Current Affairs 8th Class

Mensuration

Mensuration His chapter deals with the concept of finding the surface area and volume of the regular figures. By regular figures we mean to say the figures whose parameters are known to us. Previously we have learnt to find the perimeter and area of the rectilinear figures, but now onwards we will learn to find the area of some polygons and also discuss the surface area and volume of some solid shapes namely cuboid cube, cone and cylinder. Area of a Polygon Area of a given polygon can be found by dividing the given polygon into non-over lapping rectilinear figures. The area of the polygon will be equal to the sum of the areas of non-overlapping figures.

Example:

Find the area of the polygon given below if AP = 10 cm, BP = 20 cm, CP = 50 cm, DP = 60 cm and PS = 90 cm

(a) \[4080\,c{{m}^{2}}\] (b) \[5050\,c{{m}^{2}}\] (c) \[6060\,c{{m}^{2}}\] (d) \[7070\,c{{m}^{2}}\] (e) None of these Answer (b) Explanation: Area of the given figure \[=\frac{1}{2}\times 30\times 10+\frac{1}{2}\times 20\times 20+\frac{1}{2}\times 40(30+40)+\frac{1}{2}\times 40\]\[(20+60)+\frac{1}{2}\times 40\times 40\times \frac{1}{2}\times 60\times 30c{{m}^{2}}\] \[=(150+200+1400+1600+800+900)c{{m}^{2}}\] \[=5050c{{m}^{2}}\] Cuboid and Cube A cuboid is a closed rectangular solid bounded by six rectangular faces. A cuboid has 12 edges and 8 vertices. The length, breadth and height of a cuboid is generally denoted by I, b and h respectively. A cuboid whose length, breadth and height are equal is called a cube and each equal side is called edge of the cube.

Example:

Determine the time in which the level of the water in a rectangular tank which is 50 m long and 44 m wide will rise by 7 cm if water is flowing through a cylindrical pipe of radius 7 cm at the rate of 5 kilometre per hour. (a) 4 hours (b) 3 hours (c) 2 hours (d) 5 hours (e) None of these Answer (c) Explanation: Let x hours be the time taken by volume of water flowing through the pipe. Then, \[\frac{22}{7}\times {{\left( \frac{7}{100} \right)}^{2}}\times 5000\times X\] \[=\frac{22}{7}\times \frac{7}{100}\times \frac{7}{100}\times 5000X\] \[=77x{{m}^{3}}\] Volume of tank \[=\frac{50\times 44\times 7}{100}{{m}^{3}}=154{{m}^{3}}\] Therefore, \[77x=154\Rightarrow x=2hours\] Cone and Cylinder Cone is a solid form which is generated by the revolution of a right angled triangle about one of the sides adjacent to the right angle. The base of a cone is always circular. While a cylinder has two identical circular ends and one curved surface and area of each circular ends are same.

Example:

The volume of a cylinder having a height of 14 m and base radius 3 m is: (a)\[792\,\,{{m}^{3}}\] (b) \[99\,\,{{m}^{3}}\] (c) more...

Catgeory: 8th Class

Data Handling

Data Handling Statistics is the formal science of making effective use of numerical data relating to group of individuals or experiments. It deals with all aspects, including the collection, analysis and interpretation of data and also the planning of the collection of data in terms of the design of surveys and experiments. A statistician is someone who is particularly versed in the ways of thinking necessary for the successful application of statistical analysis. Often such people have gained this experience after starting work in number of fields. This is also a discipline called Mathematical Statistics, which is concerned with the theoretical basis of the subject. Types of Data The data may be in the form of raw or grouped. The data which is not arranged in any form is known as the raw data and data which is arranged in a definite pattern is known as the grouped data.is normally classified into two types. Primary data and Secondary data. The primary data is that data which is collected by the person himself for his own personal use, while secondary data is that data which is collected by others and used by someone else for his or her use. It may be data collected form the books, newspaper internet or any other sources. Pie Chart A pie chart is the pictorial representation of the given data with the help of non-intersecting sectors of different areas and different central angles. The magnitude of the central angles depend on the magnitude of the data. In a pie chart, the arc length of each sector and consequently its central angles and area, is directly proportional to the quantity it represents. It is named for its resemblance to a pie which has been sliced. The following pie chart represent the population of English native speakers in different countries.

Co-ordinates of a Point The pair of points which is used to describe the location of a point in two dimensional system are called co-ordinates of a point. The x-coordinate of a point is horizontal distance of the point from origin and y-coordinate of the point is the vertical distance from the origin. Line Graph A line graph is very useful for displaying data or information which changer continuously over a certain period of time. A line graph compares two variables. One variable is plotted along x-axis while another variable is plotted along y-axis. Linear Graph A linear graph is a graph which is used to represent the linear relationship between two variables. To draw a linear graph we use co-ordinates along x and y axis. The difference between a line graph Or linear graph is that a line graph display information as a series of points joined by line segments while a linear graph is always a straight line.

Example:

The quantity of petrol filled in a can and the cost of more...

Catgeory: 8th Class

Crop Production and Microorganisms

Crop Production and Microorganisms Cultivation Cultivation is the act of raising plants.

Land preparation and cultivation methodology is among the basic factors which affect the yield of crops. For getting better yield, it is important to prepare land thoroughly, so that the weeds are destroyed and water absorption capacity of the soil is increased. Crops When the same kind of plants are grown in the fields on a large scale to obtain foods like cereals, pulses, vegetables and fruits, etc., it is called a crop. Agriculture The growing of plants (or crops) in the fields for obtaining food is called agriculture. Crop production involves various agricultural practices such as:

Preparation of soil- It includes tilling, levelling and maturing of the fields.

Sowing - good quality seeds are sown in the fields either with the help of a seed drill or manually.

Adding manure and fertilisers - the deficiency of plant nutrients and organic matter in the soil is fulfilled by adding manures and fertilizers to the soil.

Irrigation - Crops can be irrigated by using traditional methods or modern methods of irrigation.

Weeding - removal of unwanted plants which grow along with a cultivated Crop can be done by hand or by using weedicides.

Harvesting - is the last step of crop production in which cutting and gathering of the mature crop is done.

Micro-organisms

A microorganism or microbe is an organism that is microscopic (usually too small that cannot be seen by the naked human eye). The study of microorganisms is called microbiology. Microorganisms live in all parts of the biosphere where there is liquid water, including soil, hot springs, on the ocean floor, high in the atmosphere and deep inside rocks within the earth's crust. Major Groups of Micro-organisms

Fungi - fungi are plant like organisms which do not have chlorophyll and do not perform photosynthesis. Some examples of fungi are: Yeast Moulds, Mushrooms, Toadstools and puff balls.

Virus - Viruses are the smallest microorganisms which can reproduce only inside the cells of the host organisms (which may be animal, plant or bacterium).

Bacteria - Bacteria are unicellular prokaryotes which have cell walls but do not have an organised nucleus and other structures. On the basis of shapes bacteria can be classified as: bacillus (rod-shaped), coccus (spherical), spirillum (spiral-shaped, vibrio (comma shaped).

Protozoa - protozoa are unicellular eukaryotes. Some examples of protozoa are: Amoeba, Paramecium and Plasmodium.

Algae - algae are plant-like organisms that have cell walls and chlorophyll within the cells. They make their own food by photosynthesis. Some examples of algae are: Chlamydomonas, Spirogyra and Blue green algae.

Useful Micro-organisms Microorganisms are vital to more...

Catgeory: 8th Class

Material

Material Material is a physical or chemical substance of which things can be made. Fibres A fibre is a piece of fabric that is long, thin and flexible. Plant fibres are the basis of fabric such as cotton, silk and wool fibres come from animals. Many artificial fibres have also been invented such as rayon, nylon, polyester, acrylic etc.

Nylon - is entirely made of chemicals. It is very strong, elastic, light and water-resistant fibre. It is lustrous in appearance. It is used in making ropes, tents, fishing nets and parachutes.

Polyester - is made from 'petroleum'. It is very strong, crease resistant, light, elastic and absorbs very little water. It is used in making pants, shirts, suits, jackets, etc.

Rayon - is also known as "artificial silk7. Cellulose which is obtained from wood pulp is the raw material to prepare rayon. It is used in home furnishings, suits, ties, blouses, sportswear, etc.

Acrylic - is made from a chemical called 'acrylonitrile'. Due to its wool like feel, acrylic fibre is often used as a substitute for wool. It is used for making sweaters, shawls, blankets, sportswear, socks, carpets, etc.

Plastics A plastic is a synthetic material which can be moulded into desired shape when it is soft and then hardened to produce a durable article. For example, polyethene, PVC, Bakelite, Melamine and Teflon. Types of Plastics:

Thermosetting plastics - which cannot be moulded by heating. For example, Bakelite and Melamine.
Thermoplastics - which can be moulded into different shapes again and again by heating. For example, Polyethene, PVC.

Metals Metals are defined as elements which form positive ions by losing electrons. For example, sodium, potassium etc. Chemical Properties

Usually have 1-3 electrons in their outer shell.
Lose their valence electrons easily.
Form oxides that are basic.
Are good reducing agents.
Have lower electronegativity’s.

Physical Properties

Good electrical conductors and heat conductors.
They are malleable i.e.can be beaten into thin sheets.
They are ductile i.e.can be stretched into wires.
Possess metallic lustre.
Opaque in nature.
Solid at room temperature (except Hg).

Nonmetals: Nonmetals are defined as elements which form negative ions by gaining electrons. For example, chlorine, oxygen, carbon, etc. Chemical Properties

Have higher electronegativity’s.
Are good oxidizing agents.
Usually have 4-8 electrons in their outer shell.
Gain or share valence electrons easily.
Form oxides that are acidic.

Physical Properties

Solids, liquids or gases at room temperature.
They are non-ductile.
Poor conductors of heat and electricity.
Transparent as a thin sheet.
Do not possess metallic lustre.

They are brittle.

Catgeory: 8th Class

Data Handling

DATA HANDLING FUNDAMENTALS The word 'statistics' is derived from the Latin word 'status' which means political state. Political states had to collect information about their citizens to facilitate governance and plan for development. Then, in course of time, statistics came to mean a branch of mathematics which deals with collection, classification, presentation and analysis of numerical data. In this chapter, we shall learn about the classification of data, i.e., grouped data and ungrouped data, measures of central tendency, and their uses. Data The word 'data' means, information in the form of numerical figures or a set of given facts. For example, the percentage of marks scored by 10 students of a class in a test are: 36, 80, 65, 75, 94, 48, 12, 64, 88 and 98. Statistics is basically the study of numerical data. It includes methods of collection, classification, presentation, analysis of data and inferences from data. Data as can be qualitative or quantitative in nature. If one speaks of honesty- beauty, colour, etc., the data is qualitative, while height, weight, distance, marks, etc., are quantitative. Data can also be classified as: raw data, and grouped data. Raw Data Data obtained from direct observation is called raw data. The marks obtained by 100 students in a monthly test are an example of raw data or ungrouped data. In fact, little can be inferred from this data. However, arranging the marks in ascending order in the above example is a step towards making raw data more meaningful. Grouped Data To present the data in a more meaningful way, we condense the data into convenient number of classes or groups, generally not exceeding 10 and not less than 5. This helps us in perceiving at a glance, certain salient features of data. Some Basic Definitions Before getting into the details of tabular representation of data, let us review some basic definitions: Observation: Each numerical figure in a data is called an observation. Frequency: The number of times a particular observation occurs is called its frequency. Tabulation or Presentation of Data A systematical arrangement of the data in a tabular form is called 'tabulation' or 'presentation' of the data. This grouping results in a table called ''frequency table' that indicates the number of scores within each group. Many conclusion about the characteristics of the data, the behavior of variable, etc., can be drawn from this table. The quantitative data that is to be analyzed statistically can be divided into three categories: Individual Series: Any raw data that is collected forms an individual series. Example: The weights of 10 students. 36, 35,32,40,65,48,54,71, 62 and 33 Percentage of marks scored by 10 students in a test: 46, 66, 96, 99, 90, 36, 81, 73, 59, 48 Discrete Series: A discrete series is formulated from raw data. Here, the frequency of the observations is taken into consideration. Example: Given below is the data showing the more...

Catgeory: 8th Class

Percentage, Profit and Loss

PERCENTAGE, PROFIT & LOSS FUNDAMENTALS

Gain = Selling price (S.P.) - Cost Price (C.P.)
Loss \[=C.P.-S.\text{ }P.\]
\[Gain%\text{ =}\frac{Gain}{C.P.}\times 100%;\text{ }S.P.\]

=\[C.P.+Gain=C.P.+C.P.\times \frac{Gain(in%)}{100}\] \[=C.P.\left[ 1+\frac{gain(in%)}{100} \right]\].

In case of loss, \[S.\text{ }P.=C.P.\left[ 1-\frac{loss%}{100} \right]\]
\[CP=\left( \frac{100}{100+gain%} \right)\times S.P.=\left( \frac{100}{100-loss%} \right)\times S.P.\]

Elementary Question:

Price of a book was decreased by 10% and then increased by 10%. If the original price of book is Rs. 100, what is its current price?

Decreased by 10%

Ans. \[Rs.100\xrightarrow{decreased\,by\,10%}10%\,Rs.100\,means\] \[100-100\times \frac{10}{100}=100-10=90\]

\[Increased\,by\,10%\,Rs.90\xrightarrow{Increased\,by\,10%}\]

\[=90+90\times \frac{10}{100}=90+9=99\] Example: Gurpreet sells two watches forRs.1980/- each, gaining 10% on one and losing 10% on the other. Find her gain or loss per cent in the whole transaction. Solution: SP of the first watch = Rs 1980/-; Gain = 10% \[\therefore \]CP of the first watch \[=\left\{ \frac{100}{(100+gain%)}\times SP \right\}=\left\{ \frac{100}{(100+10)}\times 1980 \right\}\] \[=Rs\left( \frac{100}{110}\times 1980 \right)=Rs1800/-\] SP of the second watch = Rs 1980/-; Loss% = 10%; CP of the second watch Total CP of the two watches \[=Rs\left( 1800+2200 \right)=Rs4000/-\] \[\therefore \]Total SP of the two watches \[=Rs\left( 1980\times 20 \right)=Rs\,3960/-\] Since \[\left( SP \right)<\left( CP \right),\] there is less in the whole transaction. Loss \[=Rs\left( 3960-4000 \right)=Rs40/-;\] \[~\therefore Loss%=\frac{40}{4000}\times 100%=1%\,loss\] Hence, Gurpreet loses 1 % in the whole transaction. Further concepts on percentage:

A number can be split into two parts such that one part is P% of the other. Then the two parts are:

\[\frac{100}{100+p}\times number\text{ }and\times number.\]

If the circumference of a circle is increased (or) decreased by P%, then the radius of a circle increases (or) decreases by P%.

Elementary question 1: The circumference of a circle is 88 cm, if the circumference is increased by 50%, find percentage increase in radius. Answer: \[{{C}_{1}}=88\,cm\,{{C}_{2}}={{C}_{1}}+\frac{50}{100}\times {{C}_{1}}=88+44=132\] \[{{r}_{1}}=\frac{88}{2\pi }=14\,\,and\,\,{{r}_{2}}=\frac{132}{2\pi }=21\] \[\therefore \]% age increase in radius \[\frac{{{r}_{2}}-{{r}_{1}}}{{{r}_{1}}}\times 100%=\frac{21-14}{14}\times 100%=50%\] OVERHEADS: Suppose you go to market and purchase a bed set and a TV. Naturally, you need to pay transportation charges and also labour cost and sometimes repairing charges, if there is damage. These extra expenses are called overheads. For calculating the total cost price, we add overheads to the purchase price. Elementary question 2: Mohan Kumar purchased and old bike for Rs.20000 and spent Rs. 4000 on its overhauling. Then, he sold it to his friend Jatin for Rs. 21000. Find his loss/gain percentage. Solution: Cost price = Rs. 20000, overheads = Rs. 21000; Total cost price \[=Rs\left( 20000+4000 \right)=24000;\] Selling price = Rs. 21000. Since\[\left( SP \right)<\left( CP \right)\], Mohan Kumar makes a loss. \[\text{Loss}=Rs(24000-21000)=Rs.3000;\] \[\text{Loss}%=\frac{3000}{24000}\times 100%=12\frac{1}{2}%\] Elementary question 3: By selling 36 m of cloth, a cloth-maker loses an amount equal to the selling price of 4m of cloth. Find her gain or loss percent. Solution: Let C. P. of 1 m cloth \[=x\Rightarrow CP\] of 36 m cloth = 36x Let her sell at p% loss \[\Rightarrow SP=36x\left( 1-\frac{p}{1000} \right)\] \[\therefore Loss=\frac{36xp}{100}\]But this equal S. P. of 4m cloth \[\Rightarrow more...

Catgeory: 8th Class

Logarithms

LOGARITHMS FUNDAMENTALS

Logarithm:- Let a be a positive real number other than 1 and \[{{a}^{x}}=m\], then x is called the logarithm of into the base and written as \[{{\log }_{a}}m\].

Example 1:- \[{{10}^{4}}=10000\] \[\Rightarrow \text{lo}{{\text{g}}_{10}}10000=4\] Example 2:- If \[{{3}^{-3}}=\frac{1}{27}\] \[\Rightarrow {{\log }_{3}}\frac{1}{27}=-3\]

\[(I)\,\,\text{lo}{{\text{g}}_{a}}(mn)={{\log }_{a}}m+\text{lo}{{\text{g}}_{a}}n\]
\[(II)\,\,\text{lo}{{\text{g}}_{a}}\frac{m}{n}=\text{lo}{{\text{g}}_{a}}m-\text{lo}{{\text{g}}_{a}}n\]
\[(III)\,\,{{\log }_{a}}a=1\]
\[(IV)\,\,{{\log }_{a}}1=0\]
\[(V)\,\,\text{lo}{{\text{g}}_{a}}m\,({{m}^{p}})=p(\text{lo}{{\text{g}}_{a}}m)\]
\[(VI)\,\,\text{lo}{{\text{g}}_{a}}m=\frac{1}{{{\log }_{m}}a}\]
\[(VII)\,\text{lo}{{\text{g}}_{a}}m=\frac{{{\log }_{b}}m}{{{\log }_{b}}a}=\frac{\log m}{\log a}\]

Catgeory: 8th Class

Solid Shapes

SOLID SHAPES FUNDAMENTALS

Description of Some basic shapes:

(a) Square

It has four sides and four comers. All its sides are of the same length. (b) Rectangle.

It has four sides and four comers. The opposite sides of a rectangle are parallel and of the same length. Every interior angle is a right \[\angle \]le. (c) Triangle

It has three sides and three vertices. (d) Cuboid

It has 6 flat faces, 12 straight edges and 8 vertices. (e) Cube

It has 6 flat faces, 8 vertices and 12 straight edges. (f) Cylinder

It has 3 faces \[\to \] 1 curved face and 2 flat faces. (g) Cone

It has 2 faces \[\to \] 1 curved face and 1 flat face. It has 1 curved edge.

Two- dimensional shapes have only length and breadth.
Three dimensional shapes have length, breadth and height or depth.
Three-dimensional (or 3-D) shapes can be visualized on a two dimensional (or 2-D) surface.
A net is a skeleton-outline in 2-D, which when folded results in a 3-D shape. The same solid can have several types of nets.
Dice are cubes with dots on each face: Opposite faces of a die always have a total of seven dots on them. Generally, dice have number, 1 to 6 on their faces.
A solid can be sketched in two ways.

(a) An oblique sketch which does not have proportional faces but gives a realistic feel of the 3-D solid. e.g., Oblique sketch of a cuboid.

(b) An isometric sketch, drawn on an isometric dot paper, which has proportional measurements of the solid.

Different sections of a solid can be viewed in many ways:

(a) Slicing the shape results in the cross - section of the solid. (b) Observing a 2 - D shadow of a 3 - D shape. (c) Looking at the shape from different angles, i.e., the front-view, the side-view and the top-view. Front and top view of a cylinder

Description of few more solid shapes

S. No.

Name of the figure

more...

Catgeory: 8th Class

Number System

NUMBER SYSTEM FUNDAMENTALS

A number r is called a rational number if it can be written in the form \[\frac{p}{q}\], where p and q are integers and \[q\ne 0.\]

Example:- \[\frac{1}{2},\frac{1}{3},\frac{2}{5}\] etc.

Representation of Rational Number as Decimals.

Case I :- When remainder becomes zero \[\frac{1}{2}=0.5,\frac{1}{4}=0.25,\frac{1}{8}=0.125\]

It is a terminating Decimal expansion.

Case II :- When Remainder never becomes zero.

Example:- \[\frac{1}{3}=.3333,\frac{2}{3}=.6666,\]it is a non - terminating Decimal expansion.

There are infinitely large rational numbers between any two given rational numbers.

Irrational Number:- The number which cannot be expressed in form of \[\frac{p}{q}\]and neither it is terminating nor recurring, is known as irrational number.

Examples:- \[\sqrt{2},\sqrt{3}\] etc. Rationalization :- Changing of an irrational number into rational number is called rationalization and the factor by which we multiply and divide the number is called rationalizing factor. Example:- Rationalizing factor of \[\frac{1}{2-\sqrt{3}}\] is \[2+\sqrt{3}\]. Rationalizing factor of \[\sqrt{3}+\sqrt{2}\,is\,\sqrt{3}-\sqrt{2}\] Low of exponents for real numbers.

\[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
\[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]
\[{{({{a}^{m}})}^{n}}={{a}^{mn}}\]
\[{{a}^{o}}=1\]

Some useful results on irrational number

Negative of an irrational number is an irrational number.
The sum of a rational and an irrational number is an irrational number.
The product of a non - zero rational number and an irrational number is an irrational number.

Some results on square roots

\[{{\left( \sqrt{x} \right)}^{2}}=x,x\ge 0\]
\[\sqrt{x}\times \sqrt{y}=\sqrt{xy},\,x\ge 0\,and\,y\ge 0\]
\[\left( \sqrt{x}+\sqrt{y} \right)\times \left( \sqrt{x}-\sqrt{y} \right)=x-y,(x\ge 0\,and\,y\ge 0)\]
\[{{(\sqrt{x}+\sqrt{y})}^{2}}=x+y+2\sqrt{xy},(x\ge 0\,and\,y\ge 0)\]
\[{{\left( \sqrt{x}-\sqrt{y} \right)}^{2}}=x+y-2\sqrt{xy},(x\ge 0\,and\,y\ge 0)\]
\[\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}},(x\ge 0\,and\,y\ge 0)\]
\[\left( a+\sqrt{b} \right)\left( a-\sqrt{b} \right)={{a}^{2}}-b,(b\ge 0)\]

\[\left( \sqrt{a}+\sqrt{b} \right)\times \left( \sqrt{c}+\sqrt{d} \right)=\sqrt{ac}+\sqrt{bc}+\sqrt{ad}+\sqrt{bd},\]\[(a\ge 0,b\ge 0,c\ge 0)\]

Catgeory: 8th Class

Rational Numbers

RATIONAL NUMBERS FUNDAMENTALS Rational Number:-

A number which can be expressed as\[\frac{x}{y}\], where x and y are Integers and \[y\ne 0\] is called a rational number.

e.g., \[\frac{1}{2},\frac{2}{2},\frac{-1}{2},0,\frac{3}{-\,2}\] etc.

Set of rational number is denoted by Z.
A Rational number may be positive, zero or negative
If \[\frac{x}{y}\] is a rational number and \[\frac{x}{y}>0\], then\[\frac{x}{y}\] is called a positive Rational Number.

e.g., \[\frac{1}{2},\frac{2}{5},\frac{-3}{-2},-\left( -\frac{1}{2} \right)\]etc. Negative Rational Numbers:-

If \[\frac{x}{y}\] is a rational number and \[\frac{x}{y}<0\], then \[\frac{x}{y}\]is called a Negative Rational Number.

e.g., \[\frac{-1}{2}.\frac{3}{-2},\frac{-7}{11}......\]etc. Standard form of Rational Number:-

A Rational number \[\frac{x}{y}\] is said to be m standard form, if x and y are integers having no common divisor other than one, where \[y\ne 0\].

e.g., \[\frac{-1}{2},\frac{5}{6},\frac{8}{11}\]……etc. Note:- There are infinite rational numbers between any two rational numbers. Property of Rational Number

Let x and y are two rational number and y > x, then the rational number between x and y is\[\frac{1}{2}\left( x+y \right).\]

e.g., find 2 rational number between \[\frac{1}{3}\]and \[\frac{1}{2}\] Solution:- Let \[x=\frac{1}{3}\] and \[y=\frac{1}{3}\] and y > x. Then, Rational no. between\[\frac{1}{3}\]and\[\frac{1}{2}\]is \[\frac{1}{2}\left( \frac{1}{3}+\frac{1}{2} \right)=\frac{1}{2}\left( \frac{2+3}{6} \right)=\frac{5}{12}\] Again Let \[x=\frac{5}{12}\] and \[y=\frac{1}{2}\] and y > x. then Rational no. between \[\frac{5}{12}\] and \[\frac{1}{2}\] is \[\frac{1}{2}\left( \frac{5}{12}+\frac{1}{2} \right)=\frac{1}{2}\left( \frac{5+6}{12} \right)=\frac{1}{2}\times \frac{11}{12}=\frac{11}{24}\] Hence the Rational Numbers between \[\frac{1}{3}\] and \[\frac{1}{2}\] are \[\frac{5}{12}\] and \[\frac{11}{24}\].

Let x and y are two rational number and y > x. Consider to find n rational numbers between x and y. Let d = \[\frac{y-x}{n+1}\]

Then 'n' rational number lying between x and y are \[\left( x+d \right),\left( x+2d \right),\left( x+3d \right),\_\_\_\left( x+nd \right).\] Example:- Find 9 rational number between 2 and 3. Solution:- Let x = 2 and y = 3 then y > x Now \[\mathbf{d}=\frac{y-x}{n+1}=\frac{3-2}{9+1}=\frac{1}{10}\] Then, rational number are, 2 + 0.1, 2 + 0.2, 2 + 0.3, 2 + 0.4, 2 + 0.5, 2 + 0.6, 2 + 0.7, 2 + 0.8, 2 + 0.9 = 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8 and 2.9. Representation of Rational Number on the Number line

To represent - on the number line first we draw a number line

Let O represent 0 (zero) and A represent 1. So divide OA into 4 equal parts, each point in the middle representing P, Q and R. Point R represent\[\frac{3}{4}\].

Operations on Rational Numbers

Addition of Rational Numbers:

Example: Find the sum of the rational numbers \[\frac{-4}{9},\frac{15}{12}\] and \[\frac{-7}{18}\]. Solution: \[\frac{-4}{9}+\frac{15}{12}+\frac{-7}{18}=\frac{-16+45-14}{36}=\frac{15}{36}=\frac{5}{12}\] Properties of Addition of Rational Number