###### 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 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.

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)

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.

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.

###### 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.

1. All ways the LHS and RHS will be same in value.
2. Equations will still valid if we add same number to RHS and LHS.
3. Equations will still valid if we subtract same number from LHS and RHS
4. 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
1. The adding value will subtract in opposite side.
2. The subtracting value will add in the opposite side.
3. The multiplication value will divide in the opposite side.
4. 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.

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 abc. Example abc, cab, bca. 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.

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.

## 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”;
\$dbname = “database_name”;
// Create 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.

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 m can represent as , then m is a perfect cube, where m and n 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.

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.

Cube 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, so³√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.

Decimals

Decimals are the fraction with denominator 10, 100, 1000 etc.  Or we can also say that decimals are the number with decimal points. In decimal number they are not write with denominator but with decimal point. ex. 2.3, 5.67 and it is read out two point 3 and five point six seven. So the decimal number have a whole part and fraction part. please see the fig. for more details.

Decimal fractions

Consider an graph paper with 10 CM square. And if we cut the square in equal 10 parts, then we can say that each part is 1/10 of the whole. And if we again divide the 1/10 parts in 10 equal parts then each part will become 1/100 of the whole. And we can again divide the 1/100 part in equal 10 parts then it will become the 1/1000 of the whole. This one we can practice in graph paper by shading. This 1/10, 1/100, 1/1000 is called decimal fractions.

We can also write the same thing in 0.1, 0.01, 0.001, because we know that to divide a number with 10, 100, 100. we have to just shift the decimal point from write to left. Or we can also do the normal division.

Impotent notes

If there is no whole number part in a decimal number, then we can put one zero on the left side of the decimal point. This type of decimal number is called proper decimals. ex 0.763

If the decimal number have both whole number part and fractional part then it can called improper decimals. ex 4.345

If write zeros to the extreme left (in integer number part) , the value of the number will not change. ex. 00678.4 . And if add zeros on the extrm right ( fractional Part) the value will not change.ex. 45.650000 .

Tenths

the place value of the first number after the decimal point from left Is 1/10. And  this is called Tenths. So the value of the tenths place number 1,4,7 are 1/10, 4/10, 7/10 as the place value is 1/10. It can also write as 0.1, 0.4, 0.7.

Hundredths

the place value of the Second number after the decimal point from left Is 1/100. And  this is called Hundredths. So the value of the hundredths place number 1,4,7 are 1/100, 4/100, 7/100 as the place value is 1/100. It can also write as 0.01, 0.04, 0.07.

Thousandths

the place value of the Third number after the decimal point from left Is 1/1000. And  this is called Thousandths. So the value of the thousandths place number 1,4,7 are 1/1000, 4/1000,7 /1000 as the place value is 1/1000. It can also write as 0.001, 0.004, 0.007.

Representing decimals on a number line

How we can represent 0.6 on the number line. We know that 0.6 is greater than 0 and less than one. So we have to draw a number line with one unit length. And we also know that 0.6 is on the tenths place. So we need to divide our number line in equal 10 part, and we can mark the sixth division as 0.6. Please see the bellow fig.

in fig-2 we can see the 1.2 is marked on the number line. Here the 1.2 is grater than 1 and less than 2. So we have to draw a number line with 2 units of length. And the .2 is in the tenths place so we have to divide the each unit in equal 10 parts. As the 1.2 Have one once, we have to take one complete unit in the number line first. Then we have  to take 2 one by ten unit. And can mark 1.2 as shown in the fig. Now we can try with  some more number in number line.

Fraction as decimals

We all ready know how fraction with denominator 10,100,1000 can write as decimals. Now we can study how we can make all fractions in decimals.

Ex. 11/5, solution, First we can convert this fraction with 10 denominator by multiplying both numerator and denominator by 2. So now it will become 22/10 and which =2.2 in decimal form. Now yo can try with any fractional number. We can also convert the fraction to decimal by normal division. When we divide 11/5 then we will get 2.2 as quotient.

Decimals are fractions

Now we can study how can convert a decimal to fraction .

Ex. 1.2 solution: 1.2=1+2/10, which = 10/10+2/10 = 12/10.

Now we can try with other numbers…

Comparing Decimals

In order to compare two decimals we have to compare the whole part first. If the whole part of one decimal is big then that number will be the bigger one. If whole part is same then we have to compare the tenth place. If the tenth place is bigger for one number then it will be the bigger number. If the tenth place is also same then we have to check the hundredth place and the thousandth place so on.

compare 32.55 and 32.5

32.55 = 32+5/10+5/100,  32.5=32+5/10+0/100, Here the whole part and tenth part are same for both number but the 100th part is bigger for the first number. So the number 32.55 is the bigger one. Now we can try with more numbers.

Length

In length we can say

1m =100cm Or 1cm = 1/100m =0.01m

1cm=10mm 0r 1mm=1/10cm=0.1cm

In order to convert m to cm we have to multiply with 100. And from centimeter to meter we have to divide with 100. (from big unit to small unit multiplication and from small unit to big unit division).

By this we can convert a given length in mm,cm,m or combination of any two.

ex. 156 cm = 1.56m 246cm= 2m and 46 cm or 2.46m.

We can also try with many other length.

Weight

In weight

1000g=1kg or 1gram = 1/1000kg

2350g can be write as 2000g + 350g That means 2kg 350g.

To convert from kg to gram we have to multiply with 1000. I f we need convert fro gram to kg we have to divide with 1000. (from big unit to small unit multiplication and from small unit to big unit division).

Now we can try with different weight.

Measurement of capacity

1000ml=1l  or  ml=1/1000l

In order to convert litter to ml we have to multiply with 1000. And to convert ml to l we have to divide. (from big unit to small unit multiplication and from small unit to big unit division).

Rupees and paise

100 paise = 1 rupee or 1paise = 1/100 rupee..

to convert rupee to paise we have to multiply with 100. In order to convert paise to rupee wee need to divide with 100. (from big unit to small unit multiplication and from small unit to big unit division). Try with exercise