 Regular Solid Bodies Notes

Read And Learn More WBBSE Solutions For Class 6 Maths

Geometry Chapter 1 Regular and Irregular figures or objects

Regular objects:

1. The objects which have consistency in shape are called Regular objects.
2. If we place a string over the object and when it is stretched along the body of the object, the string is symmetrical with the body then the object is regular.

For example:

1. Books,
2. Football,
3. The wall of a building,
4. Bricks,
5. Boxes,
6. Pencils,
7. Scales,
8. Drums,
9. Benches,
10. Chairs,
11. Tables,
12. Plates,
13. Glass, etc. are regular objects.

Important Definitions Related to Solid Geometry

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Irregular objects:

1. The objects which do not have consistency in shape are called Irregular objects.
2. In the case of an irregular object, a stretched string when placed along its body of it is not symmetrical along the body of the object.
3. Examples of irregular objects are A piece of broken glass, a piece of tattered paper, a broken wall, coal, etc.

Regular Objects:

Simplification Maths Class 6

Irregular Objects

 

Understanding Regular Solids

Geometry Chapter 1 Solids

1. The common property of each of the objects is that each space in the atmosphere has some weight.
2. Let us consider a wooden box. It has some weight. Also, it has length, breadth, and height.
3. If we consider the upper face of a table that has only length and breadth. It has no thickness or height.
4. A single hair has only length.
5. Dimensions: The length, breadth, and thickness of a body are called the Dimensions of the body.
6. In the above examples, a wooden box has three dimensions Length, breadth, and height or thickness.
7. The upper face of a table is two-dimensional in length and breadth.

Simplification Maths Class 6

1. A hair is a unidimensional length. The shadow of a building has two dimensions
2. An object or a body having three dimensions length, breadth, and height, which occupies some space and has some weight is called a Solid.
3. Examples of solids: Chair, table, drum almirah, cube, book, brick, a ludo die, football, sphere cone, prism, pyramid, tetrahedron, etc.

 

 

Geometry Chapter 1 Different Types Of Solids

Solids are of different shapes.

Class 6 Math Solution WBBSE

Solids of different shapes are discussed below:

 

Rectangular Parallelopiped :

1. A rectangular parallelopiped is a solid bounded by three pairs of parallel surfaces. It has 6 surfaces, 8 vertices, and 12 edges.
2. The dimension of a rectangular parallelopiped is 3 and its two adjacent surfaces are at right angles to each other.
3. Bricks, books, boxes, etc. are examples of rectangular parallelopiped.

 

 

Cube :

1. If all the surfaces of a parallelopiped are squares,’ then the parallelopiped is called a Cube.
2. The length, breadth, and height of a cube are equal. It is a three-dimensional solid body.
3. It has six surfaces, eight vertices, and twelve edges.
4. A ludo die is an example of a cube
5. parallel and congruent. Each of them has five sides.
6. The is a prism, its five side faces are parallelograms.
7. The congruent end faces may be triangle, quadrilateral, or any polygon and their names are given accordingly as a triangular prism, quadrilateral prism, polygon prism, etc.
8. The straight line obtained by the intersection of any two side faces is called a side edge.
9. If the side edges are perpendicular to the end faces, then the prism is called a Right Prism.
10. If the side edges are not perpendicular to the end faces, then the prism is called Oblique Prism.

Short Questions on Solid Geometry

Pyramid

1. A solid body bounded by some plane faces is such that its base face is a polygon and the side faces are triangles having a common vertex is called a Pyramid.
2. The common vertex of a pyramid is a point in space that lies outside the base face.
3. The base face of a pyramid may be a triangle, quadrilateral, or polygon, and the side faces are some triangles having a common vertex.
4. The adjoining figure is a pyramid. Its base face is a polygon of six sides (called hexagons) ABCDEF; it is called the base of the pyramid.
5. P is the common vertex of the side faces which are six triangles. P is the vertex of the pyramid.
6. The perpendicular drawn from the vertex, P upon the base is called the height of the pyramid.
7. In the figure, PO is the height. The intersecting straight line by any two triangles in the side faces is called the side edge.
8. If the perpendicular drawn from the vertex of a pyramid upon the base passes through the center of the base, the pyramid is called a Right Pyramid
9. If the base of a right pyramid is a rectangle or a Square, then the perpendicular from the vertex upon the base passes through the point of intersection of the diagonals of the base.
10. If the base of a right pyramid is a regular polygon i.e., the sides of the base are of equal length, the right pyramid is called a Regular Right Pyramid.
11. The side faces of a right pyramid are congruent isosceles triangles.
12. A pyramid is. not a right pyramid is called a Transverse Pyramid  The perpendicular drawn from the vertex of a pyramid upon any side of the base is called the slant height.
13. In PK is the slant height of the pyramid.

Class 6 Math Solutions WBBSE English Medium

Tetrahedron:

1. If the base of a pyramid is a triangle, then it is called a Tetrahedron.
2. A tetrahedron has four vertices, four plane faces, and six side edges.
3. If the base of the tetrahedron is an equilateral triangle then it is called a Right Tetrahedron.
4. If the four faces of a tetrahedron are equal equilateral triangles, then it is called a Regular Tetrahedron.

Cone:

1. The solid generated by the revolution Of a right-angled triangle about one of the sides containing the right angle as an axis is called a Cone.
2. It is also called a right circular cone.
3. ZAOP is the right angle of the right-angled triangle AOP.
4. Revive the triangle about OP as an axis, then point A forms a circle.
5. This circle is the base of the cone and OA is its radius, ZAPB is the vertical angle and P is the vertex of the cone.
6. OP is perpendicular to the base and it is the height of the cone.
7. AP is the slant height.
8. The foremost part of the plantain flower (Mocha of banana), the sharpened end of a pencil, conical tent, etc. are examples of cones.
9. The dimension of a cone is three.

Common Questions About Geometrical Shapes

Sphere :

1. The solid generated by the revolution of a semi-circle about its diameter as an axis is called a Sphere.
2. It is bounded by a surface.
3. The radius of the semi-circle is the radius of the sphere.
4. The dimension of the sphere is three.
5. Football, cricket ball, marble, etc. are examples of spheres.AB is the diameter and OA = OB = radius of the sphere, the center of the sphere.

 

Practice Problems on Regular Solids

Cylinder:

1. The solid generated by the revolution of a rectangle about one of its sides as an axis is called a Cylinder.
2. It is called a right circular cylinder. Its dimension is three.
3. Tin caskets, drums, full pencils, water pipes, candles, etc. are examples of cylinders.
4. Considering the side AB of the rectangle ABCD as the axis, revolving the rectangle about it, CD forms a curved surface.
5. In one complete revolution, a right circular cylinder is generated.
6. The CD is the generating line and AB is the axis of the cylinder.
7. Two end faces of it are two parallel circles.
8. AB is the height and AD is the radius of the base circle.

Algebra Chapter 3 Statistical Data Its Handling And Analysis

Algebra Chapter 3 Introduction

1. We are having generally some statements like literacy in our country is below 40%; the number of child workers in our country is 60%: rupee value during January 2018 in all of India is 12 paisa with the base year of 1960.
2. We find these words in newspapers, seminars, classrooms, on radios, T.V., etc. These statements may be expressed as numerical statements in figures, which are simple, precise, meaningful, and suitable for communication.
3. These facts and figures of the population of a place, birth, death, income, expenditure, etc. are known as Statistics.
4. The word “Statistics” seems to be derived from the Latin word “Status” the Italian word “Statista” or the German word “Statistics”.
5. The word “Statistics” is used in singular or plural.
6. In the second case, it means a collection of facts i.e., figures relating to population, national income, number of public schools, and production of tea coffee in different years.
7. Percentages, averages, and coefficients derived from numerical facts are also known as statistics in the plural sense.
8. As a singular, statistics refers to various methods adopted for the collection, classification, analysis, and interpretation of figures or data.
9. So the term statistics is defined in two different senses
   1. Statistics is a collection of information in numerical terms.
   2. For example, marks obtained by the students of a class, monthly wages of the workers in a factory, numbers indicating births, deaths, and marriages in different states, etc. are statistics in this sense and they are called statistics as Statistical Data.
   3. Statistics is the science that deals with the collection, analysis, and interpretation of numerical data.
   4. In this sense, statistics is defined as statistical methods which are used for the collection, analysis, and interpretation of numerical observations.

Algebra Chapter 3 Statistical Data

Definition:

1. Data is a collection of observations expressed in numerical figures.
2. This collection may be done either by measurement or by counting.
3. The word “data” which is the plural form of the word Datum refers to a collection of observations of characteristics of individuals or items and is expressed in numerical figures obtained through measurement or counting.
4. The collection of facts or data is the very first step in an investigation.
5. The data to be collected can broadly be classified into two types:
   1. Primary Data
   2. Secondary Data.
6. Primary data refer to those data which are collected by the investigator either on his / her own or through some agency, set up for a specific purpose, directly from the field of enquiry for the first time.
7. Examples: Reserve Bank of India Bulletin (monthly)
   Coal Bulletin (monthly)
   Railway Board Annual Bulletin etc.
8. Secondary data refer to those data which have been previously collected by some other agencies, private or public for one purpose and which are usually available in journals, magazines or research publications which are used for another purpose.
9. Examples: Annual statement of the Foreign Trade
   International Labour Bulletin (monthly)
   Annual Statistical Abstract of India etc.
10. Let us suppose someone asks you “What is the total number of members of your very neighboring family?
11. Your answer is 6 How many boys and girls are there in the family?
12. Your answer is that the family has 2 girls and one boy.
13. How many boys and girls are there in the family, who have passed Madhyamik Examination?
14. Your answer is that only one girl has passed Madhyamik Examination and she is the eldest girl.
15. What class does the boy read in?
16. You say that the boy reads in class six.
17. Actually, the person was not known about your very neighboring family and now he knows something about the family through you.
18. These figures (numbers) are known to him though you are called “Datas”.
19. The collection of facts (which are previously unknown and now known) through an investigation is called Data.

WBBSE Class 6 Statistical Data Notes

Algebra Chapter 3 Collection Of Data

Actually, the data are collected by two methods:

1. Census or Complete Enumeration:

1. Census or Complete Enumeration refers to the study of all the items (or observations) in the population.
2. In this connection, the two terms namely Population and Sample are to be well acquainted.
3. A Population is defined as an aggregate or whole of objects possessing certain common characteristics.
4. So all the observations under consideration of a statistical inquiry constitute a population.
5. A sample is a selected number of objects or observations, each of which is a part of the population.
6. So a sample is defined as a part of the population selected for estimating one or more characteristics of the population.
7. So in the census method, the entire population is investigated. It requires a large number of investigators and it involves much money and time.
8. But the data are much more reliable and accurate.

2. Sample Survey:

1. A sample Survey stands for the study of some specific items drawn from the populations.
2. So in the sample survey method, the entire population is not investigated, only a part of the population is investigated.
3. It involves less time and less labor.
4. If we measure the heights of all the students of the school which form the population and then calculate the average height of the population.
5. This method is called the Census or Complete Enumeration.
6. If we select 100 students from the school and measure the height of these 100 students, which form the sample.
7. The average height of these 100 students is calculated.
8. This method is known as Sample Survey.

The following methods are used to collect the data:

1. Direct Personal observation: In this method, the investigator goes to the field of inquiry to have on-the-spot information.
2. Indirect oral investigation: In this method information is collected indirectly from persons who are acquainted with the fact under study by interviewing them.
3. Questionnaires sent through mail: In this method, information is received through the mail where a set of questions with blank spaces for answers along with the instruction are sent to the investigators with a request that they should return them duly filled in.

Real-Life Applications of Statistical Data

Algebra Chapter 3 Raw Data

Suppose you have collected the facts of the numbers of members of 10 families of a locality of Bishnupur village in the district of Bankura:

The collected facts or data are:

 

 

1. These data are called Raw Data.
2. These data are not organized.
3. Definition: When statistical data is arranged in an arbitrary manner, we call it to be raw data.
4. So statistical data may originally appear in a form, where the collected data are not organized numerically.
5. We call them raw data.

 

Algebra Chapter 3 Tally Mark

1. A tally mark is a slanted stroke (/) for counting.
2. We know that when data are arranged in an arbitrary manner, then the collected data are called raw data.
3. If the data be arranged in ascending order of magnitudes, then the presentation of the data is called Array.
4. An array does not reduce the bulk of records. To reduce the large bulk of data present them with the help of Tally marks.
5. A tally mark is an upward-slanted stroke (/) that is put against each occurrence of a value.
6. For every occurrence of the value, we put each time a tally mark.
7. When a value occurs more than four times, for every fifth occurrence a cross (\) tally mark is put which is running diagonally, across the four tally marks.
8. This facilitates the counting of tally marks at the end.
9. From the table of the previous article, we see that 4 families have a number of members 4.
10. For this, we have to put 4 tally marks as shown in the following table.

 

 

In the same way, we can construct tally marks for all other families:

 

 

The rule for putting tally marks :

1. For counting any raw data if the number of one kind of quantity is very large
2. Suppose that the number of families having 4 members is 22
3. Then put 4 tally marks, the fifth tally mark should be a cross tally mark (\) running diagonally across the 5 tally marks.
4. Putting the tally marks is shown below:
5. This facilitates the counting of tally marks and the possibility of wrong counting of tally marks will be less.

 

Practice Questions on Statistical Measures

Algebra Chapter 3 Frequency

1. Suppose the marks obtained in Mathematics by 30 students of class VI in a certain school are collected by an investigator from the official records.
2. The collected data are as follows (out of a total of 40 marks):
3. 28, 20, 30, 10, .25, 28, 17, 28, 28, 30, 9, 18, 19, 28, 30, 20, 18, 26, 17, 10, 9, 25, 17, 10, 26, 20, 17, 25, 20, 28.
4. These are the raw data collected by the investigator and the variable is the “marks obtained in Mathematics”.
5. The recorded raw data appear in a complex and arbitrary manner. One cannot fully grasp the true significance of the figures.
6. So some modification in the arrangements of the data is necessary. Therefore the data should be arranged in a definite order either ascending or descending.
7. The above data are arranged in ascending order as follows:
8. 9, 9, 10, 10, 10, 17, 17, 17, 17, 18, 18, 19, 20, 20, 20, 20, 25, 25, 25, 26, 26, 28, 28, 28, 28, 28, 28, 30, 30, 30.
9. The 30 observations are not all different, some of them are repeated. The distinct observations are known as the values of the variable.
10. The above arrangement can also be represented in the form of a table as shown below.
11. From the above table, we see that 2 students have got 9 marks
    3 students got 10 marks
    4 students got 17 marks
    2 students got 18 marks
    1 student got 19 marks
    4 students got 20 marks
    3 students got 25 marks
    2 students got 26 marks
    6 students got 28 marks and
    3 students got 30 marks.
12. Here the number of students 2, 3, 4, 2, 1, 4, 3, 2, 6, 3 are called the Frequency of the marks 9, 10, 17, 18, 19, 20, 25, 26, 28, 30 respectively (the marks obtained are the values of the variable).
13. A characteristic that can be expressed numerically is called the variable.
14. The number of times that each value of variable occurs is known as its frequency.
15. Here an important characteristic of the frequency is that the sum of the frequencies of the variable values is the total number of sample values over which the data are collected.
16. In the above table, the sum of the frequencies i.e. the total frequency = 30 which is the sample value i.e. the total number of students.

Examples of Bar Graphs and Pictographs

Algebra Chapter 3 Frequency Distribution

1. We know that when the collected data from the field of inquiry are arranged in an arbitrary manner is called Raw Data.
2. Then these raw data are arranged properly in ascending order.
3. A classification showing the different values of a variable and their respective frequencies i.e., the number of times each value occurs, side by side is called a Frequency Distribution of the values.

For this observe the following table:

 

 

 

 

1. A table containing the title: The members of a family, tally mark, frequency, is prepared.
2. This table is called the Frequency Distribution Table.
3. So the table which is prepared when the raw data are arranged with the help of tally marks and the frequency is called the Frequency Distribution Table.
4. How is the Frequency Distribution Table prepared?
5. The frequency distribution table is prepared according to the following rule:
6. Step-1: The collected raw data are arranged in ascending order of magnitude
7. Step-2: Determine the number of occurrences of the same kind of quantity or number.
   In the above table, the families containing 4 members are one kind and the families containing 5 members are another kind, etc.
8. Step-3: Construct different horizontal rooms or spaces of each kind of quantity or number in the table.
9. Step-4: Now construct different vertical columns for each sample, tally mark, frequency, etc. in the table.
10. Step-5: Insert tally marks of each of the horizontal rows in the tally column.
11. Step-6: Then insert the frequency of each of the horizontal rows in the frequency column.
12. Step-7: Give a title of the table (according to the data available) above the table.
    Then prepare a space below the frequency column and add the frequencies of the rows.
    This will give the total frequency of this whole data.
    Completing all the steps described above, you are able to prepare or construct a complete frequency distribution table.

Important Definitions Related to Data Handling

Algebra Chapter 3 Bar Graph Or Bar Chart

1. Bar Graph or Bar Chart Simply a Bar diagram is a popular method of graphical presentation of data. In graph paper, the bar diagram is constructed.
2. A bar implies a thick line having a small breadth.
3. A bar diagram consists of parallel bars, each of which has the same breadth.
4. All these bars are drawn on a common baseline and the distance between two consecutive bars is always the same.
5. The height of each bar represents the frequency of each item of the data.
6. Bar Graphs are classified into two types: Horizontal Bar graphs or charts; Vertical Bar graphs or charts.
7. In the horizontal bar diagram (or graph of chart), a set of parallel bars are drawn horizontally on a vertical baseline called the Y-axis.
8. In the vertical bar diagram (or graph or chart), a set of parallel bars are drawn vertically on a horizontal baseline called the X-axis.
9. Suppose a frequency distribution table of the dolls prepared by Binaybabu of Kumor Para in the last week is obtained as follows
10. We have to prepare a vertical bar diagram using the above data.
11. In a graph paper, we draw two perpendicular straight lines in the space available in the graph paper; one is taken as X-axis and the other is taken as the Y-axis.
12. This Vertical Bar graph is showing weekly production of dolls on different dates
13. In the above vertical bar graph, the days are represented along the horizontal axis and the number of dolls prepared to be represented along the vertical axis., 6 vertical rectangular bars are drawn which are placed at equal distances and they are standing on the horizontal axis for 6 days respectively. Observing these vertical bar graphs we can easily determine how many dolls are prepared on which days.
14. So a bar graph or bar diagram consists of rectangular parallel bars which are drawn according to the collected data, each bar has the same breadth and they are drawn on graph paper horizontally of vertically on a common baseline so that the distance between two consecutive bars is always the same.

Characteristics of Bar Diagram:

1. One variable (in the above diagram the variable is the days) is plotted along the horizontal axis i.e., along the X-axis.
2. The other variable (in the above diagram the variable is the number of dolls produced) is plotted along the vertical axis i.e., along the Y-axis.
3. The bars will be rectangular in shape.
4. The bars can be drawn either horizontally or vertically.
5. The bars are of equal widths or breadths.
6. The distance between any two consecutive bars is always the same.
7. The scale of units along the X-axis and along the Y-axis must be shown separately.
8. Generally, the scale of units both along the X-axis and along the Y-axis is expressed in terms of 1 small square division in the graph paper.
9. The height of each bar represents the value (or frequency) of each item of the data.
10. How the bar diagram be drawn?
11. The following working steps are to be noted while drawing a bar
12. Step-1: We first draw two perpendicular straight lines on the same graph paper, one is drawn horizontally to indicate the X-axis, and the other is drawn vertically to indicate the Y-axis.
    Let the X-axis be denoted by OX and the Y-axis be denoted by OY.
    The point of intersection of these two perpendicular straight lines i.e., the two axes is O which is taken as the origin.
    Straight lines can be drawn with the help of scale and pencil.
13. Step-2: Omitting one or 2 small square divisions in the graph paper from O, mark the breadth or widths of the consecutive bars with a pencil along, OX.
    Special care is to be taken here that the widths of each bar should be the same and the distance between any two consecutive bars is always the same.
14. Step-3: Similarly insert the numbers which are to be plotted along the Y-axis i.e., along OY taking the unit of scale in terms of one square division in the graph paper or any convenient unit of scale.
    Actually, the frequencies are plotted along Y-axis.
15. Step-4: Now mark the numbers involving frequencies on graph paper with a pencil.
    According to these, mark the extremities or endpoints of the bars.
16. Step-5: Draw the rectangular bars clearly.
    This completes the drawing of bar graphs.

 

 

 

Algebra Chapter 2 Concept Of Directed Numbers And Numbers Line

Algebra Chapter 2 Directed Numbers

1. The numbers which have both magnitudes and directions are called Directed Numbers.
2. For example, + 2, + 3, + 6, , – 1, – 2, – 3, etc. are directed numbers.
3. The directed number (+2) has magnitude 2 and the direction is from 0 towards the right; (+2) is a directed number.
4. Similarly (-3) is also a directed number because its magnitude is 3 and its direction is from 0 towards the left.

 

Algebra Chapter 2 Absolute Value Of A Number

Absolute Value Of A Number:

1. If we omit the sign of a directed number, then only the magnitude of the number is called its absolute value.
2. The absolute value of a directed number is a pure number which is always positive.
3. We write the absolute value of a number x as |x| (Modulus x).
4. By definition,
5. \(|x|=\left\{\begin{array}{c}
   x, \text { if } x>0 \\
   -x, \text { if } x<0 \\
   0 \text { if } x=0
   \end{array}\right.\)
6. |+2| = 2     (∵ +2 > 0)
7. |0| = 0
8. |- 3| = – (- 3) = 3        (∵ – 3 < 0) 
9.  |- 5| = – (- 5) = 5.
10. Absolute value is always positive.

WBBSE Class 6 Directed Numbers Notes

Algebra Chapter 2 Opposite Number

Opposite Number:

1. In our daily life, we use an infinite number of opposite words.
2. For example, the opposite word of Long is Short, and the opposite word of More than is Less than.
3. Similarly, the opposite of Small is Large
   1. The opposite of Income is Expenditure
   2. The opposite of Deposit is Sepnd
   3. The opposite of Up is Down
   4. The opposite of an Increase is a Decrease
   5. The opposite of East is West
   6. The opposite of North is South
   7. The opposite of the Right side is the Left side
   8. The opposite of Credit is Debit, etc.
4. If one of all these words is taken as positive, then the other will be taken as negative.
5. Generally to make out more than or increase ‘+’ (positive or plus sign or addition) sign is used and less than or decrease (negative or minus sign or subtraction sign) sign is used.
6. In the case of numbers,
   1. The positive of + 2 is – 2
   2. The opposite of -2 is + 2
   3. The opposite of – 3 is + 3
   4. The opposite of + 8 is – 8, etc.
7. So if we keep the magnitude of a directed number but take the direction of the number in the opposite sense, then we get the opposite directed number.
8. The two numbers with opposite signs whose absolute values are the same then one is called the opposite of the other number.
9. In general, we can determine the opposite number by putting a ‘+’ sign in place of the sign and by putting a sign in place of the ‘+’ sign.
10. Since 0 is not with a positive or negative sign, the opposite number of 0 is 0.

Short Questions on Directed Numbers

Algebra Chapter 2 Opposite Quality

Opposite Quality:

1. When we express any number of any value with a unit or by using any sign or symbol, then it is called a quantity.
2. For example, x, y, z,   a, b, c, ………………., p, q, r,……………., etc. are quantities.
3. So ₹ 4, -7 kg, 10 meters, x, y,………etc. are quantities.
4. Now we want to determine the opposite quantity of a quantity.
5. + ₹ 8 is a quantity.
6. Since the opposite number of (+8) is (-8).
7. So the opposite quantity of (+ ₹ 8) is (- ₹ 8).
8. Again the opposite quantity of ₹ 10 more is ₹ 10 less.
9. The opposite quantity of 20 metres North is 20 metres South
10. The opposite quantity of 12 metres above is 12 metres down
11. The opposite quantity of 14 metres East is 14 metres West
12. The opposite quantity of ₹  6 profit is ₹  6 loss.

 

Algebra Chapter 2 Synonymous Quality

Synonymous Quality:

1. The synonymous quantity of 50 metres above is (- 50) metres down
2. Similarly, the synonymous quantity of 5 metres long is (- 5) metres short
3. The synonymous quantity of ₹ 25 profit is (- ₹ 25) loss.

Algebra Chapter 2 General Rule of Addition and Subtraction of Directed Numbers

General Rule of Addition and Subtraction of Directed Numbers:

1. Rule 1: If the signs of two given directed numbers are the same, then first add the absolute values of the directed numbers and then put the same sign as that of the given directed numbers before the obtained sum.
2. Rule 2: If the signs of two given directed numbers are opposite, then first subtract the absolute values of the given directed numbers and then put the sign of that directed number whose absolute value is greater and that will be the required sum. If the absolute values are the same, then the sum will be 0.
3. Rule 3: When a given directed number is to be subtracted from another given directed number, then the result of subtraction will be obtained by the addition of the second number and the absolute value of the first directed number.

 

Algebra Chapter 2 Number Line

Number Line:

1. The numbers 1, 2, 3, 4,  etc. are positive integers and -1, -2, -3, -4,  etc. are negative integers.
2. We also take 0 (zero) as an integer (even integers).
3. We take a point O on the straight line XX’ and point O is taken as the origin or 0 (zero).
4. With O as the centre, we have to place the respective positive integers at equal intervals on the right side of O on the line OX and the respective negative integers are also to be placed at equal intervals on the left side of O on the line OX’.
5. Line XX’ is called the Number Line.
6. Now, place the points A, B, C, D, etc. on the right side of 0 on the XX’ line
7. such that OA = AB = BC = CD =  OA is taken as the. unit length.
8. Denote the points A, B, C, D, etc. by the positive integers 1, 2, 3, 4, …….
9. Again, in the same way, point A’ is placed on the left side of 0 such that OA’ = OA. A’ is denoted by (- 1).
10. Now place the points B’, C’, D’,  on OX’ such that OA’ = A’B’ = B’C’ = C’D’ =……….., and they are denoted by – 2, – 3, – 4, …… etc., the negative integers.

Properties Of Number Line:

1. The number Line is a straight line
2. A point O is marked on the number line, generally, it is placed in the middle of the number line and it is taken as 0 (zero)
3. All the positive numbers are written on the right side of the O
4. All the negative numbers are written on the left side of the O
5. The number 0 (zero) is neither positive nor negative
6. As we proceed from O towards the right, the magnitudes of the numbers are increasing
7. As we proceed from O towards the left, the magnitudes of the numbers are decreasing
8. The value of any number towards the right of a number on the number line is always greater than the number and the value of any number towards the left of a number on the number line is less than the number
9. Two numbers which are indicated by two equidistant points from O on both sides of it on the number line are equal in magnitude (absolute values) but with opposite signs.
10. These numbers are called opposite numbers.
11. The density of real numbers on the number line is so large that between any two numbers on the number line within a small distance, there exist an infinite number of real numbers.

 

Algebra Chapter 2 Addition And Subtraction Of Directed Numbers With The Help Of Number Line

 

You have already learned how to place a directed number on the number line. Now we shall discuss the addition and subtraction of the directed numbers with the help of a number line.

Addition with the help of a number line :

The addition of directed numbers may be of four types:

1. Addition of a positive number to a positive number ;
2. Addition of a negative number to a positive number ;
3. Addition of a positive number to a negative number ;
4. Addition of a negative number to a negative number.

Common Questions About Positive and Negative Numbers

1. Addition of a positive number to a positive number :

Example : (+4) + (+6) =?

Here we shall find the sum of two directed numbers (+4) and (+6).

Place 0 which indicates the number 0 (zero) on the number line XX’ as shown in the figure.

 

 

Moving 4 units to the right of O (i.e., in the positive direction), we get the place of the directed number (+4).

Let this point be denoted by A.

Then move further 6 units in the same direction. So total units of movement from O is 10 units in the positive direction and this is the position of the directed number (+10).

Let this point be denoted by B.

OA = + 4 and AB = + 16

(+ 4) + (+ 6) = OA + AB = OB (according to the figure) = + 10

∴ (+ 4) + (+ 6) = (+ 10).

So the required sum = +10.

(+4) + (+6) = +10.

2. Addition of a negative number to a positive number :

Example : (+ 7) + (- 3) = ?

Here we have to add a negative directed number (- 3) to a positive directed number (+ 7).

 

Place O which indicates the number 0 (zero) on the number line XX’ as shown in the figure.

Moving 7 units to the right of O (i.e., in the positive direction), we get the position of the directed number (+ 7).

Let this point be denoted by A.

hen moving 3 units to the left of this point A (here we have to move back 3 units because the addition of (- 3) implies subtraction of (+ 3) or in other words come back 3 units in the negative direction), we get a point B (in the figure) which is 4 units from O in the positive direction and this is the position of the directed number (+4).

(+ 7) + (- 3) = (+ 4) = 4.

(+ 7) + (- 3) = OA + (- AB)

= OA – AB = OB

= (+ 4) = 4.

So the required sum = 4.

(+ 7) + (- 3) = 4.

Practice Questions on Directed Numbers and Number Line

3. Addition of a positive number to a negative number

Example : (- 7) + (+ 4) =?

Here we have to add a positive directed number (+4) to a negative directed number (-7).

 

Place O which indicates the number 0 (zero) on the number line XX’ as shown in the figure.

Moving 7 units to the left of O (i.e., the negative direction), we get the position of the directed number (- 7).

Let this point be denoted by A.

Then we move back 4 units to the right of this point (Here we have to move back 4 units because the addition of (+ 4) implies the movement of 4 units towards the right i.e., in the positive direction), and we get a point which is 3 units from O in the negative direction.

This is the position of the directed number (- 3). Let this point.

∴ (- 7) + (+ 4) = [- (7 – 4)]

= (- 3)

= – 3.

(- 7) + (+ 4) = – OA + AB

= – (OA – AB)

= – OB = – 3

The required sum = – 3.

(- 7) + (+ 4) = – 3.

4. Addition of a negative number to a negative number :

Example : (- 3) + (- 2) = ?

 

 

Place O which indicates the number 0 (zero) on the number line XX’ as shown in the figure.

Moving 3 units to the left of O (i.e., in the negative direction), we get the position of the directed number (- 3).

Let this point be denoted by A.

Then we move 2 units from point A on the number line in the same direction i.e., in the negative direction.

Let this point be denoted by point B. So total units of movement from O is 5 in the negative direction and this is the position of the directed number (- 5).

(-3) +(-2) = [-(3.+ 2)]

= (-5)

=-5

∴ (- 3) + (- 2) = – [OA + AB]

= (- OB)

= (- 5)

= – 5

The required sum = – 5.

(- 3) + (- 2) = – 5

Conceptual Questions on Number Line Operations

2. Subtraction of Directed Numbers

The subtraction of directed numbers may be of 4 types :

1. Subtraction of a positive number from a positive number
2. Subtraction of a negative number from a positive number
3. Subtraction of a positive number from a negative number
4. Subtraction of a negative number from a negative number.

Since the operation of subtraction is a reverse operation of addition, the operation of the subtraction can be performed through the operation of addition by changing the sign only.

1. Subtraction of a positive number from a positive number :

Example : (+ 10) – (+ 3) = ?

Solution : (+ 10) – (+ 3) = (+ 10) + (- 3)

[∵ The operation of subtraction is a reverse operation of addition and opposite number of (+ 3) = (- 3)]

= [+ (10 – 3)]

= (+ 7)

= 7.

 

(+ 10) – (+ 3) = 7.

 

2. Subtraction of a negative number from a positive number:

Example : (+ 3) – (- 9) =?

Solution : (+ 3) – (- 9) = (+ 3) + (+ 9)

(∵ Opposite number of – 9 = + 9)

= [+ (3 + 9)]

= (+ 12)

= 12.

 

(+ 3) – (- 9) =12.

Examples of Operations with Directed Numbers

3. Subtraction of a positive number from a negative number :

Example : (- 5) – (+ 7) = ?

Solution : (- 5) – (+ 7)

= (- 5) + (- 7)     [∵ opposite number of + 7 is – 7].

= [- (5 + 7)]    (∵ according to the rule of addition)

= (-12)

= – 12

 

(- 5) – (+ 7) = – 12

 

4. Subtraction of a negative number from a negative number

Example : (- 10) – (- 8) =?

Solution : (- 10) – (- 8) = (- 10) + (+ 8)     [∵ opposite number of – 8 is + 8]

= [- (10 – 8)]      [∵ according to the rule of addition]

= (-2)

= -2.

 

(- 10) – (- 8) = -2.

 

Algebra Chapter 2 Natural Numbers Positive Integers Negative Integers Integers

1. The numbers which are positive integers i.e., the numbers 1, 2, 3, 4, …… up to infinity is called Natural Numbers.
2. The natural numbers consecutively are placed equidistantly on the number line to the right side of the number 0 (zero).
3. The integers which are greater than zero (0) are called Positive Integers.
4. The positive integers are 1, 2, 3, to infinity.
5. In fact, natural numbers and positive integers are the same.
6. So the positive integers are also placed on the number line after the number zero.
7. The integers which are less or smaller than zero (0) are called Negative Integers.
8. The negative integers are written by putting a minus sign (-) towards the left side of the positive integers.
9. So the negative integers are:
10. -∞ (minus infinity), – 3, – 2, – 1.
11. The negative integers start from (-1) and are going towards the left side of zero up to infinity.
12. These numbers are placed on the number line towards the left side of the number 0 (zero).
13. The number 0 (zero) is neither positive nor negative.
14. The negative integers, zero, and positive integers together are called Integers.
15. All the natural numbers including zero together are also Integers.
16. Zero (0) is called an even integer.

 

Important Definitions Related to Directed Numbers

Algebra Chapter 2 Verification of Associative law and Commutative law of addition

Associative Law of Addition :

1. If a, b, and c are integers (positive or negative), then (a + b) + c = a + (b + c).
2. This is the Associative law of addition.
3. For example (+ 4), (- 5) and (+ 2) are 3 integers, then we get,
4. {(+ 4) + (- 5)} + (+ 2)
   = (4 – 5) + 2
   = – 1 + 2
   = 1 ; and
   (+ 4) + {(- 5) + (+ 2)}
   = (+ 4) + (- 5 + 2)
   = 4 + (- 3)
   = 4 – 3
   = 1.
   ∴ {(+ 4) + (- 5)} + (+ 2) = (+ 4) + {(- 5) + (+ 2)}
5. So, (+ 4), (- 5) and (+ 2) be 3 given integers and they obey the associative law of addition.
6. In the same way, we can prove that any three integers obey the additive associative law.

Commutative law of addition:

1. If a and b be any two integers, then a + b = b + a.
2. This is the commutative law of addition.
3. For example, (+ 4) and (- 7) be two integers.
4. Then we get, (+ 4) + (- 7) = 4 – 7 = – 3 and
   (- 7) + (+ 4)
   = – 7 + 4
   = – 3.
   ∴  (+ 4) + (- 7) = (- 7) + (+ 4).
5. So the given two integers obey the commutative law of addition.
6. In the same way, we can prove that any two integers always obey the commutative law of addition.

 

 

 

Algebra Chapter 1 Concept Of Algebraic Variables Or Quantities Or Symbols

Algebra Chapter 1 What Is Constant

Constant:

1. In mathematics, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 these ten symbols are called digits.
2. Arranging these digits in different ways we get an infinite number of numbers.
3. The magnitudes of these numbers are Definite.
4. In mathematics wherever they are used, the magnitude of these numbers never changes.
5. For example, the magnitude of the digit 2, anywhere in mathematics it is used, its value is the same in the whole world, never its value is changed.
6. So the number 2 is a constant.
7. These types of numbers are called constants.
8. So all the mathematical symbols each of whose magnitudes are always the same and definite and never value changed, are called constants.
9. For example, 1, 2, 3, 4, etc. are constants.
10. Using any mathematical symbol before or after or above or below a constant, we can change the direction of any other measure of the mathematical symbol but its magnitude or absolute value will always remain the same.
11. For example, +2, – 2, 2⁺, 2^–, —>2, etc. represent the different measures of 2 but its magnitude will be the same as 2.
12. This definite magnitude of the number is called its absolute value and the absolute value of any number is always positive.
13. For example | + 2 | = 2 ; | – 2 | = 2.
14. The definition of the absolute value of any number x is | x | which is
    | x | = x if x > 0
    = – x if x < 0
    = 0 if x = 0

WBBSE Class 6 Algebraic Variables Notes

Algebra Chapter 1 What Is Variable

Variable:

1. The variable is a quantity whose value is not fixed or definite and which accepts different values in different mathematical problems.
2. For example: In the mathematical problem 2x = 4, 2 and 4 are constants but x is a variable.
3. Here x = 2.
4. Again in the mathematical problem x + 1 = 0, x = – 1.
5. In any mathematical problem, x can take any value, for this reason, x is called a variable.
6. In general, we use the English alphabets a, b, c, x, y, z, etc. to express the variables.
7. In the mathematical problem 2n + 2 = 0, n is a variable quantity.
8. The variables obey the rules of mathematical operations like constants or real numbers.
9. For example—
   1. Associative law of addition:
   2. If x, y, z be any 3 variables, then x + (y + z) = (x + y) + z
   3. Commutative law of addition:
   4. If x, and y be any two variables, then x + y = y + x.
   5. Associative law of multiplication:
   6. If x, y, and z be any 3 variables, then x. (y.z) = (x.y).z.
   7. Commutative law of multiplication:
   8. If x, y be any two variables, then x x y = y x x.
   9. Distributive law:
   10. If x, y, z be any 3 variables, then x.(y + z) – x.y + x.z.

Understanding Algebraic Symbols for Kids

Algebra Chapter 1 Use Of Variables

Use Of Variables:

1. In an algebraical problem, for any unknown quantity or number, we use a variable.
2. For example “The present age of the father is twice that of the son”—In this type of mathematical problem, we take the present age of the son or father as a variable.
3. Let the present age of the son be x
4. We use a variable to express a general quantity which is denoted for different values of a quantity.
5. For example—to express the quantities 2¹, 2², 2³, etc. a general quantity, we write 2ⁿ, n = 1, 2, 3,…………….., where n is taken as a variable.
6. The branch of mathematics in which we can solve the problems of mathematics using English alphabetic symbols is called Algebra.
7. A big branch of mathematics Algebra is formed based on different use of variables.
8. Algebra is a more generalized form of the problems of Arithmetic.
9. For example, in Arithmetic,
   (2 + 3)² = 2² + 2.2.3 + 3²
   (3 + 4)² = 3² + 2.3.4. + 4⁴
   i.e., the square of the sum of two numbers = The square of the first number + 2 x the first number x the second number + the square of the second number.
10. This formula in arithmetic can be written through the variables as (a + b)² = a² + 2ab + b²
11. This is the algebraic formula which is a more simplified form.
12. There is a fantastic use of variables in modem mathematics.
13. For these reasons he or she who will learn the use of variables correctly and accurately would be able to show his or her credit in solving the mathematical problem.

Important Definitions Related to Algebra

Algebra Chapter 1 Algebraic Sign And Symbol

Algebraic Sign And Symbol:

1. ‘+’: Addition sign: For example, x + y, where x and y are two variables.
2. ‘-‘: Subtraction sign: For example, x – y.
3. ‘x’: Multiplication sign: For example, x x y.
4. ‘÷’: Division sign : For example, x -f y or ~.
5. ‘=’: Equal sign : For example, x = y i.e., the values of .x and y are same.
6. ‘>’: Greater sign: For example, x > y means that the value of x is greater than the value of y (or simply x is greater than y).
7. ‘<’: Less (smaller sign) : For example, x < y means that the value of * is less than the value of y (or simply x is less than y).
8. ‘≥’: Greater than or equal sign: For example, a: > y means that the value of x is greater than the value of y or x is equal to y.
9. ‘≤’: Less than or equal sign: For example, x < y means that the value of x is less than the value of y or x is equal to y.
10. ‘>≠’: Not greater than a sign: For example, x > y means that the value of x is not greater than the value of y.
11. ‘<≠’: Not less than a sign: For example, x <≠ y means that the value of x is not less than the value of y.
12. ‘≠’: Not equal to sign: for example, x * y means that the value of x is not equal to the value of y.
13. ‘∼’: Difference sign: For example, x ~ y means that the smaller number between x and y is to be subtracted from the greater number.
    x ~ y means that
    1. x – y if x > y
    2. y – x if y > x.
14. ‘≡’: Equivalent to sign: For example, x = y means that x is equivalent to y.

 

 

Arithmetic Chapter 13 Fundamental Concept of Ratio And Proportion

Arithmetic Chapter 13 What is meant by the ratio

1. In our daily life, we divide different materials in some definite ratio. For example, let your father divide some amount of money between you and your brother in the ratio 2 : 3. So what actually is meant by this 2 : 3? In the present chapter, we shall discuss a fundamental theory regarding this.
2. In general, Ratio is the comparative relationship of two quantities of the same kind expressed in the same unit. The ratio is without any unit.
3. While comparing similar kinds of quantities, sometimes we take the difference of them i.e., one quantity is less or more than the other by how much, or sometimes, we find how many times or parts is one of the other quantities and we always express it in ratio. Generally this last type of comparison between two quantities we call ratio.
4. So by the ratio 2 : 3, we mean that if the first number is 2, then the second number will be 3. Similarly, if the first quantity is Rs. 2 or 2 gm or 2 kg or 2 km or
5. liters or 2 m etc., then the second quantity will be Rs 3 or 3 gm or 3 kg or 3 km or
6. liters of 3 m etc.

WBBSE Class 6 Ratio and Proportion Notes

Example:

1. If the ratio of Ram’s money and Shyam’s money is 4: 5, then it means that Ram has Rs 4 or a multiple of Rs 4, then Shyam has Rs 5 or the same multiple of Rs. 5.
2. If you have 7 balls and your brother has 9 balls, then the ratio of balls = is 7:9 (writing your ball numbers first). If we write your brother’s ball numbers first then the ratio becomes 9: 7.
3. If the first quantity of the two quantities is and the second quantity is b, then their ratio will be-a:
4. But its converse may not be correct i.e. if the ratio of two quantities is a: b then it is not necessary that the first quantity is a and the second quantity is b.
5. Now a question arises if a ratio is 2 : 3, then what are the values of the first quantity and how many are its numbers? Similarly, what are the values of the second quantity and how many are its numbers?
6. Now the first quantity may be:2 x 1 = 2, 2 x 2 = ,2 x 3 = 6, 2 x 4 = 8   etc.
7. Under this conditions the second quantity will be respectively 3×1=3, 3×2 = 6, 3 x 3 = 9, 3 x 4 = 12,  etc.
8. Similarly, the 1st quantity may be any one of the numbers 2/1, 2/2, 2/3, 2/4, 2/5, 2/6, …………., etc.
9. Under this condition, the second quantity may be any one of the numbers 3/1, 3/2, 3/3, 3/4, 3/5, 3/6, ……., etc.
10. Again the first quantity 2 is a real number. Now the question is that “Is the first number any one of variables values a or b or c or y or z?
11. Then the answer is that it is sure that it may be. So what is the condition?
12. The condition is that if the first quantity be a or x, then the second quantity will be \(\frac{3}{2} \times a=\frac{3 a}{2}\)
13. or, \(\frac{3}{2} \times x=\frac{3 x}{2}\).
14. Because, \(a: \frac{3 a}{2}=1: \frac{3}{2}=2: 3\)
15. or, \(x: \frac{3 x}{2}=1: \frac{3}{2}=2: 3\)
16. So the first quantity may be any real number of any quantity and its number is infinite; the second quantity only follows the first quantity under the condition that its number is also infinite.
17. Here the first quantity is only open and the second quantity is closed under the condition.
18. Again the same thing conversely holds for the second quantity and in that case, the second quantity is open and the first quantity is closed under the condition.
19. So the ratio is a multiplicative relation with respect to a condition of an indefinite number or quantity with another number or quantity of the same kind.

Understanding Ratio and Proportion

Arithmetic Chapter 13 Characteristics of Ratio

1. The ratio is the quotient of two quantities of the same kind expressed in the same unit.
2. The ratio may be between two or among more quantities of the same kind expressed in the same unit.
3. When the ratio is between two quantities of the same class expressed in the same unit, the first quantity is called the Antecedent and the second quantity is called the consequent.
4. If the ratio is among more than two quantities or numbers, then the quantities or numbers are called respectively the first element, second element, third element, fourth element, , etc.
5. The value of any given ratio will be different when any real number is added to or subtracted from the elements of the given ratio.
6. For example, 2 : 3 ≠ (2 + 1) : (3 + 1) or, 3 : 4
7. 2 : 3 ≠ (2 – 1) : (3 – 1) or, 1 : 2.
8. The value of any given ratio will remain unchanged when the elements of the given ratio is multiplied or divided by any real numbers other than zero.
9. For example, 2 : 3 = (2 x 2) : (3 x 2) = (2 x 3 : 3 x 2) = ………………. etc.
10. 2: 3 = 2/2 : 3/2
    = 2/3 : 3/2
    = 2/4 : 3/4 = ……………. etc.
11. Generally, the ratio may be between two or among more than two numbers or quantities of the same kind or class while expressing in the same unit only.
12. The original value or quantity of the elements in a given ratio may or may not be indicated by the individual values of the ratios.
13. For example, if the given ratio is 2 : 3, then the values of the Antecedent and Consequent may or may not be 2 and 3.
14. A given ratio is only a pure number, it has no unit.
15. A given ratio can be expressed into a vulgar fraction and a vulgar fraction can also be expressed into a ratio.
16. A given ratio can equivalently be expressed into another ratio.
17. For example, the ratio 2 : 3 is also equivalently expressed into 10: 15.

 

Arithmetic Chapter 13 Conversion Of A Given Ratio Into A Vulgar Fraction And A Vulgar Fraction Into A Ratio

1. We express a given ration as a vulgar fraction.
2. suppose we are a ratio 4: 7
3. If we express it into a vulgar fraction, then it will be 4/7.
4. So we have \(4: 7=\frac{4}{7}\)
5. Similarly, we get, \(5: 6=\frac{5}{6}\)
   \(8: 9=\frac{8}{9}\)
   \(a: b=\frac{a}{b}\)
   \(x: y=\frac{x}{y}\),………. etc.
6. Conversely, \(\frac{1}{2}=1: 2\),
   \(\frac{2}{3}=2: 3\)
   \(\frac{7}{8}=7: 8\)
   \(\frac{p}{q}=p: q\)
   \(\frac{m}{n}=m: n\) etc.
7. Therefore, we can express a vulgar fraction into a ratio.

Short Questions on Ratios

Arithmetic Chapter 13 To Express in Lowest Form Of The Rato

1. If it is possible to divide each of the antecedent and consequent of a given ratio by an integral real number (other than zero), then we say that the ratio is not in the lowest form.
2. Then we divide each of the antecedent and consequent by that integral real number, then the ratio formed is said to be expressed in the lowest form.
3. For example 12: 15 = 4: 5 (Divide by 3)
   20: 25 = 4: 5 (Divide by 5)
   27: 30 = 9: 10 (divide by 3)
   a²: ab = a: b (Divide by a)
   xy : xz = y : z (Divide by x)
   a²bc: ab²c = a: b (Divide by abc)

Important Definitions Related to Ratios

Arithmetic Chapter 13 Classification Of Ratio

1. The different types of ratios are given below
2. Simple Ratio: The ratio of two quantities of the same kind expressed in the same unit is called a Simple Ratio.
3. So the ratio whose two terms (antecedent and consequent) are simple quantities of the same kind is called Simple Ratio.
4. For example ₹ 4: ₹ 9 = 4: 9
   5 m: 6 m = 5: 6
   7 km: 11 km = 7:11 etc.
5. Simple Ratios are of three types:
   1. The ratio of greater inequality
   2. Ratio of lesser inequality, and
   3. Ratio of equality.
6. The ratio of greater inequality: A ratio in which the antecedent is greater than the consequent is called a ratio of greater inequality.
7. The ratio a: b is said to be a ratio of greater inequality if a > b.
8. For example 9: 8 (9 > 8); 13: 7 (13 > 7) etc. are the ratio of greater inequality.
9. The ratio of lesser inequality: A ratio in which the antecedent is less than the consequent is called a ratio of lesser inequality.
10. The ratio a: b is said to be a ratio of lesser inequality if a < b.
11. For example 6: 11 (6 < 11), 8: 15 (8 < 15), etc. are the ratio of lesser inequality.
12. The ratio of equality: A ratio in which the antecedent and consequent are equal to each other is called a ratio of equality.
13. The ratio a: b is said to be a ratio of equality if a = b.
14. For example 4: 4, 7: 7, 10: 10, etc. are the ratio of equality.
15. Compound ratio: The ratio whose antecedent is obtained by the continued production of the antecedents of the given two or more ratios and the consequent is obtained by the continued production of the consequents of the aforesaid ratios is called the compound ratio of the given ratios.
16. For example 4: 5 and 6: 7 is 4 x 6: 5 x 7 = 24: 35.
17. Similarly, the compound ratio of 2 : 3, 4: 7, 8: 11, 10: 13 is (2 x 4 x 8 x 10): (3 x 7 x 11 x 13) = 640: 3003.
18. The compound ratio of. a: x, b: y, c: z is (a x b x c) : (x x y x z) = abc : xyz.
19. Inverse ratio: If two ratios are such that the antecedent and consequent of one are respectively the consequent and antecedent of the other, then they are said to be the Inverse ratio of one another.
20. For example, the inverse ratio of 6: 7 is 7: 6; the inverse ratio of 10: 11 is 11: 10; the inverse ratio of a: b is b: a; the inverse ratio is x: y is y: x, etc.
21. Duplicate Ratio: A ratio, which is obtained in such a way that its antecedent is obtained by the square of the antecedent of a given ratio and its consequent is obtained by the square of the consequent of the given ratio, is called the Duplicate ratio of the given ratio.
22. A ratio x²: y² is the duplicate ratio of the ratio x: y.
23. For example the duplicate ratio of 2: 5 is 2²: 5 ² = 4: 25
24. The duplicate ratio of 5: 7 is 5²: 7² = 25: 49
25. The duplicate ratio of a: b is a²: b² etc.
26. Sub-duplicate Ratio: A ratio, whose antecedent and consequent are obtained by the square root of the antecedent and consequent respectively of a given ratio, is called the sub-duplicate ratio of the given ratio.
27. The sub-duplicate ratio of the ratio x: y is √x: √y
28. For example The sub-duplicate ratio of 4: 9 = √4: √9 = 2 : 3
29. The sub-duplicate ratio of 16:25= √16 : √25 =4:5
30. The sub-duplicate ratio of a²: b² = a: b etc.
31. Triplicate Ratio: A ratio whose antecedent and consequent are obtained by the cube of the antecedent and consequent respectively of a given ratio, is called the Triplicate ratio of the given ratio.
32. The triplicate ratio of x: y is x³: y³.
33. For example the triplicate ratio of 3: 4 is 3³: 4³ = 27: 64
34. The triplicate ratio of 1: 7 is 1³: 7³ = 1: 343
35. The triplicate ratio of a: b is a³: b³ etc.
36. Sub-triplicate Ratio: A ratio, whose antecedent and consequent are obtained by the cube roots of the antecedent and consequent respectively of a given ratio, is called the sub-triplicate ratio of the given ratio.
37. The sub-triplicate ratio of x: is 3√x: 3√y.
38. For example the sub-triplicate ratio of 1: 27 is 3√1; 3√27 =1:3
39. The sub-triplicate ratio of 8: 125 is 3√8: 3√125 = 2: 5
40. The sub-triplicate ratio of x³ : y³ is 3√x³ : 3√y³= x : y; etc.

 

Arithmetic Chapter 13 Proportion

1. When the values of two ratios, expressed in the lowest term, are equal, they are said to be in proportion and one is called proportional to the other.
2. For example 4: 6 and 10: 15 be two given ratios and they are equal in their lowest terms.
3. These two ratios are said to be in proportion and one is called proportional to the other and we write as 4: 6:: 10: 15.
4. Similarly, 8: 12: : 14: 21; T 10: ₹ 15:: 6 meters: 9 meters, etc. are examples of proportions.
5. When four quantities are so related that the ratio between the first and the second quantities is equal to the ratio between the third and the fourth quantities, then the four quantities are said to be in proportion.
6. Here it is necessary to be mentioned that the first and second quantities are of the same kind, the third and fourth quantities may not be the previous same kind but may be different types of same kind quantities.
7. One important formula :
8. If four quantities are in proportion, then we have
9. So,
   1. First quality = \(=\frac{Second quantity \times Third quantity}{Fourth quantity}\)
   2. Second quantity = \(=\frac{First quantity \times Fourth quantity}{Third quantity}\)
   3. Third quantity = \(=\frac{First quantity \times Fourth quantity}{Second quantity}\)
   4. Fourth quantity = \(=\frac{Second quantity \times Third quantity}{First quantity}\)
10. If any three terms of a proportion of four terms are known then the remaining term(the unknown term) can be determined by the above rule

Examples of Real-Life Applications of Ratios

Arithmetic Chapter 13 Different Types Of Proportion

1. There are three types of proportions:
2. Simple proportion (or Direct proportion):
   1. Definition: Two quantities are so mutually related that the increase (or decrease) of the values of one results in the increase (or decrease) of the values of the other and the ratio between the two values of the first quantity is equal to the ratio between the corresponding two values of the second quantity
   2. Then it is said that the two ratios are in a simple proportion (or Direct proportion).
3. Inverse (or Reciprocal) Proportion:
   1. Two mutually related quantities are such that the increase (or decrease) of the values of one results in the decrease (or increase) of the values of the other, then the ratio between the two values of the first quantity is equal to the inverse or reciprocal ratio between the values of the second quantity, then it is said that either of the two ratios is in inverse (or reciprocal) proportion to the other.
   2. Definition: If two ratios are such that one ratio is equal to the reciprocal of the other, then either of them is said to be in inverse or reciprocal proportion of the other.
4. Continued Proportion:
   1. Definition: If three quantities are such that the first quantity: Second quantity = Second quantity: Third quantity, then they are said to be in continued proportion.
   2. The second quantity is called the Mean proportional between the first and third.
   3. Here the second quantity or the Mean proportional = √(First quantity x Third quantity )
   4. First quantity = (Second quantity)² ÷ third quantity;
   5. Third quantity = (Second quantity)² ÷ first quantity.

 

 

Arithmetic Chapter 12 Measurement Of Time

Arithmetic Chapter 12 Different units of measurement of time

1.

1. Day: The time is taken by the Earth to make one complete rotation around
2. its own axis is called one day.
3. Hour: If one day is divided into 24 equal parts, then each part is called one
4. hour.
5. Minute: When one hour is divided into 60 equal parts, then one part is called one minute.
6. Second: When one minute is divided into 60 equal parts, then one part is called one second.
7. So 1 day = 24 hours
8. 1 hour = 60 minutes
   1 minute = 60 seconds

Important Definitions Related to Time Measurement

2.

1. Week: 7 days make one week.
2. Date: 7 days in a week are denoted by 7 dates.
3. The dates are
4. Monday
   Tuesday
   Wednesday
   Thursday
   Friday
   Saturday
   and Sunday.
5. Fortnight: 2 weeks together is called a Fortnight. In Bengali 15 days together are called a Fortnight.
6. The number of fortnights two
   1. The bright fortnight (Sukla fortnight)
   2. The dark fortnight (Krishna Fortnight).
7. Month: Generally a month is said to have 30 days, although all the months of a year do not have 30 days.
8. The total number of days in different months is given below
9. January = 31 days
   February = 28 days (In a leap-year, the month of February will have 29 days)
   March = 31 days
   April = 30 days
   May = 31 days
   June = 30 days
   July = 31 days
   August = 31 days
   September = 30 days
   October = 31 days
   November = 30 days
   December = 31 days.
10. Year: Usually a year contains 12 months or 365 days.
11. But a Leap-Year contains 366 days.
12. A period: of 12 years altogether is called a period.
13. So, 7 days = 1 week 1 Fortnight (Paksha) = 2 weeks (In Bengali 15 days)
    1 month = 30 days
    1 year = 365 days (General year) = 366 days (Leap-year)
    1 month = 2 Fortnights
    1 year =12 months
    1 period (Years) =12 years
    1 century =100 years.
14. Leap-year: If the number denoting a year is divisible by 4, then the year is known as Leap-year.
15. For example 1908, 1912, and 1916, years, etc. are Leap years,
16. A leap year has 366 days and the month of February has 29 days.
17. But all the years denoting the century are not leap years.
18. A century year will be a leap year if the number denoting the year is divisible by 400.
19. The years 1624, 1736, 1984, and 2004 are leap years as these numbers are divisible by 4.
20. Again the century years 1600, 2000, and 1200 are leap-years as these numbers are divisible by 400 whereas the century years 1700, 1800, and 1900 are not leap-years as they are not divisible by 400.
21. Century-year: 100 years altogether is known as a Century year.
22. For example Nineteenth century (from 1801 year to 1900 year)
23. The Twentieth century (From 1901. year to 2000 year)
24. Twenty oneth century (From 2001 year to 2100 year) etc.

WBBSE Class 6 Measurement of Time Notes

Arithmetic Chapter 12 Addition, Subtraction Multiplication and Division in the Measurement of Time

 

1. In our daily life, sometimes we are to do addition, subtraction, multiplication, and division two or more times.
2. In this regard, we shall discuss illustrative examples.
3. In general, following the rules discussed below, we add, subtract, multiply, and divide the problems regarding times.
4. Rule 1. In the case of a second or minute 60 is taken as a unit; i.e., in two or more mathematical operations about seconds or minutes which are more than 60 seconds or 60 minutes, then we write the remaining seconds or minutes taking every 60 seconds or 60 minutes as units.
5. For example, 52 seconds + 44 seconds = 96 seconds. We take 60 seconds = 1 minute, so there is the remaining 96 – 60 = 36 seconds.
6. ∴ The addition can be written as 52 seconds + 44 seconds = 1 minute 36 seconds.
7. Rule 2. In the case of hours and days, 24 hours is taken as a unit i.e., 24 hours = 1 day.
8. Rule  3. In the case of days and weeks, 7 days are taken as a unit, i.e., 7 days = 1 week.
9. Rule 4. In the case of weeks and years, 52 weeks is taken as the unit, i.e., 52 weeks = 1 year.
10. Rule 5. In the case of days and months, 30 days are taken as a unit, i.e., 30 days = 1 month.
11. Rule 6. In the case of days and years, 365 days are taken as a unit, i.e., 365 days = 1 year.
12. Rule 7. In the case of months and years, 12 months is taken as a unit, i.e., 12 months = 1 year.
13. Rule 8. In the case of years and periods, 12 years is taken as a unit, i.e., 12 years = 1 period.

Understanding Time Measurement

Arithmetic Chapter 12  Day of a particular date

1. If we know the day of a particular date of any year, then we can easily calculate the day of any date preceding or following year.
2. First, we shall calculate the number of days between the two given dates including only one day of these two given dates.
3. Then divide the number of days by 7.
4. If there is no remainder, then the day of the required date will be the same day as the given date.
5. If the remainder is 1, then the day of the required date will be the first day after or before the given day in the case of the following year or the preceding year.
6. Again if the remainder is 2, then the day of the required date will be the second day after or before the given day in case of the following or preceding year and so on.

 

Arithmetic Chapter 11 Square Root

Arithmetic Chapter 11 Square and square root :

Definition:

1. If a number is multiplied by the same number then the product obtained is called the square of that number and the number is called the square root of the product.
2. For example, when 2 is multiplied by 2, the product is 4 and it is written as 2 x 2 = 4.
3. Then 4 is the square of 2 and the square root of 4 is 2.
4. In order to indicate the square of a number, a small 2 is written on the right top of that number.
5. For example, the square of 4 is written as 4² = 16 and the square of 5 is written as 5² = 25.
6. Again, in order to express the square root of a number; we write the sign √ on the left-hand side of that number.
7. For example; the square root of 4, we write it as √4 i.e., the square root of 4 = √4, and similarly, the square root of 9 = √9.
8. Remember, √4 = 2 and √9 = 3.
9. Mathematically, the square of any number a is a x a = a^2, and the square root of a² is √a=a

WBBSE Class 6 Square Root Notes

Arithmetic Chapter 11 Mathematical Significance of square and square root

1. All of you know that if you add 2 twice then the result of the addition is 4.
2. If you add 3 thrice then the result of addition is 9,
3. If you 4, four times then the result of addition is 16.
4. ∴ The square of 2 is 4, the square of 3 is 9, and the square of 4 is 16.
5. So, the meaning of doing the square of any number is to perform a mathematical process where a number is added up to at a number of times.
6. Again 2 can be subtracted from 4, two times, and the result of the final subtraction is 0; 3 can be subtracted from 9, three times, and the result of-final subtraction is 0; 4 can be subtracted from 16, four times and the result Of final subtraction is 0.
7. The square root of 4 is 2, the square root of 9 is 3 and the square root of 16 is 4, These square roots are whole numbers.
8. The number from which the subtraction is done must be a perfect square whole number. ,
9. So, the meaning of doing the square root is to obtain a whole number so that this whole number can be subtracted from the given number, the same whole number of times.

 

Arithmetic Chapter 11 Perfect Square Numbers

1. We know that if an integer is multiplied by the same integer, then the product obtained is a square number. Again it is not only a square number, but also an integer. So the square roof of this perfect square number is an integer.
2. Therefore, we can give the following definition of a perfect square number
3. An integer that is a whole number is said to be a Perfect Square Number if the square root of that integer (whole number) is an integral whole number.
4. Finally, we can give the following definition of a perfect square number
5. An integral whole number is called a Perfect Square Number if it can be expressed as the product of two same-directed integral numbers.
6. For example, the integral whole number 4 can be expressed as the product of two same-directed integers (+2), i.e., 4 = (+2) x (+2).
   ∴ 4 is a perfect square number
7. or, the square root of any integral whole number is an integer, then the integral whole number is called Perfect Square Number.
8. For example, √9 = 3 and is an integer.
9. ∴ 9 is a perfect square number.

Understanding Square Roots

Arithmetic Chapter 11 The Perfect Square Numbers from 1 to 1000 and their list

 

 

Arithmetic Chapter 11 Characteristics of Perfect Square Numbers

1. The digit in the unit’s place of a perfect square number must be any one of the digits 0, 1, 4, 5, 6, or 9.
2. The digit in the unit’s place of any perfect square number never is 2, 3, 7, or 8.
3. Any perfect square number cannot contain an odd number of zeroes at the end.
4. Any perfect square number can be expressed as the product of two equal (both magnitude and sign) integers.

5.

1. The square number of a number containing one digit is a number having 1 digit or 2 digits.
2. The square number of a number containing 2 digits is a number having 3 digits or 4 digits.
3. The square number of a number containing 3 digits is a number having 5 digits or 6 digits.
4. The square number of a number containing 4 digits is a number having 7 digits or 8 digits.
5. The square number of a number containing 5 digits is a number having 9 digits or 10 digits.
6. The square number of a number containing 6 digits is a number having 11 digits or 12 digits etc.

6.

1.  If the unit’s place digit of a number is 0, then the units place digit of the square of that number is 0.
2. If the unit’s place digit of a number is 1, then the unit’s place digit of the square of that number is 1.
3. If the unit’s place digit of a number is 2, then the unit’s place digit of the square of that number is 4.
4. If the unit’s place digit of a number is 3, then the unit’s place digit of the square of that number is 9.
5. If the unit’s place digit of a number is 4, then the unit’s place digit of the square of that number is 6.
6. If the unit’s place digit of a number is 5, then the unit’s place digit of the square of that number is 5.
7. If the unit’s place digit of a number is 6, then the unit’s place digit of the square of that number is 6.
8. If the unit’s place digit of a number is 7, then the unit’s place digit of the square of that number is 9.
9. If the unit’s place digit of a number is 8, then the unit’s place digit of the square of that number is 4.
10. If the units place digit of a number is 9, then the unit’s place digit of the square of that number is 1.

 

Arithmetic Chapter 11 Determination of two special types of the square and square root

1.

1. We know that (11)² = 11 x 11 = 121
   (111)² = 111 x 111 = 12321 .
   (1111)² = 1111 x 1111 = 1234321
2. We observe the squares of the above numbers. We can find the square of a number containing 1111….etc i.e., the number formed only by 1 at case (without multiplicities)

Step 1.

1. First, find the number 1 in the given number. Then write the consecutive natural numbers starting from 1 as many 1 are there in the given number.
2. For example, suppose you have to find the square of 1111 i.e. (1111)².
3. The number 1 in the number is 4.
4. At first, you write the first four consecutive natural numbers starting from 1 i.e., 1234.

 

Step 2.

1. Then you write the numbers in a reverse way up to 1.
2. The number so obtained is the required square of the number.
3. For the above example, after 1234 you write 321 i.e., after 1234 you put 321 so that the number obtained is 1234321. This is the square of 1111.
   (1111)² = 1234321.
4. Similarly, We get (111111)² = 12345654321.
5. Here the number 111111 contains 6 ones. So we write first 123456 and then 54321 is written in a reverse way, so that we get 12345654321 which is the square of 111111.
6. Again, the square root of 12321 is 111 because the given number 12321 contains the first 3 consecutive natural numbers 1,2,3, and then 2, 1 are written in a reverse way.
   712321 = 111.
7. Similarly, 7123454321 = 11111.

 

2.

We know that 9² = 81, (99)² = 9801, (999)² = 998001.

We observe the squares of the above numbers. We can find the square of a number of form 999…… etc i.e., the number formed by 9 only at ease (without multiplication).

The process of obtaining the square of such a number is discussed below

Step 1.

First, count the number of 9s’ i.e. first find how many 9s’ are there in the given number.

Then subtract 1 from the number obtained.

So the square number must contain one 9 less than the number of 9s that the given number.

 

Step 2.

1. Then put one 8 on the right side of the last 9 and then put as many zeroes as the number of 9 written and lastly put 1.
2. Then the required square number is obtained.
3. For example, in the square of 999999 i.e., (999999) the given number contains 6 nines, so the square number contains 6-1 = 5 nines, then put one 8 and 5 zeroes and then put 1 in the last
4. ∴(999999)² = 999998000001. .
5. In this way, we can find easily the square of any number formed by 9 only.
6. Again in a reverse way, we can find the square root of the numbers satisfying the above conditions easily.
7. For example, √99980001 = 9999,
   √999999998000000001= 999999999.

Important Definitions Related to Square Roots

Arithmetic Chapter 11 Method of finding the square root

1. Here we shall discuss two methods of finding the square root of any positive integer.
2. The methods are
3. Determination of square root by Factorisation Method
4. The working rule of this method can be discussed step-wise as follows
5. Step 1. The given perfect square number is factorized into prime factors.
6. Step 2. Write the same prime factors arranging them in pairs by multiplication.
7. Step 3. For each pair of the same prime factors, take or select only one prime factor. The selection of a single factor can be done for each pair of different factors.
8. Step 4. Then obtain the product of these selected prime factors.
9. Step 5. This required product will be the square root of the given perfect square number.
10. Observe the following example :

Short Questions on Finding Square Roots

Example 1: Find the square root of 14400.

Solution:

Step 1

Step 2

14400 = (2×2) x (2×2) x (2×2) x (3×3) x (5×5)

Step 3

∴ √14400 = 2 x 2 x 2 x 3 x 5 = 120

step 4

The required square root =120.

2. Determination of square root by division method :

The square root of each perfect square number can be determined easily by the division method. Generally, we use the division method for the calculation of the square root of large perfect square numbers.

The working rule of this method can be discussed step-wise as follows :

Step 1:

1. At first, we mark each pair of digits starting from the extreme right digit towards the left (i.e., starting from the digit that lies in the units’ place towards the left) by putting short lines over them.
2. For example,
3. If the number of digits of the given number is an odd number, we go on marking by short lines over each pair of digits then at the extreme left end, we are left with a single digit.
4. Therefore, there will be a short line marking over this single digit at the left end in this case.
5. If the number of digits of the given number is an even number, then we go on marking by short lines over each pair of digits till the end.
6. So marking will be done over all the digits.

 

Step 2.

1. Now give two division signs in two sides of the given number.
2. Then below the digit (in the case of a number containing an odd number of digits) or the pair of digits (in the case of a number containing an even number of digits), we write a perfect square number equal to or nearer to but less than the number above it.
3. Here the first digit i.e., the extreme left digit (which is put in the quotient place of this division process) in the required square root of the given number will be the square root of that perfect square number.
4. Now subtract this perfect square number from the number above it.
5. For. example

 

 

Common Questions About Perfect Squares

Step 3.

1. Put the next pair of marking digits on the right side of the result of subtraction.
2. For example,

 

 

 

Step 4.

1. Now take this number so obtained as a dividend and for the divisor, we take the number which is twice the number already put in the first place of the required square root.
2. For example,

 

 

 

Step 5.

1. Then on the right side of this number (i.e., the divisor), we put a maximum digit by our choice of selection such that when the divisor (which is obtained after putting this digit) so obtained is multiplied by that digit obtained by our selection produces the maximum number not exceeding the dividend number.
2. The product so obtained is put below the dividend and subtracted.
3. For example,

 

 

Step 6.

1. Proceed step 5 repeatedly till the last pair of marking digits in the extreme right end.
2. For example,

 

 

Step-7.

1. After the completion of the division process, the obtained number in the quotient is the required square root of the given number.
2. ∴ The square root of 55225
   = √55225
   = 235
3. And the square root of 853776
   = √853776
   = 924.

Practice Problems on Square Roots

Arithmetic Chapter 11  Some special learning things

When the given number is factorized:

1. if it is seen that after pairing the same factors one extra factor is remaining which is not paired up, then we note that :
2. the given number is not a perfect square number; if the given number is multiplied by or divided by that factor, then the number so obtained in either of the cases will be a perfect square number.
3. For example, we take the number 180
4. Here factor 5 is only one and it is remaining.
5. It is unpaired.
6. So the given number 180 is not a perfect square number
7. If the given number 180 is multiplied by 5, then the product is 180 x 5 = 900 which is a perfect square number.
8. Again if the given number 180 is divided by 5, we get 180 ÷ 5 = 36 which is also a perfect square number.

If 2 or more factors are remaining which are not paired up, then:

1. The given number is not a perfect square number.
2. If the given number is multiplied by the product of the remaining factors or if the given number is divided by the product of the remaining factors.
3. Then the numbers obtained in both cases will be perfect square numbers.
4. For example, we take the number 1260.
5. 1260 = 2x2x3x3x5x7 = (2 x 2)x(3 x 3)x5x7.
6. Here 5 and 7 these two factors are remaining unpaired.
7. So the given number 1260 is not a perfect square number.
8. When the given number 1260 is multiplied, by 5 x 7 = 35, then the product is 1260 x 35 = 44100 which is perfect or if the given number 1260 is divided by 35, we get, 1260 35 = 36 which is also a perfect square number.

 

If there is a remainder in the division process of finding the square root of a given number, then:

1.  The given number is not a perfect square number
2. If we subtract the remainder from the given number, then the number obtained after subtraction will be a perfect square number
3. The square of the next integer number obtained in the quotient will be the perfect square number next to the given number.

Examples of Square Roots in Real Life

1. For example, let us take the number 3250
2. Here the remainder is 1.
3. ∴ The given number 3250 is not a perfect square number.
4. 3250 – 1 = 3249, which is a perfect square number.
5. The quotient in the division of the square root is 57. Its next integral number is 57 + 1 = 58.
6. The square of the number 58 is 3364.
7. 3364 is the perfect square number next to the given number 3250.

 

 

 

Arithmetic Chapter 10 Highest Common Factor And Least Common Multiple Or 3 Numbers

Arithmetic Chapter 10 Factor and Multiple

1. In the operation of division, a number is divided by another number.
2. A number that is divided is called the Dividend and the number by which division is performed is called the Divisor.
3. The result of the operation of division is called the Quotient.
4. The consecutive positive integers starting from 1 i.e., the numbers 1, 2, 3, 4, … . are called Natural Numbers.
5. When a natural number is divided by another second number and leaves no remainder, then we say that the first number is completely divisible by the second number.
6. Here the first number is called the dividend and the second number is called the divisor.
7. The result of the division is called the quotient.
8. For example, when 27 is divided by 3, then there will be no remainder and so we say that 27 is divisible by 3.
9. Here 27 is the dividend, 3 is the divisor and the result of the division is 9 which is the quotient.
10. If a number is divisible by another, then the dividend is called the Multiple of the divisor and the divisor is called a Factor of the dividend. In the above example, 27 is a multiple of 3, and 3 is a factor of 27.
11. Similarly, if 65 is divided by 13, then there will be no remainder and so 65 is divisible by 13. Here 65 is a multiple of 13 and 13 is a factor of 65.
12. The natural numbers which are divisible by 2 are called Even numbers and which are not divisible by 2 are called Odd Numbers. 2, 4, 6, 8, etc are even numbers and 1, 3, 5, 7, etc. are odd numbers.

WBBSE Class 6 HCF LCM Notes

Arithmetic Chapter 10 Common Factor

1. A number that is a factor of two or more numbers is called a Common Factor of those numbers.
2. For example : 15 = 3 x 5; 25 = 5 x 5; 35 = 5 x 7.
3. Here we see that 5 is a factor of 3 numbers 15, 25, and 35.
4. So 5 is a common factor of the numbers 15, 25, and 35.

 

Arithmetic Chapter 10 Greatest Or Highest Common Factor

1. A composite number has two or more factors. The highest (or greatest) among all possible common factors of two or more numbers is called the Highest (or Greatest) Common Factor of those numbers and abbreviated as H.C.F. or G.C.F.
2. For example, we consider the numbers 36 and 48.
3. ∴ 36 has factors: 1, 2, 3, 4, 6, 9, 12, 18, 36;
4. ∴ 48 has factors: 1, 2, 4, 6, 8, 12, 16, 24, and 48.
5. 36 and 48 have common factors: 1, 2, 6, and 12.
6. Among these factors, the highest common factor is 12. So, H.C.F. of 36 and 48 is 12.

Understanding HCF and LCM 

Arithmetic Chapter 10 Determination Of Highest Common Factor  Of Three Numbers

1. In order to determine the H.C.F. of three or more numbers, first we find the prime factors of the numbers.
2. Then choose the common factors of the numbers. The product of these common prime factors is the required H.C.F. of the numbers.
3. For example 84 = 2 x 2 x 3 x 7
   126 = 2 x 3 x 3 x 7
   210 = 2 x 3 x 5 x 7
4. 2, 3, and 7 are the common factors of 84, 126, and 210.
5. ∴ The H.C.F. = 2 x 3 x 7 = 42.

 

Arithmetic Chapter 10 Highest Common Factor Of Compound Quantities

1. In order to obtain the H.C.F. of two or more compound quantities, they are expressed in the lowest- order of units. Then determine the H.C.F. of them and put the unit.
2. Observe the following example

Short Questions on HCF and LCM

Example: Find the H.C.F. of 8 kg 981 gm; 14 kg 113 gm and 16 kg 679 gm.

Solution: 8 kg 981 gm = 8981 gms.

14 kg 113 gm = 14113 gms.

16 kg 679 gm = 16679 gms.

First, find the H.C.F. of 8981 gms and 14113 gms

 

 

H.C.F. of 8981 gms and 14113 gms = 1283 gms.

Now we shall find the H.C.F. of 1283 gms and 16679 gms.

 

 

H.C.F of 1283 gms and 16679 gms = 1283 gms.

H.C.F. of 8981 gms., 14113 gms., 16679 gms. = 1283 gms.

So the required H.C.F. = 1283 gms. = 1 kg = 283 gm.

 

Arithmetic Chapter 10 Common Multiple or Least (Lowest) Common Multiple

1. There are many multiples of a natural number.
2. Two or more natural numbers may have an infinite number of common multiples.
3. The last (or lowest) common multiple of all the common multiples of two or more natural numbers is called the L.C.M. of the Least (or lowest) Common Multiple.
4. For example, we consider the numbers 12, 15, 20, and 30.
5. Multiple of 12 is 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, etc.
6. Multiple of 15 is 15, 30, 45, 60, 75, 90, 105, 120, etc.
7. Multiple of 20 is 20, 40, 60, 80, 100, 120, 140, 160, etc.
8. Multiple of 30 is 30, 60, 90, 120, 150, 180, 210, 240, etc.
9. From above we see that the common multiple of 12, 15, 20, and 30 are 60, 120, 18.0, etc.
10. Among them, 60 is the smallest common multiple.
11. ∴ The lowest common multiple of 12, 15, 20, and 30 = 60, and hence L.C.M. of then = 60.
12. It is clear that the L.C.M. of two or more given numbers must be divisible by the numbers.

 

Arithmetic Chapter 10 Determine Of Least Common Multiple Of Three Numbers

1. In order to determine the L.C.M. of three or more given numbers, first of all, find the prime factors of the numbers individually and then determine the common prime factors of the numbers.
2. Next, take other factors (other than the common factors) remaining of all the numbers and omit the factor or factors which have already been taken.
3. The product of these factors and the common prime factors will give the L.C.M. of the numbers.
4. For example, we take the numbers 12, 16, and 20.
   12= 2 x 2 x 3;
   16= 2 x 2 x 2 x 2;
   20= 2 x 2 x 5.
5. The common prime factors = 2, 2, and the other factors remaining in the numbers are 3; 2, 2; 5 respectively.
6. Therefore the L.C.M. of 12, 16 and
   20 = 2 x 2 x 3 x 2 x 2 x 5
   = 240.

Arithmetic Chapter 10 Determination of Least Common Multiple of compound quantities

1. In order to determine the L.C.M. of two or more compound quantities, the quantities should be expressed in the lowest order or units.
2. Then obtain the L.C.M. of them in the usual process and put the unit to this L.C.M.
3. This gives the required L.C.M.
4. Observe the following example

Common Questions About Finding HCF of Three Numbers

Example: Find the L.C.M. of 30 min. 48 sec.; 46 min. 12 sec. and 1hr. 17 min.

Solution: 30 min. 48 sec. = 1848 sec.

46 min. 12 sec. = 2772 sec.

1 hour. 17 min. = 4620 sec.

 

 

L.C.M. of 1848, 2772, 4620 = 2 x 2 x 3 x 11 x 7 x 2 x 3 x 5 = 27720

The required L.C.M. = 27720 sec.

= 462 min.

= 7 hr. 42 min.

The required L.C.M = 7 hr. 42 min.

Important Definitions Related to HCF and LCM

Arithmetic Chapter 10 Relation between Highest Common Factor and Least Common Multiple of two numbers

1. Let 22 and 33 be two given numbers.
2. Then H.C.F. of 22 and 33 = 11; L.C.M. of 22 and 33 = 66.
3. H.C.F. x L.C.M. = 11 x 66
   =11 x 2 x 33
   = 22 x 33
   = Product of two given numbers.
4. ∴ Product of two given numbers = Their H.C.F. x L.C.M.
5. ∴ L.C.M. = Product of two numbers ÷ H.C.F.
6. H.C.F. = Product of two numbers ÷ L.C.M.
7. Thus dividing the product of two numbers by their H.C.F. we can find the L.C.M. of the numbers.
8. Again dividing the product of two numbers by their L.C.M., we can find H.C.F.
9. This formula is not true for three (3) numbers.

 

 

 

Arithmetic Chapter 9 Recurring Decimal Number

Arithmetic Chapter 9 Recurring Decimal Numbers

1. To convert a vulgar fraction into a decimal fraction, in some cases, it is observed that the operation of division comes to an end i.e., there is no remainder left and the decimal fraction obtained contains a finite number of digits after the decimal point.
2. These decimal fractions are called Finite Decimal Fractions.
3. Again in some cases, it is observed that the operation of division never comes to an end i.e., there is an infinite number of digits after the decimal point, these decimal fractions are called Infinite Decimal Fractions.
4. For infinite decimal fractions, in some cases, it is observed that the same remainder or a set of the same remainders occurs repeatedly in the operation, of division.
5. So a figure (digit) or a set of figures (digits) appears repeatedly in a definite order in the quotient during the operation of division in some cases of infinite or non-terminating decimal fractions.
6. Such a type of decimal number is called a Recurring Decimal Number.

WBBSE Class 6 Recurring Decimal Notes

Arithmetic Chapter 9 Conversion of Vulgar fraction into Recurring Decimal Fraction

Let us consider the Vulgar fraction 1/3

1. 

∴ 1/3 = 0.333….

= 0.3.

The given fraction is 1/3 Due to the presence of 1 in the numerator and 3 in the denominator of the given vulgar fraction, when it is converted into a decimal fraction a non-terminating decimal fraction is obtained.

In the process of division, the dividend i.e., the numerator 1 which is less than the divisor 3 (i.e., the denominator), we put a decimal point in the quotient and put one zero after 1 in the dividend so that the new dividend becomes 10 and dividing it by 3, we get 3 as the quotient and it is placed after the decimal point in the quotient.

Understanding Recurring Decimals

The remainder obtained is 1.

We put one zero on the right side of the remainder every time of division.

Thus the new remainder is 10 after putting one zero after 1 and it is divided by 3, the quotient obtained is 3.

But every time the same remainder 1 occurs i.e., the same remainder 1 occurs repeatedly and the division never comes to an end.

The quotient is 0-333  and we write this as 0-3. We place a dot above 3 after the decimal point and it indicates that after the decimal point, 3 recurs endlessly in the quotient.

Such a type of decimal fraction is called a Recurring Decimal Fraction.

Again, we consider the vulgar fraction 2/7 and it is to be converted into a decimal fraction

∴ 2/7 = 0.285714285714….

= 0.285714.

We have to divide 2 by 7 (here numerator is 2 and the denominator is 7) in the vulgar fraction 2/7.

We see that the remainder of 2 occurs in the seventh step.

Thus the repetition starts at the seventh step. The successive quotients after the decimal point are 2, 8, 5, 7; 1, 4, 2, 8, 5, 7, 1, 4, 2,

∴ 2/7 = 0.285714.

Two dots are placed one is above the digit 2 and the other is above the digit 4.

This indicates that all the digits present between 2 and 4 will recur in the same order as they occur in 285714.

The decimal conversion of the vulgar fraction 2/7 is 0.285714, which is a recurring decimal number.

Similarly, we get

Short Questions on Recurring Decimal Numbers

,35/12 = 2.91666….= 2.916

.

421/330 = 0.127575….

= 0.1275

From the above discussions, we conclude that the order to convert a vulgar fraction into a decimal fraction, the numerator of the given vulgar fraction is divided by its denominator if the operation of division never comes to an end

i.e., Always there is a remainder left, the same remainder or a set by the same remainders occurs repeatedly then a digit or a set of digits recurs repeatedly or endlessly in a definite order in the quotient, and the fraction thus obtained is a non-terminating decimal fraction which is called a Recurring Decimal Fraction.

Arithmetic Chapter 9 Recurring Period

1. In a recurring decimal fraction, some or all the digits recur.
2. The portion of the decimal fraction which recur is called the Recurring period or Recurring part.
3. The portion of a recurring decimal fraction after the decimal point which does not recur is called it’s Non-recurring Part.
4. For example, we consider the recurring decimal fraction 2.43725, portion 725 of it is the Recurring Period, and portion 43 of the given recurring decimal fraction (after the decimal point) is the Non-recurring part.

Arithmetic Chapter 9 Pure Recurring Decimal Fraction And Mixed Recurring Decimal Fraction

1. A recurring decimal fraction that does not contain any non-recurring part is called a Pure Recurring Decimal Fraction.
2. A recurring decimal fraction that contains a non-recurring part is called a Mixed Recurring Decimal Fraction.
3. For example, 0.36, and 021578 are Pure recurring decimal fractions,s and 0.231, and 0.065217 are Mixed recurring decimal fractions.

Important Definitions Related to Recurring Decimals

Arithmetic Chapter 9 Conversion Of A Few Special Vulgar Fractions Into Recurring Decimals

 

Common Questions About Converting Recurring Decimals

∴ 1/7 = 0.142857

2/7 = 0.285714

3/7 = 0.428571

4/7 = 0.571428, etc.

From above, we get, when the vulgar fractions having 7 as the denominator are converted into decimal fractions, then the converted decimal fractions are pure recurring decimal fractions and they obey some rules.

In the recurring part of the decimal conversion of a vulgar fraction whose denominator is 7, six digits 1, 2, 4, 5, 7, 8 (excluding the digits 3, 6, 9) are arranged in some special definite order.

For example, in the recurring part of the decimal conversion of 1/7, six digits 1, 2, 4, 5, 7, 8 recur in the order 1, 4, 2, 8, 5, 7.

We place the digits 1, 4, 2, 8, 5, and 7 along the circumference of a circle in a clockwise direction.

Then we can get equivalent recurring decimal fractions of the vulgar fractions

1/7, 2/7, 3/7, 4/7,….

1/7 = 0.142857

2/7 = 0.285714

3/7 = 0.428571

4/7 = 0.571428

5/7 = 0.714285

6/7 = 0.857142.

There are 6 digits in the recurring part of all these pure recurring decimal fractions.

Now we shall discuss the decimal conversion of vulgar fractions whose denominator is 13.

Here we get two types of periods.

In the periods of decimal conversion of 1/13, 3/13, 4/13, 9/13, 10/13, and 12/13, we get the digits 0, 7, 6, 9, 8, and 3 in the cyclic order

∴ 1/13 = 0.076923

∴

1/13 = 0.076923

3/13 = 0.230769

4/13 = 0.307692

9/13 = 0.692307

10/13 = 0.769230

12/13 = 0.923076.

There are 6 digits in the periods of all these pure recurring decimal fractions.

Again in the periods of decimal conversion of six vulgar fractions 2/13, 5/13, 6/13, 7/13, 8/13, and 14/13, we get the digit  1, 5, 3, 8, 4, 6 in the cyclic order.

∴ 2/13 = 0.153846

 

 

∴ 2/13 = 0.153846

5/13 = 0.384615

6/13 = 0.461538

7/13 = 0.538461

8/13 = 0.615384

11/13 = 0.846153.

Arithmetic Chapter 9 Number of digits in the Non-recurring part of a Mixed recurring decimal

1. The decimal conversion of a vulgar fraction is a mixed recurring decimal if the denominator of the vulgar fraction when it is expressed in its lowest term has a factor of 2 or 5 or both.
2. The number of digits in the non-recurring part will be the highest index of 2 or 5 present in the denominator of the vulgar fraction.
3. Observe the following examples:

Examples of Real-Life Applications of Recurring Decimals

Example 1:  \(\frac{1}{12}=\frac{1}{2^2 \times 3}\)

The index of 2 in the denominator is 2.

So the number of digits in the nonrecurring part of the mixed recurring decimal fraction = The index of 2 in the denominator = 2.

1/12 = 0.083.

Non-recurring part = 08.

Example 2: \(\frac{1}{15}=\frac{1}{3 \times 5}\)

The index of 5 in the denominator = 1.

So the number of digits in the nonrecurring part of the mixed recurring decimal fraction = The index of 5 in the denominator = 1.

1/5 = 0.06.

Non-recurring part = 0.

∴ The number of digits in the non-recurring part = 1.

Example 3: \(\frac{1}{24}=\frac{1}{2^3 \times 3}\)

The index of 2 in the denominator = 3.

So the number of digits in the nonrecurring part = The index of 2 in the denominator = 3.

1/24 = .0416.

Non-recurring part = 3.

Practice Problems on Recurring Decimals

Example 4: \(\frac{1}{60}=\frac{1}{5 \times 2^2 \times 3}\) 

The indices of 2 and 5 in the denominator are 2 and l respectively.

So the number of digits in the non-recurring part = The greater of the indices of 2 and 5 = The greater of 2 and 1=2.

∴ 1/60 = 0.016.

No-recurring part = 01.

∴ The number of digits in the non-recurring part = 2.

Arithmetic Chapter 9 Conversion of Recurring Decimal into Vulgar Fraction

1. There are two types of recurring decimals pure recurring decimals and mixed recurring decimals.
2. First, we shall discuss about the conversion of pure recurring into a vulgar fraction.

1. Conversion of pure recurring decimal into vulgar fraction

Example 1: Convert 0.1 into vulgar fractions.

Solution :

Given

0.1

0.2 x 10 = (.2222 ) x 10

= 2.2222 …..  (1)

0.2x 1 = (0.2222 ) x 1 = 0.2222………… (2)

(Multiplying 0.2 by 1)

Subtracting (2) from (1), we get,

0.2 (10 – 1) = (2.2222 ) – (.2222  ) = 2

or, 0.2 x 9 = 2

or 0.2 = 2/9

0.1 into vulgar fractions = 2/9

Example 2: Convert 0*35 into a vulgar fraction.

Solution:

Given

0*35

0.35 = 0.353535

Multiplying both sides by 100 and 1 respectively, we get,

0.35 x 100 = (0.353535 ) x 100 = 35.353535…………(1)

0-35 x 1 = (0.353535 ) x 1 = 0.353535………(2)

Subtracting (2) from (1), we get,

0.35 (100 – 1) = 35

or, 0.35 x 99 = 35

or, 0.35 = 35/99.

From the above examples, we get the following rule of conversion of pure recurring decimal into vulgar fraction

For example 0.54632 = \(\frac{54632}{99999}\)

0.205 = \(\frac{205}{999}\)

0.51

= 51/99

= 17/13.

0.35 into a vulgar fraction 35/99

2. Conversion of mixed recurring decimals into vulgar fractions:

For this observe the following examples

Example 1: Convert 0.1275 into a vulgar fraction.

Solution :

Given

0.1275

0-1275 = 0-12757575……..(1)

Multiplying both sides by 10000, we get,

0.1275 x 10000 = (0.12757575 ) x 10000 = 1275.757575………(2)

Multiplying both sides of (1) by 100, we get,

0.1275 x 100 = (0.12757575 ) x 100

= 12.757575……….(3)

Subtracting (3) from (2), we get,

(10000 – 100) x 0.1275 = 1275 – 12

or, 9900 x 0.1275 = 1275 – 12

or, 0.1275 = \(\frac{1275-12}{9900}\)

= \(\frac{1263}{9900}\)

= \(\frac{421}{3300}\)

0.1275 into a vulgar fraction = \(\frac{421}{3300}\)

Conceptual Questions on the Difference Between Terminating and Recurring Decimals

Example 2. Convert 0-26321 into a vulgar fraction.

Solution:

Given

0-26321

0.26321 = 0.26321321321…………(1)

Multiplying both sides of (1) by 100000, we get,

0.26321 x 100000 = (0.26321321321 ) x 100000 = 26321.321321

Again multiplying both sides of (1) by 100, we get,

0.26321 x 100 = (0.26321321321……) x 100 = 26.321321321

Subtracting (3) from (2) we get,

0.26321 (100000 – 100) = 26321 – 26

or, 0.26321 x 99900 = 26321 – 26

or, 0.26321 = \(\frac{26321-26}{99900}\)

= \(\frac{26295}{99900}\)

= \(\frac{8765}{33300}\)

0-26321 into a vulgar fraction = \(\frac{8765}{33300}\)

Example 3. Convert 3.128 into a vulgar fraction.

Solution:

Given

3.128

3.128 = 3.1282828..  (1)

Multiplying both sides of (1) by 1000, we get,

3.128 x 1000 = (3.1282828 ) x 1000 = 3128.282828..

Again multiplying both sides of (1) by 10, we get,

3.128 x 10 = (3.1282828 ) x 10 = 31.282828…… . .

Subtracting (3) from (2), we get,

(1000 – 10) x 3.128 = 3128 – 31 or, 990 x 3.128 = 3128 – 31

or, 3.128 = \(\frac{3128-31}{990}\)

= \(\frac{3097}{990}\)

= \(3 \frac{127}{990}\)

∴ 3.128 = \(3 \frac{127}{990}\)

From the discussions of the above examples, we get the following rule

Some examples are given below:

1. 0.02028 = \(\frac{2028-2}{99900}\)

= \(\frac{2026}{99900}\)

= \(\frac{1013}{49950}\)

2. 10293 = \(\frac{10293 – 102}{9990}\)

= \(\frac{52424}{9990}\)

= \(5 \frac{2474}{9990}\)

= \(5 \frac{1237}{4995}\)