Thursday, November 24, 2011

Huge Calculations are Easy now...!!

Hey friends!, as i was doing calculations mainly some huge multiplications, divisions and squares, i found that they are consuming longer time than the whole problem has to be solved. So i tried to browse for the better shortcuts of doing them. After sparing some time in practicing those methods i gradually reduced the time of doing those calculations. So here in this post i will write the methods of using them.



  • NOTE:

Before entering into the explanation, i want the reader of this article has to remember one key point before starting implementing. That is, almost all methods uses the technique of excess carried over. As one single place will have only one digit, if the method gives the result in 2 digits for one place then the excess will be forwarded to the next place. Ex: If units place has 12 as a result, then 2 will be places in units place and 1 will be forwarded to the next cell i.e., tens place.


Firstly, we will deal with MULTIPLICATION:

To multiply two numbers (of two or more digits), split each number into two parts. If the first number is a1 + b1 and the second number is a2 + b2, then the product of the two numbers is:



(a1 x a2) + (a1 x b2 + b1 x a2) + (b1 x b2)
The solution comprises three parts (as shown by the boxes and arrows above): the head, the middle, and the tail.
  1. The digits on the right are multiplied vertically to get the tail part: b1 x b2 (excess carried over)
  2. All digits are multipled crosswise and added together to get the middle part: a1 x b2 + b1 x a2 (excess carried over)
  3. The digits on the left are multiplied vertically to get the head part: a1 x a2
Here is a simple example to illustrate this technique.


23 x 41 = 943
The steps are:
  1. 3 x 1 = 3
  2. 2 x 1 + 3 x 4 = 14, put down 4 and carry over 1
  3. 2 x 4 = 8, plus the 1 carried over, is 9
The speed gain using this technique (over the conventional method of multi-line long multiplication) becomes more apparent when handling larger numbers. Here is another example involving excess carryover at each stage.


108 x 64 = 6912
The steps are:
  1. 8 x 4 = 32, put down 2 and carry over 3
  2. 10 x 4 + 8 x 6 = 88, plus the 3 carried over, is 91; put down 1 and carry over 9
  3. 10 x 6 = 60, plus the 9 carried over, is 69
Maybe this method looks some what crazy and funny but it really works on every calculation.









Now we will deal with DIVISIONS:

It is very easy to understand with an example rather than understanding with the theoretical method.

Take for example, 716769 ÷ 54. Yes, you too CAN work it out manually -- and in one line -- without having to reach for the calculator! "on top of the flag". The trick is to reduce the divisor to a mentally manageable value by putting its other digits "on top of the flag". In this example, the divisor will be reduced to 5 (instead of 54) by pushing the 4 up the flagpost, as shown below. Corresponding to the number of digits flagged on top (in this case, one), the rightmost part of the number to be divided is split to mark the placeholder of the decimal point or the remainder portion.

Now observe carefully as we walk through the steps of this example:


716769 ÷ 54 = 13273.5
  1. 7 ÷ 5 = 1 remainder 2. Put the quotient 1, the first digit of the solution, in the first box of the bottom row and carry over the remainder 2
  2. The product of the flagged number (4) and the previous quotient (1) must be subtracted from the next number (21) before the division can proceed. 21 - 4 x 1 = 17

    17 ÷ 5 = 3 remainder 2. Put down the 3 and carry over the 2
  3. Again subtract the product of the flagged number (4) and the previous quotient (3), 26 - 4 x 3 = 14

    14 ÷ 5 = 2 remainder 4. Put down the 2 and carry over the 4
  4. 47 - 4 x 2 = 39

    39 ÷ 5 = 7 remainder 4. Put down the 7 and carry over the 4
  5. 46 - 4 x 7 = 18

    18 ÷ 5 = 3 remainder 3. Put down the 3 and carry over the 3
  6. 39 - 4 x 3 = 27. Since the decimal point is reached here, 27 is the raw remainder. If decimal places are required, the division can proceed as before, filling the original number with zeros after the decimal point

    27 ÷ 5 = 5 remainder 2. Put down the 5 (after the decimal point) and carry over the 2
  7. 20 - 4 x 5 = 0. There is nothing left to divide, so this cleanly completes the division.

This takes a little bit time to understand. But its easy to solve rather than BALDING your head :P










As we are clear about the Multiplications and Divisions, without any waste of time we will deal with

SQUARES:

The HUGE knowledge needed to solve the following method is  (a + b)2 = a2 + 2ab + b2
Yeah you heard right....... above formula is enough to solve the following method. Te method is as follows.

First, a nifty shortcut! The square of a number ending in 5 is almost a no-brainer.
If n is the number formed by the preceding digit/s (before the 5), get the product of n and n+1.
Then just append 25 (i.e. 5 x 5) to this product.
For example, 752:


7 x 8 = 56; therefore solution is 5625.
Another example, 1152:


11 x 12 = 132; therefore solution is 13225
For other cases of squaring, the same shortcut techniques used in multiplication may be utilised.
Especially the general-purpose Urdhva Tiryagbhyam (Vertically and Crosswise) formula.
To get the square of a number (of two or more digits), simplify by splitting it into at least two parts, a and b.
Thus (a + b)2 = a2 + 2ab + b2
Squaring shortcut technique
The solution comprises three parts, neatly fitting the three boxes shown above. Just adjust for excess carry over.
  1. the head: a2
  2. the middle: crosswise multiplication and doubling a x b x 2
  3. the tail: b2
Here is a simple example to illustrate this technique.


232 = 529
The steps are:
  1. tail: 32 = 9, put it down in the rightmost box
  2. middle: 2 x 3 x 2 = 12, put down the 2 in the middle box and carry over the 1
  3. head: 22 = 4, plus the 1 carried over, is 5 in the left box
Another example.


1082 = 11664
The steps are:
  1. tail: 82 = 64, put down the 4 and carry over the 6
  2. middle: 10 x 8 x 2 = 160, plus the 6 carried over, is 166; put down the 6 and carry over the 16
  3. head: 10 x 10 = 100, plus the 16 carried over, is 116.

So i hope that  the reader has understood these crazy methods and makes his MATH simple and avoid ambiguity and avoid wastage of time.


Thanks for reading my post..... :) :)










                                                                               --- PaVaN KoUndiNyA                                                         


2 comments: