## Comparing Quantities-grade 08

###### Ratios

Ratios can be use to compare quantities. Example. If we have 5 orange and 20 apple the we can write the ratio 5:20. If we want to write the ratio of apple and orange the it will write as 20:5. The ratio is writing respectively.

###### Ratio as fraction

The ratio can also write as in terms of fractional number. Here the first ratio will come as numerator and and the second ratio will come as denominator. So in our previous example we can write the ratio of orange and apple is 5:20 = 5/20.

If we are saying the ratio of orange and apple is 5:20, that means if there is 5 orange then there is 20 apple. The ratio will always write as simplest form. So 5:20 will write as 1:4. That means there will be 1 orange for 4 apple. Like this we can compare the quantity.

###### Ratio as percentage

From the ratio we can find the percentage of quantities.For example. 5 orange and 20 apple can write as 5/20. now we will make the denominator as 100. then it will become (5×5)/(20×5)= 25/100. So the percentage of orange is 25%. (that means there will be 25 orange when apple is 100). We can also find the fraction by unitary method like ( 5/20)x100 = 500/20= 25%.

###### Finding the increase or decrease percentage

To find the increase percentage of thing, for example. The price of a scooter was 34000 in last year. if the price increased by 20 % what will be the price of the scooter this year.

**Method -1:** Here first we will find the 20% of the current price and then add that to the existing price then we will get the new price.

(20/100) x 34000 = 6800

increased price = 34000+6800= 40800

Method2: Unitary method. (120/100)*34000 =40800 ( here the 120 = 100% + the increased 20% =120%)

To find the decrease percentage of thing, for example. The price of a scooter was 34000 in last year. If the price decreased by 20 % what will be the price of the scooter this year. Here also we can find the 20% of the current price and can subtract the amount form the current price, then we will get the present price.

In unitary method we can find the 80 % of the current value (because the decreased percentage is 20%). which is (80/100)*34000.

###### finding discounts

Discount is the difference of the actual price(MRP) and the sales price.

Discount= market price- sales price.

If an item Have the market price 840 and it is selling at 714 then

Discount = 840-714=126.

we can also find the discount % by

Discount %= (discount amount / market price) *100.

(remember (% /100)*total=value)

We can also find the discount amount if the discount % is given by

Discount amount= (discount%*market price)/100.

## WordPress Elementor Astra basic tutorial

By the use of elemetor and astra theme we can make very stylish wordpress site. Please follow below instruction to install and work with elementor and astra.

###### Installation

First we have to install wordpress in our domain. I Am using virtualmin control panel to manage my server. In virtual min we can install wordpress in single click from the install script option. After the installation, we can do the preliminary setup for the site by the link provide.

###### Installing elementor

After completing the wordpress installation we can login into wordpress using the username and password we created during the installation. The login link will be domain name/wp-admin.

After log in from the dashboard, go for the plugin option and click on add new plugin. Then search for the elements in side the plugin option. you can see that the elementor page builder plugin. And click on install and the activate. your element is ready.

###### Installing astra theme

The astra is the stylish theme. But in order to run astra we need elementor. That is the reason we install elementor previously. To install astra theme we have to go appearance then theme. Click on add new. then search for astra. Then install and activate astra. Now everything is finish to work with elementor and astra.

###### creating new pages

For creating new pages just go to page option from the dashboard. In the new installation we can see two default page , sample and privacy page. By placing the cursor over the page name edit delete and other option will pop up and we can use that if needed. We can also use bulk action option after selecting multiple pages.

To create a new page just click on add new and give the page title and click to publish an select the public option. So that the page will saved and publish. And now, again we can go to the pages option and click on the page title and use edit with elementor . So that we can start to build the page with elementor. In order to exit the elementor, just click on the menu bar on the left top corner and we can see exit elementor.

## Triangles grade-09

Triangl is a three sided polygon. Triangle is made up with the parts of sides, angles and vertex.

###### congruence of triangle

If an object is Congruence with other object, then the both object can be perfectly site one over another. So if two triangle are congruent then all of the parts of the two triangle should be same.

Consider the triangle i, ii, we can say both triangle are congruent because all the side and angle are same for bot triangle. But here we have to take care to say the name of both congruent triangle. Because in the figure i and ii, the two triangles will be congruent only if we keep the side AB of the one triangle over the side PQ of the another triangle. So the name of both congruent triangle should be in the congruent order.

here ΔABC≅ΔPQR. It can be write in many ways corect as well as wrong. so please take care and write the name of other congruent triangle in the fig. with the triangle ABC.

**Important Notes.** Corresponding parts of congruent triangle are same (**CPCT**).

In Order to find two triangles are congruent we have to check by keep both triangle one over another. This is not possible in all time, so now we can study some criterias to check the congruence of triangles.

**SAS Congruence rule**

*if two side and one include angle are same for two triangle then the two triangle will be in congruence*

This can’t be proven by previously known result. So it is accepted true as an **axiom.**

###### ASA congruence rule

If two angles and included side of two triangles are same then they are in congruent.

This can be proved so that it can be called as a theorem.

**Theorem 7.1 (ASA) **:

In the Above fig ∠B=∠E, ∠C= ∠F and BC=EF (Given)

consider AB=DE Then we can say that ΔABC≅ΔDEF (SAS RULE)

now jus we have to prove that AB=DE

Suppose BA greater than DE, Then we can construct a line PC as shown in the fig. 7.12

Then PB=DE (by construction)

∠B=∠E (given)

BC=EF (given)

Ther for ΔPCB≅ΔDEF (SAS)

As we know CPCT are same

∠PCB=∠DEF (CPCT)

∠ACB=∠DEF (given) there for ∠PCB=∠ACB and this is only possible when P is coincide with A

There for BA =ED and ΔABC≅ΔDEF (SAS RULE) so ASA rule is proved for the congruence of triangles.

###### AAS CONGRUENCE RULE

This congruence rule says that if any two angle and any one side of two triangle is same then the triangle will be in congruence.

This is happening because, we know that the sum of all the angle of a triangle is 180º. There for if two angle is same for two triangle then the third angle will also same. (same thing is subtracted from equal then the result will also same). If three angle and one side is same the these triangles will in congruence by ASA rule. And hence we can say that AAS rule is valid.

###### Some property of a triangle

As we know in an isosceles triangle the two side are equal. By constructing many Isosceles triangle we can check the angle opposite to the same side. then we can see that the opposite angles of the same sides are equal. This one can be prove by the congruence rule.

**Theorem 7.2 : **Angles opposite to equal side of an *isosceles triangle are equal*

Consider the triangle ABC where AB=AC , Now constructing the bisector of the angle A with the line AD. Now consider the two triangles BAD and CAD

AB=AC (Given)

∠BAD=∠CAD (By construction)

AD=AD (common line)

ther for ΔBAD≅ΔCAD

Ther for ∠B=∠C (CPCT)

So it is proved.

By the observation, trial and error we can say that converse of this theorem is also true. That is if two angle of any triangle equal then the opposite side will also equal. It can also proof as a theorem.

**Theorem 7.3 : **The side opposite to equal angle of a triangle are equal.

proof

Consider the triangle ABC. And draw the bisector of the ∠A by the line AD, is also perpendicular to BC.

∠BAD=∠CAD (By construction)

AD=AD (Common Side)

∠ADB=CDA=90 So,

ΔABD≅ΔACD (ASA rule) So,

AB=AC (CPCT)

###### SSS congruence rule

**Theorem 7.4 :** *If three side of one triangle are equal to three side of other triangle then two triangle are congruent*

As we know if three side of two triangle are equal then there three angle are also equal. so this theorem can proven by SAS SSA rules.

###### RHS congruence rule

**Theorem 7.5 : ***In two right angle triangle if the hypotenuse and one side is same then two triangle are congruent.*

Here RHS stands for Right angle, Hypotenuse, Side. But here this three can be any order.

###### Inequalities in triangle

By observing we can understand that any one side of the triangle increased then the opposite angle also increase.

**Theorem 7.6 :*** If two side of the triangle are unequal, the angle opposite to the larger side is larger.*

From this fig. we can see that, when the one side of the triangle increased, the angle opposite to that side will also increase. thus the theorem is proved.

**Theorem 7.7** *We can also say that the side opposite to the biggest angle will be the logestside. *

**Theorem 7.8** *The sum of any two side of a triangle is greater than the third side.*

## Ruben

## Simple Equations -Grade-07

###### Equations

Equation is a mathematical expressions connected with equal operator.

Normally an equation always have a variable. Variable is a letter in an equation and it can vary its value n different time. So now we can says that an equation is the condition on variable.

for example 4x +5=17, Here x is the variable.

An equation always have a equal operator and one LHS and one RHS. The equation will only satisfied (true) when LHS =RHS.

In order to satisfy an equation, the variable in the equation should have a particular value.

The value of the variable in which the equation will satisfy is called solution of the equation.

The process to finding out the solution of the equation is called solving the equation.

**Now we can practice to convert some conditional statement to equation**

The sum of three times x and 11 is 32

3x+11=32

If you subtract 5 from 6 times a number., you get 7

6x-5=7

One fourth of m is 3 more than 7

m/4=3+7 or m/4-7=3

One third of a number plus 5 is 8

n/3+5=8

**Now we can practice to make conditional statement from an equation **

x-5=9, 5p=20, 3n+7=1, (m/5)-2=6

First one is, subtracting 5 from a number is 9. Other equation you can practice.

Raju’s father’s age is 5 more than 3 times raju’s age. If raju’s father have 44 year old . Set Up an equation and find raju’s age.

We can also practice same type of other questions.

###### Solving an equation

The following points we can keep it in our mind to solve an equation.

- All ways the LHS and RHS will be same in value.
- Equations will still valid if we add same number to RHS and LHS.
- Equations will still valid if we subtract same number from LHS and RHS
- Equations will still valid if we multiply or divide same number in LHS and RHS

If we move any value fRom LHS to RHS or RHS to LHS we have to follow the below mentioned

###### Transposition rules

- The adding value will subtract in opposite side.
- The subtracting value will add in the opposite side.
- The multiplication value will divide in the opposite side.
- The dividing value will multiplied in the opposite side.

First we have to practice solving the equation by separate the variable and solve the equation. For this method we are not using the transposition method, but we will add, subtract, multiply or divide both side in such a way that the variable will be separated by cancel the other terms with the variable.

for example x-1=0, here we can add 1 to both side, so that the -1 will be cancelled in the LHS and we will get the value of the x. we can practice such a exercise from test book.

We can also solve the equation by transposition method. For solving by this method please follow the transposition rules mentioned above. We can also practice the exercises for this method.

###### From solution to equation

Finding a solution from the equation is called normal path. So we move on reverse path, that means from the solution steps back to equation.

If we have a solution like x=5 or y=9, from this we can obtain their equations. To find the equation from solution we can add, subtract, multiply or divide equal numbers on the both side of the solution.

for example x=5, x+2=5+2 , which can write as x+2=7 (is an equation).

We can also see that a single solution can have any number of equations. Now please practice the exercises.

###### application of simple equation in practical situation

As we already seen we can convert the actual situation to an equation. And hence we can solve the equations as we studied. Now we can do the exercise for dealing with more practical situations and learn how to solve them.

## Playing with numbers Grade-8

**Number in general form**

Consider a number 52, It can also write as 50+2 or 10*5+2. Same way 37 can write as 10*3+7. In general a two digit number *ab *made of digit *a* and *b, then*

*ab=10*a+b or ab=10a+b*

writing the number in *10a+b* is called generalised form of a number. If a number is write in normal form such as *52* then it is calle usual form of the number.

Similarly a three digit number 351 can write as 100*3+10*5+1*1. This can generalise as follows. If a three digit number *abc *made up with digit *a,b,c *, Then

*abc=100*a+10*b+1*c or abc= 100a+10b+c*

Now we can do the exercise with more two digit and three digit number.

**Reversing the two digit number**

COnsider a two digit number *ab. *If we reverse the number we will get *ba. *If we add both number together we will get as follows.

ab= 10a+b (generalise form)

ba=10b+a

ab+ba=10a+b+10b+a

11a+11b

11(a+b)

From above expression we can understand that, The sum of a two digit number and it reverse will be the multiple of 11. And if divide the sum we will get the sum of the digit as quotient. now we can try with some more numbers.

Now we can check the subtraction of a two digit number with its reverse. **Here we have to subtract the smaller number from the big one**. The number *ab*

ab-ba=(10a+b)-(10b+a)

10a+b-10b-a=9a-9b

9(a-b)

From the above expression we can understand that the subtraction of two digit number with it reverse will be the multiple of 9. and if we divide the result we will get the different of the two digits. Now we can do this exercise wit more two digit numbers.

**Reversing of three digit number**

As explained above we can check for a three digit number. Here the three digit number cant participate in addition. but it can do with the subtraction. Find ** abc-cba **( here we have to subtracts small number from big number)

abc=100a+10b+c, cba=100c+10b+a

(100a+10b+c)-(100c+10b+a)

100a+10b+c-100c-10b-a

99a-99c

99(a-c) if c>a then 99(c-a)

From the above expression we can under stand, the the difference of a three digit number and its revers will be the multiple of 99. And if we divide the difference with 99 we will get the difference of fist and third digit. Now we can practice with more numbers.

**Forming three digit numbers with given three digit.**

If we have a three digit number like ** abc, **Then we can make two more three digit numbers by shift the digit of

*Example*

**abc.****Now we can add this three numbers, then divide the result with 37 we will get no reminder. By cheeking with different numbers we can under stand that the sum of any possible three digit number with same digits (like abc) is divisible with 37.**

*abc, cab, bca.***Letters for digit**

In this we will study how can find the value of a letter which is used as a digit in a number. It can be find, when such a Numbers are participating in any arithmetical operations like addition multiplication etc.

To find the value of a letter in a number as said above there is two rules they are as follows.

1.Each letter in a number must stand for just one digit. Each digit must be represent by just one letter.

2. The first digit of an number cant be zero.

Example. 31Q + 1Q3=501, From this expression, we can under stand Q+3 is a number whose once place is 1. So we can see that Q is 8. Now place the value 8 instead of Q and check expression is correct. Now we can find the value of the letters in the following expressions.

1. A+A+A=BA find the value of A and B.

Solution: since the once place of A+A+A=A, it will only happen when A=0 or 5. If A =0 then A+A+A =0 only. Then there is no need of B. So we can try A=5. So 5+5+5=15. That is correct. There for A=5 and B=1

2. Find value of A &B in the expression.

BA * B3 = 57A

As the once place of the result (3*A) is A itself, the can only become 0 or 5. Now we can look at B, if B is 1 then BA*B3 can become maximum 19*19=361. But the the actual result is 57A. that means the actual result is more than 500. So B is not 1. Now consider B is 3. So BA*B3 can get minimum 30*30=900. But the actual value is less than 600. So we can say that B = 2. Now we can put the value of B=2 and A= 0 and do the multiplication. 20*23=460. But this value is not correct. Noe we can take the value of A =5, then 25*23=575. which is correct. So A=5 and B=2.

**Test of divisibility**

In last years we study the divisibility rule. But in this chapter we will see how them works.

**Testing the divisibility of 10**. We know that 10 is divisible if the number have 0 in its once place.

Let us consider any number **ABC.**Then the general form can write as

100A+10B+C. since 100A and 10B are divisible by 10, now only C is responsible for divisibility of ABC. That is only possible when A=0. So this is the reason for divisibility of 10.

**Testing the divisibility of 5. **This one can also enplane as above. Here the 100C and 10B are also divisible by 5 because 10=5*2. Then only A is responsible for divisibility of the number with 5. So the A can only be 0 or 5.

**Testing the divisibility of 2. **This one is also same as above. The 100C and 10 B are also divisible by 2 because 2*50=100. So here is also A is only responsible for the divisibility of the number. So A should be the multiple of 2 in order to get the number divisible with 2.

**Divisibility of 9 and 3. **The explanation of this one cannot work as same as above. We know that the sum of all digit of a number is divisible with 9 0r 3 then the number is divisible with 3 and 9. So here the number r is **ABC.**

100A+10B+C

A(99+1)+B(9+1)+C

99A+9B+(A+B+C)

**9(11A+B**)+(A+B+C)Here the bold and underline part is already multiple of 9 and hence it is divisible with 3 and 9. Then if the other part (A+B+C) is divisible with 3 or 9 then the number is divisible by 3 and 9. Now we can do the exercise for the Patrice.

## Computer & IT

## PHP code for various mysql statement

In Order to read or write data in **mysql **server, first we have to connect with mysql server. The bellow PHP code can use to connect the mysql server.

$servername = “localhost”;

$username = “your_user_nmae”;

$password = “password”;

$dbname = “database_name”;

// Create connection

$conn = new mysqli($servername, $username, $password, $dbname);// Check connection

if ($conn->connect_error) {

die(“Connection failed: ” . $conn->connect_error);

}

When ever the above code execute, The** $conn** variable will have the connection with the particular database. If we want to read the database name from file we can use the following PHP code

`$dbname = file_get_contents("database_name.txt");`

**Reading data from a mysql table**

After making connection with particular database , we can read the data from any table in the database with following PHP code.

`$sql="SELECT * FROM table_name WHERE column_name LIKE '%$q%'";`

$result = $conn->query($sql);

Here the **SELECT **will tell the sql instruction to select the all the rows from the table’ *table_name*‘. And this instruction will select all the row which contain the word in the **$q** variable in the column ‘*column_name*‘. % sumble before the$q variable says that that any words before the $q variable will not consider. Sme way % symbol on the right side of the $q variable says that any words after the $q variable will not consider. After executing the first line of code The instruction will be in the **$sql **variable. During the execution of the second line of the code, all the rows as per the instruction will store in the variable **$result**. Please see the various row selection instructions below.

`$sql="SELECT * FROM table_name WHERE column_name LIKE '%$q%' ORDER BY column_name DESC LIMIT 100 ";`

$result = $conn->query($sql);

The above selection will same as previous one. But the row will be arrange in the varble in descending order of any column name. For making ascending order chege the keyword DESC to ASC.

**Another selection**

`$sql1 = "SELECT * FROM table_name WHERE column_name = '$invoice_number' ";`

$result1 = $conn->query($sql1);

In the above selection the row will select if the exact value of ‘*$invoice_number*‘ variable found in the column_name.

**Another selection**

`$sql1 = "SELECT * FROM table_name WHERE time_stamp >= '$from_time_stamp' AND time_stamp <= '$to_time_stamp' ";`

$result1 = $conn->query($sql1);

In the above selection the variable is compare with logical operator. Here we are checking the variable with multiple column by connecting the instruction with ‘AND ‘operator. We can also use the ‘OR’ operator. Once the rows are selected and stored in a variable by executing the above code, we can get the number of rows in the variable by the following code.

`$num1=$result1->num_rows;`

Now we can get the all values in the selected rows by following code.

`$i=0;`

while ($i < $num1) {$row1 = $result1->fetch_assoc();

$id=$row1['id'];

$date =$row1['date'];

$disc=$row1['disc'];

$invoice_number=$row1['invoice_number'];

$trn=$row1['trn'];

$code1=$row1['code1'];

$code2=$row1['code2'];}

In the above code during the execution of the while loop, in each step the `fetch_assoc()`

will assign the first row values to the variable $row1 and in the second step row2 and in the third step row 3 and complete the all row selected. Now we can assign the values of each column name in that particular row to avariabel by `$code2=$row1['code2'];`

**Inserting a new row in to a table**

To insert a new row into the table first we have to connect the database with the connecting PHP code. Then we can use the following PHP SQL statement to insert a new row.

`$sql = "INSERT INTO table_name (partno, dis, make, price, qty, date, unit, vat, search, orginal_price)`

VALUES ('$partno', '$disc', '$make', '$price', '$qty', '$d/$m/$y', '$unit', '$vat', '$partno $disc $make', '$cost_price')";

$result = $conn->query($sql);

$last_id = $conn->insert_id;

The above code will insert the values of the variables inside the **value **brackets. Each value will inserted in the column described in the table_name bracket respectively. The Table name tels in which table the values will enter. And the result of the insertion will assign to the **$result** variable. We can also get the inserted row id with last line of code.

**Editing a row In a SQL table**

By using the following code we can edit a particular row in a table.

`$sql4="UPDATE table_name SET column_name='$last_id' WHERE id='$last_id'";`

$result4 = $conn->query($sql4);

The above code will update the table ‘*table_name*‘ by setting the new value in the column name ‘*column_name*‘ with the **$last_id** variable. with matching condition of row with WHERE keyword. We can also use various conditions with different logical operators.

**Deleting a row from SQL table**

By using the following code we can edit a particular row in a table.

`$sql18 = "DELETE FROM table_name WHERE id='$id'";`

$result18 = $conn->query($sql18);

The above code will delete the rows from the table_name table, where the column name is **id** and the values of the rows equal to the value in the **$id** variable. We can also use various type of logical operators in the WHERE keyword.

**Creating a new table in a SQL database**

Please use the below code to create anew table in the connected database.

`$sql = "CREATE TABLE table_name(id INT(4) NOT NULL PRIMARY KEY AUTO_INCREMENT, partno VARCHAR(300), disc VARCHAR(300), qty BIGINT(15), unit VARCHAR(50), price FLOAT(12,2), tprice FLOAT(12,2), vat FLOAT(12,2),vat1 FLOAT(12,2), itemcode VARCHAR(300), gtprice FLOAT(12,2), cost_price FLOAT(12,2), total_cost_price FLOAT(12,2) )";`

$result12 = $conn->query($sql);

By the above code we can create a new table ‘*table_name*‘ in the connected database. We should take care the new table name is not already in the same database.

**Notes: **

As we open the connection to a database, we need to close the connection after the use. By using the following code we can close thee database connection.

`mysqli_close($conn);`

**$conn** variable name should be same as used in the connection code.

All the above codes will execute in side a php opening and close tag as shown below.

`<?php ?>`

If any one need more details or help in this topic, Please give a comment bellow.

## Data Handling Grade-6

*A data is collection of numbers gathered together to give some information.*

For example The score of each batsman in a cricket team in a match can be stored and show a an information about the score. This type of lot of examples we can find out.

**Recording data**

We can record the data in different ways. But depending upon the method used to record data, the information from the data will be vary. And the difficulty to get the information will also vary.

**Organisation of data**

**Tally marks : **This method is used to organize data, by the use of tally mark. In the method the name of the item to be write in one column and the number of the corresponding item will marked on the other column. When marking the number it will show as a group of 3,5,10. so that the number can easily read. Please see the below fig. for tally ark table.

We can make different tally mark table for different information. Once we make a tally mark table we can also get the following informations. 1. The minimum number of time 2. The maximum number of time, 3. Same number of time.

**Pictograph**

A pictograph represent the data through pictures of object. It helps answer the question on the data at glance. Seethe picture of pictograph.

In pictograph the item name will listed in one column and the number of itm will show in other column in the form of the picture of the item. In the graph there will be a mark for how many number contain one picture. So we have to multiply the number of the picture with this mark to get the actual number of the item. Please read the below picture for know more about pictograph.

**Drawing a pictograph**

Drawing the pictograph is interesting. But if one picture is the representation of 10, we can make half picture for 5 and 1.4 for 2.5. But for representing 1 or 6 we have to round of the nearest 5 or 0. By doing the exercise we can understand this one easily.

**Bar Graph**

In a bar graph each item will represent as a vertical or horizontal bar with equal width suprated with equal distance. And the length of the each bar will the represent the number of each item.

** Scale of the bar graph : **The scale represent the each unit of length of bar graph is how much number . See below picture of the bar graph. In this the scale is 1 unit length =100 nos. (cars).

From the bar graph we can get lot of information. Just practice with more bar graph so that we can understand well about the use of bar graph.

while drawing the bar graph to get the length of the bar we have to divide the number of items with the scale of the bar graph.

## Cubes and Cube Roots

**Cubes**

In geometry cube is a solid figure with all side equal. And we know that the volume of the cube is * side * side *side* or we can say that side cube. So we can say that a cube is a number which is the product of a same number in three times. ex. 1 * 1* 1= 1³ =1, 2 *2 * 2= 2³=8, 3 * 3* 3= 3³=27. So here 1, 8, 27 are cube numbers. And can also call as perfect cubes. In general we can say that, If a number

*can represent as*

**m****, then**

*n³***is a perfect cube, where**

*m**and*

**m***are natural numbers. Now we can find out the cubes of the numbers from 1 to 10. Then we can understand , there are only 10 perfect cubes between 1 to 1000.*

**n**Now consider the above table, we can see that the cubes of even number is even and the cubes of odd number is odd.

We can also consider some numbers with one it once place and observe the ones place of the cubes, we can see that that will also one. We can also check other numbers ending with 2,3,4 etc and check the ones place of the cubes.

**Some interesting patterns**

From above patterns we can under stand if we add consecutive odd numbers we will get the cubes. As we know one cube is one only. If we want to find cube for 2 from above pattern we can do as follows. 2* (2-1)=2, as the answer is 2, the first odd number to be started is 2+1=3, from 3 we have to add 2consecutive odd number we will get the cubes. That is 3+5 =8 = 2³.

More more clarity we will try for the cube of 3. 3*(3-1)= 6, so First odd number to be started is3+1 =7 and 3 consecutive odd number to be added.

7+9+11=27=3³.

Now we can find the cube of any number using this pattern. try for 6,8 and 7 cube.

**Another pattern**

In the above pattern the 1 and 3 in the beginning and end of the RHS of the all the equation is same. Here the smallest and biggest number is multiplying together and the product is again multiplying with 3 and adding one along with the result. This patterns can be use to find out the differences of cubes of two consecutive numbers.

**Cubes and their prime factors**

If make the prime factor of perfect cube, the we can see that all the prime factor will appear exact three times. Please see the below fig.

If any number in the prime factorization, not exactly in 3 times then the number is not a perfect cube.

**Smallest multiple for a perfect cube**

This is the smallest number, which is used to multiply a non perfect cube to make them perfect cube.

For example, The number 392. When we do the prime factorization we will get 392=2*2*2*7*7. Here we have 2 in perfect three times. But 7 is only in two times. So we need one more 7 in prime factor to make the number in perfect cube. If we multiply the number 392 with seven then that number will become perfect cube. So 7 is the smallest multiple.

In the above case there is 2 sevens, so if divide 392 with (7×7)=49 then also it will become perfect cube.

C**ube root**

If the volume of a cube is 125, what will be the side of the cube. The side will be the number whose cube is 125. For finding that we can do the cube root (³√125). Please see the bellow table for cube root of 1 to 10.

**Cube root through prime factorization**

As we early discus cube root can be find by prime factorization. For that we have to find the prime factors of the given number. Then we have to club each numbers in three set. Then we can multiply the number in each set together we will get the cube root of the number. For example ** 3375=3*3*3*5*5*5, **so

**³√3375=3*5=15**

**Cube root of a cube number**

Now we will study how we can find a a cube root of a cube number by division method. But in this method we cannot find the cube root of non perfect cube number.

Findig the cube root of 857375, ³√857375

**Step1 : **First we have to group the numbers with 3number in a group form right to left. If the last group (left most) has only 2 or one number we can leave them as it is. In our example 857 375 .

Step2: In the first group the ones digit is **5**. So in the ones place number in the cube root will be the number whose cube will have **5** in its once place. the 5 is that number. Because 5 cube is 125 and its once place is 5. So we got the **once place of our cube root, which is 5**.

step3 : Now we can consider the second group of number. Which is **857. **Now we have to estimate the cube root of 857. Such that 9³ =729 and 10³=100. So the cube root of 857 will be in between 9 and 10. Now the smallest number is 9, so the tens place of our cube root is **9. **There for** cube root of 857375=95. **We can check the result by multiplying 95*95*95. We can practice with more cube numbers.