Distributive Laws

It says that multiplying a number by a group of numbers added together is same as doing each multiplication separately. 

1. a * ( b + c ) = a * b + a * c

Examples : 

1. 7 * (50 - 2 )
   = 7 * 48
   = 336
lets rearrange
( 7 * 50 ) - (7*2)        [ here, a * ( b - c ) = a * b - a * c]
=350 -14
= 336 

2. 625 * (-35) + ( -625 ) * 65 
 that is ,  [ 625 * (-35) ] +  [ ( -625 ) * 65 ]
              = (-21,875 ) + ( -40,625 )
              = - 62,500
lets rearrange
[ 625 * (-35) ] +  [ ( -625 ) * 65 ]
here, 625 is common in both , so lets take 625 out
that is, 625 * ( -35 - 65 ) 
          = 625 * -100
          = - 62,500

Associative Laws

Associative Laws

It means that it doesn't matter how we group the numbers.

1. ( a + b ) + c = a + ( b + c )
2. ( a * b ) * c = a * ( b * c ) 

Examples :

1. ( 3 + 2 ) + 5
   = 5 + 5
   = 10
lets regroup it,
 3 + ( 2 + 5 )
= 3 + 7
= 10

2. ( 3 * 2 ) * 5
   = 6 * 5
   = 30 
lets regroup it,
3 * ( 2 * 5 )
= 3 * 10
 = 30 

Commutative Law

Commutative Law

The Commutative laws which means we can swap numbers over and still get the same answer when we add or multiply.

1. a + b = b + a
2. a * b = b * a

Examples : 

1. 3 + 4 = 7
now, lets swap
  4 + 3 = 7
even if we swap we are getting same answer 7.
 
2. 3 * 4 = 12
    4 * 3 = 12
 
3. 8 * 53 * (-125)
   = 424 * (-125)
   = -53,000
lets swap
   53 * (8*(-125))
 = 53 * (-1000)
 = -53,000
even if we swap we are getting same answer -53,000

NOTE :
 - *- = + 
 + * + = +
 + * - = -
 - *+ = -

4. 15 * (-25) * (-4) * (-10)
   = [15 * (-25) * (-4) ] * (-10)
   = 1500 * (-10)
   = -15,000 
lets swap
15 * (-25) * (-4) * (-10)
   =  (-25) * [15 * (-4)  * (-10)]
   =  -25 * 600
   = -15,000

Factors and Multiples

Factors

        Factors can be defined as the numbers which we can multiply together to get another number.

Example

2 * 3 = 6 , here 2 and 3 are the factors 6

2  *4 = 8 , 2 and 4 are the factors of 8.

Note : A number can have many factors

Example

1 * 12 = 12

2 * 6 = 12

3 * 4 = 12

 so the factors of 12 are 1 , 2 , 3 , 4 , 6 and 12.

Multiples

        A multiple is the result of multiplying a number by an integer.

Example

Multiples of 2

2 * 1 = 2

2 * 2 = 4

2 * 3 = 6 ,...

so the multiples of 2 are 2 , 4 , 6 , 8 , ...

Note : Factors and multiples are different but they both involve multiplication.

Class 6 : Excersice 10.3

10. By splitting the following figures into rectangles, find their areas ( The measures are given in cms )

a.

Solution: 

First we have to broken the given figure into rectangles as shown above (in blue ).

now lets find each rectangles area

Area of 1st Rectangle :
length = 4cm , width = 2cm
Area = 4 * 2 = 8 cm square

Area of 2nd rectangle :
length = 6cm , width = 1cm
Area = 6 * 1 = 6 cm square

Area of 3rd rectangle :
length = 3cm , width = 2cm
Area = 3 * 2 = 6 cm square

Area of 4th rectangle :
length = 4cm , width = 2cm
Area = 4 * 2 = 8 cm square

Area of the given figure = area of 1st rectangle + area of 2nd rectangle
                                         + area of 3rd rectangle + area of 4th rectangle  
                                       = 8 + 6 + 6 + 8 cm square
                                       = 28 cm square

Class 6 : Excersice 10.3

12. How many tiles whose length and breath are 12cm and 5cm respectively will be needed to fit in a rectangular region whose length and breath are respectively :

a. 100cm and 144cm.

Solution

first let we find
 the total area of the region = 100 * 144
                                            = 14400 cm square
area of one tile = 5 * 12
                         = 60 cm square
Number of tiles required = 14400 / 60
                                        = 240

b. 70 cm and 36 cm

Solution

first let we find
 the total area of the region = 70 * 36
                                            = 2520 cm square
area of one tile = 5 * 12
                         = 60 cm square
Number of tiles required = 2520 / 60
                                        = 42

Class 6 CBSE : Chapter 10

11. Split the following shapes into rectangles and find their areas. ( The measurement given in cm).

a.

in the above picture,

first let we name the corners as shown in the figure
here, we have to shapes into rectangle
so, to split the given shape into rectangles lets draw a line (in orange) 
and name that as CX.
given, AB = 2cm
hence CX also 2cm 
since ABCX forms rectangle (opposite sides are equal in length)
BC = 10cm which means AX = 10cm
1st rectangle ABXC
area of reactangle ABXC = 10 * 2
                             = 20 cm square 
2nd rectangle XDFE
given, AD = 12, we knew AX = 10
so, XD = AD - AX = 12 - 10
       XD = 2cm
XD = EF = 2cm
given, DF = 10cm
XE = XC + CE = 2 + 8 =10
so, XE = 10cm
now we know all the sides measurement
lets find its area now
Area of rectangle XDFE = 10 * 2
                                        = 20 cm square
we have to find  
Area of given shape = area of rectangle ABXC + area of rectangle XDFE
                                 =              20                       +               20
                                 =   40 cm square.

b. 

in the above figure 
first let we name all the corners as ABCDEFGHIJKL
lets split the above shape into square ABLC, rectangle KDEJ and another square IFGH
 Area of square ABLC  = 7 * 7 = 49 cm square
Area of square IFGH = 7 * 7 = 49 cm square
Area of rectangle KDEJ 
here, KD = KL + LC + CD = 7 + 7 + 7 = 21
Area of rectangle KDEJ = 21 * 7 = 147 cm square   
Area of  ABCDEFGHIJKL = Area of square ABLC  +  Area of square IFGH + 
                                                                                                      Area of rectangle KDEJ  
                                                        =  49 + 49 + 147
                                                        = 245 cm square

c.  

 













In the above figure,
lets first name the corners ABCDEFGH
we split the above given shape into two rectangles
ABCD and EFGH
Area of rectangle ABCD :
CD = CE + EF + FD = 2 + 1 + 2 = 5 ( GH = 1,so EF =1 )
AB = CD = 5 , AC = BD = 1
Area of rectangle ABCD = 5 * 1 = 5 cm square

Area of rectangle EFGH :
EF = GH = 1
EG = FG = 4
Area of rectangle EFGH = 1 * 4 = 4 cm square

Area of the given shape = Area of rectangle ABCD + Area of rectangle EFGH
                                       =      5   +   4
                                       = 9 cm square

Grade 1 : Numbers

Numbers :

Numbers are the basic thing of mathematics. Numbers are used to count and to measure.

Positive Number :

A positive number is a number which is above 0. It can be written as +1, +2, +3,... or just number itself like 1, 2, 3,...

Negative Number :

A negative number is a number which is below 0. It can be written as -1, -2, -3,...

Example :

Positive and negative numbers are used  in everyday life is in measuring temperature.

If the temperature falls below 0 degree C, we can use negative numbers.

Natural Number :

The numbers which we use to count is called as Natural Number.

1, 2, 3, 4,.... are some of natural numbers

Whole Number :

Whole numbers are positive numbers including 0 without any decimal or fraction.

0, 1, 2, 3, 4,... are some of whole numbers.

NOTE : All natural number are Whole number and not vice versa because 0 is not a natural number.

Integers :

An integer is a whole number that can be positive , negative or zero.

that is, it includes

                    * counting numbers ( Natural number )

                    * Zero

                    * negative of the counting numbers (-1, -2, -3,...)

So, ...,-3, -2, -1, 0, 1, 2, 3, ...

NOTE : We can say all natural and whole numbers are integers. 

Even Number : 

Any integer that can be divided by 2 without leaving remainder. 

2, 4, 6, 8,... . Even number can also be in negative.

In simple we can say, even number is an number which we can share equally into two portions.

Odd Number :

 Any integer that cannot be divided by 2 without leaving remainder.

1, 3, 5, 7,...
Odd number can also be in negative.

In simple we can say, odd number is an number which we cannot share equally into two portions.