Examples of Real Numbers

We first arrange the product in form of column i.e hundreds, tens and ones and then proceed: 
    Hundreds       tens           ones
       4                   6               3
      X                    1               2
____________________________
                                                    We first multiply 463 by 2, we get
   
 
 Hundreds       tens           ones
       4                   6               3
      X                    1               2
____________________________
    8                   12                6.    Now we find that there is 12 on tens, so we retain 2 on tens place and carry 1 to hundreds place.
 
    Hundreds       tens           ones
       4                   6               3
      X                    1               2
____________________________
     9                    2                6    now we take 463 by 1 which is at tens place, by putting X at ones place
  46                     3                X
___________________________
 5 5                    5                6     We add both the product and get  the result

 So 463 * 12 = 5556.

We first arrange the product in form of column i.e. hundreds, tens and ones and then proceed:
             Hundreds    tens  ones
                   3            6       7
                  X                      3
____________________________ On multiplying by 3 we get
                                         21
  Here we get 21 at ones place so we retain 1 and carry 2 at tens place. This 2 is added to 18 at tens ( 6 * 3 ) + 2 = 20
 
          Hundreds    tens  ones
                   3            6       7
                  X                      3
____________________________ 
                            20          1
--------------------------------------------------
Here we get 20 at tens place so we retain 0 and carry 2 at hundreds place. This 2 is added to ( 3 X 3 ) = 9 + 2 = 11
 
 
          Hundreds    tens  ones
                   3            6       7
                  X                      3
____________________________ 
                  11          0          1
-------------------------------------------------
 So answer is 1101. 

We know  by zero property of multiplication of the numbers, that any number multiplied by 0, the result is always 0
                 So 180 X 0 = 0 

We know by identity property of multiplication, that when any number is multiplied by 1, the product is the number itself, so
=> 356 * 1 = 356. 

First we write the numbers in their proper places and get:
          hundreds            tens     ones
              3                       8          5
              X                       2          0
________________________________
             0                        0          0   we first multiply all the digits by 0, which gives 0 at all the places.
 Now we multiply 385 by 2 . Before this  we put X at ones place.  so we get
 
        hundreds            tens     ones
              3                       8          5
              X                       2          0
________________________________
             0                        0          0 
 
             6                       16         X  This 16 is at tens place, so we retain 6 at tens place and carry 1 to hundreds place
_______________________________
 
 
hundreds            tens     ones
              3                       8          5
              X                       2          0
________________________________
           1+ 0                     0          0 
 
             6                       6         X  
________________________________
             7                       6          0 
________________________________

So, answer is 760.  

In such questions, we first multiply 135 by 9 and then multiply it by 100, i.e. we put 2 zeroes with the answer on the right hand side.
 Lets see how to proceed:
   hundreds     tens      ones
     1                 3            5
      X                              9
________________________
                                  45, we find ( 5 X 9 = 45 ), we write 5 at tens place and carry 4 to be added to tens place.
 
 hundreds     tens      ones
     1                 3            5
      X                              9
________________________
                    (27 +4)      5 
 hundreds     tens      ones
     1                 3            5
      X                              9
________________________
                       31           5
_________________________
Now from 31, 1 is left at tens place and 3 is carried over to hundreds place
 hundreds     tens      ones
     1                 3            5
      X                              9
________________________
    (9+3)             1         5 
 ________________________
Adding 9 + 3 = 12 , so we get
 
 hundreds     tens      ones
     1                 3            5
      X                              9
________________________
    12               1            5  
________________________
  
Now we multiply 1215 with by 100, we get:
 = 1215 X 100,
 = 121500.

In such problems, when we have 300 * 20,
So, it can be written as 3 * 2 * 1000,
= 6 * 1000,
= 6000. 

We find the sum very complicated and rather difficult to solve. But remember when any number is multiplied by 0, the result is always 0. This property of multiplication is called the property of multiplication with 0. So we get the result of this problem without actually solving the question,
= 0. 

We know that any number multiplied by 1 gives the same number as the result. This is called the identity property of multiplication.
Thus, when 840 * 1 = 840.

 Here we first multiply 240 by 3, which is at ones place and then multiply 240 by 20 (as 2 is at tens place, then we add both the digits to get the desired result).
             Hundreds   tens   ones
               2                4        0
                                X         3
 ___________________________
              7               2          0 (we get 7   as 3 * 2 = 6 add 1, which is carried from tens place)
Now 
 
      Hundreds   tens   ones
          2                4        0
                          X         2
 ___________________________
         4               8          0
 Thus 240 X 20 = 4800
 
Now to get the result of 240 X 23, we add 4800 + 720 and get,
  
      4  8  0  0
  +      7  2  0
    ___________
      5  5  2   0  
_______________

We first arrange the product in form of column of tens and ones and then proceed: 
    Tens      Ones
    2             3
   X              5
____________
              15, we write 5 in ones place and take 1 as a carry over, so it becomes
 
    Tens      Ones
    2             3
   X              5
____________
   11          5    

Here we multiply 2 * 5 = 10 and add 1 to it, so at ones place we get 5 and at tens place we get 11.
So  we have 23 X 5 = 115.

In such cases, we need not to apply column method to get the solution to this.
As any number is multiplied by 1, the result is same number. This property is called the identity property of multiplication.
So we get:
= 6009 * 1,
= 6009.

In such problems we can write them as 
345 * 2 * 100,
So first we solve 345 * 2
Write the digits as per their place value and then multiply:
 
 
        Hundreds    Tens    Ones
            3               4         5
            X                          2
      ______________________
         6                 8         10, from 10, 0 retains the ones value and 1 is carried over to tens place,
  We get,
    Hundreds    Tens    Ones
            3               4         5
            X                          2
      ______________________
         6                 9           0
      ______________________
 Now we multiply 690 by 100,
 = 690 * 100,
 = 69000.

Hundred   tens  ones
            8              0       0
            X              1       7
_______________________
        56               0       0, we put X at ones place and then multiply 800 by 1, we get
 
 Hundred   tens  ones
            8              0       0
            X              1       7
_______________________
        56               0       0 
        8 0              0       X Now we add the two rows and get the result
   ______________________
    13  6              0        0
________________________
So, answer is 13600.

First we arrange the digits at proper place and then proceed:
 
        Hundreds     tens      ones
            4                5            0
            X                2            5
____________________________
          20            25             0    from 25 of tens place, 2 is carried to hundreds place, we get
And get:
 
         Hundreds     tens      ones
            4                5            0
            X                2            5
____________________________
          22               5            0 ,   Now we place X at once place  and then multiply 450 by 2, we get
 
         Hundreds     tens      ones
            4                5            0
            X                2            5
____________________________
          22               5            0 , 
         9 0              0            X
___________________________
      11 2              5            0 
___________________________
So, answer is 11250.