tricks for solving calculation

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dimpy.handa

Dimpy Handa
it would be nice if we collect some tricks on solving some long calculations. like some short tricks and all. i will be adding some soon. all are invited to contribute.
 
Re: short tricks

There is a trick for multiplying big numbers

when we have to multiply 998 with 997,its quiet long multiplication.

But there is short cut for this method

998*997=995006

1)Subtract the first number 998 from 1000 and the result is 002(Inorder to avoid confusions write it as 002 because the main number is the three digit number)

2)Subtract the second number 997 from 1000 and the result is 003.

3)Now multiply the two numbers that obtained as the result of the subtraction.

002*003=006

4)This result forms the last three digit of the multiplication.

5)Then subtract any one main number with the remainder of the other number

either subtract 998-003=995 or 997-002=995

this number forms the first three number.

hence the obtained six numbers are 995006...
 
Re: short tricks

Squares of numbers ending in 5 :
Now we relate the sutra to the ‘squaring of numbers ending in 5’. Consider the example 252.

Here the number is 25. We have to find out the square of the number. For the number 25, the last digit is 5 and the 'previous' digit is 2. Hence, 'one more than the previous one', that is, 2+1=3. The Sutra, in this context, gives the procedure 'to multiply the previous digit 2 by one more than itself, that is, by 3'. It becomes the L.H.S (left hand side) of the result, that is, 2 X 3 = 6. The R.H.S (right hand side) of the result is 52, that is, 25.

Thus 252 = 2 X 3 / 25 = 625.
 
Re: short tricks

The formula simply means : “all from 9 and the last from 10”

The formula can be very effectively applied in multiplication of numbers, which are nearer to bases like 10, 100, 1000 i.e., to the powers of 10 . The procedure of multiplication using the Nikhilam involves minimum number of steps, space, time saving and only mental calculation. The numbers taken can be either less or more than the base considered.

The difference between the number and the base is termed as deviation. Deviation may be positive or negative. Positive deviation is written without the positive sign and the negative deviation, is written using Rekhank (a bar on the number).
 
Re: short tricks

Multiplication of two 2 digit numbers.

Ex.1: Find the product 14 X 12

i) The right hand most digit of the multiplicand, the first number (14) i.e., 4 is multiplied by the right hand most digit of the multiplier, the second number (12) i.e., 2. The product 4 X 2 = 8 forms the right hand most part of the answer.

ii) Now, diagonally multiply the first digit of the multiplicand (14) i.e., 4 and second digit of the multiplier (12) i.e., 1 (answer 4 X 1=4); then multiply the second digit of the multiplicand i.e., 1 and first digit of the multiplier i.e., 2 (answer 1 X 2 = 2); add these two i.e., 4 + 2 = 6. It gives the next, i.e., second digit of the answer. Hence second digit of the answer is 6.

iii) Now, multiply the second digit of the multiplicand i.e., 1 and second digit of the multiplier i.e., 1 vertically, i.e., 1 X 1 = 1. It gives the left hand most part of the answer.

Thus the answer is 16 8.
 
Re: short tricks

Consider the division by divisors of more than one digit, and when the divisors are slightly greater than powers of 10.

Example 1 : Divide 1225 by 12.

Step 1 : (From left to right ) write the Divisor leaving the first digit, write the other digit or digits using negative (-) sign and place them below the divisor as shown.

12
-2
¯¯¯¯
Step 2 : Write down the dividend to the right. Set apart the last digit for the remainder.

i.e.,, 12 122 5
- 2

Step 3 : Write the 1st digit below the horizontal line drawn under the dividend. Multiply the digit by –2, write the product below the 2nd digit and add.

i.e.,, 12 122 5
-2 -2
¯¯¯¯¯ ¯¯¯¯
10

Since 1 x –2 = -2 and 2 + (-2) = 0

Step 4 : We get second digits’ sum as ‘0’. Multiply the second digits’ sum thus obtained by –2 and writes the product under 3rd digit and add.

12 122 5
- 2 -20
¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯
102 5

Step 5 : Continue the process to the last digit.

i.e., 12 122 5
- 2 -20 -4
¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯
102 1

Step 6: The sum of the last digit is the Remainder and the result to its left is Quotient.
Thus Q = 102 and R = 1
 
Re: short tricks

The Sutra 'Sunyam Samyasamuccaye' says the 'Samuccaya is the same, that Samuccaya is Zero.' i.e., it should be equated to zero. The term 'Samuccaya' has several meanings under different contexts.

i) We interpret, 'Samuccaya' as a term which occurs as a common factor in all the terms concerned and proceed as follows.

Example 1: The equation 7x + 3x = 4x + 5x has the same factor ‘ x ‘ in all its terms. Hence by the sutra it is zero, i.e., x = 0.

Otherwise we have to work like this:

7x + 3x = 4x + 5x
10x = 9x
10x – 9x = 0
x = 0

This is applicable not only for ‘x’ but also any such unknown quantity as follows.
 
The Sutra can be taken as Purana - Apuranabhyam which means by the completion or non - completion. Purana is well known in the present system. We can see its application in solving the roots for general form of quadratic equation.

We have : ax2 + bx + c = 0

x2 + (b/a)x + c/a = 0 ( dividing by a )

x2 + (b/a)x = - c/a

completing the square ( i.e.,, purana ) on the L.H.S.

x2 + (b/a)x + (b2/4a2) = -c/a + (b2/4a2)

[x + (b/2a)]2 = (b2 - 4ac) / 4a2
________
- b ± √ b2 – 4ac
Proceeding in this way we finally get x = _______________
2a
 
In the book on Vedic Mathematics Sri Bharati Krishna Tirthaji mentioned the Sutra 'Calana - Kalanabhyam' at only two places. The Sutra means 'Sequential motion'.

i) In the first instance it is used to find the roots of a quadratic equation 7x2 – 11x – 7 = 0. Swamiji called the sutra as calculus formula. Its application at that point is as follows. Now by calculus formula we say: 14x–11 = ±√317

A Note follows saying every Quadratic can thus be broken down into two binomial factors. An explanation in terms of first differential, discriminant with sufficient number of examples are given under the chapter ‘Quadratic Equations’.

ii) At the Second instance under the chapter ‘Factorization and Differential Calculus’ for factorizing expressions of 3rd, 4th and 5th degree, the procedure is mentioned as 'Vedic Sutras relating to Calana – Kalana – Differential Calculus'
 
The Sutra Ekanyunena purvena comes as a Sub-sutra to Nikhilam which gives the meaning 'One less than the previous' or 'One less than the one before'.

1) The use of this sutra in case of multiplication by 9,99,999.. is as follows .

Method :

a) The left hand side digit (digits) is ( are) obtained by applying the ekanyunena purvena i.e. by deduction 1 from the left side digit (digits) .

e.g. ( i ) 7 x 9; 7 – 1 = 6 ( L.H.S. digit )

b) The right hand side digit is the complement or difference between the multiplier and the left hand side digit (digits) . i.e. 7 X 9 R.H.S is 9 - 6 = 3.

c) The two numbers give the answer; i.e. 7 X 9 = 63.
 
Multiply any two numbers from 11 to 20 in your head.
Take 15 x 13 for example..

Place the larger no. first in your mind and then do something like this Take the larger no on the top and the second digit of the smaller no. in the bottom.

15

3

The rest is quite simple. Add 15+3 = 18 . Then multiply 18 x 10 = 180 ...

Now multiply the second digit of both the no.s (ie; 5 x 3 = 15) Now add 180 + 15

Here is the answer 180 + 15 = 195 .

Think over it. This is a simple trick. It will help you a lot
 
To multiply by 9,try this:
(1) Spread your two hands out and place them on a desk or table in front of you.
(2) To multiply by 3, fold down the 3rd finger from the left. To multiply by 4, it would be the 4th finger and so on.
(3) the answer is 27 ... READ it from the two fingers on the left of the folded down finger and the 7 fingers on the right of it.
 
This one is as easy as the previous ones but you have to pay a little more attention to this one . Read carefully :

Let the number be 35

35 x 35

Multiply the last digits of both the numbers ; thus ___ 5 x 5 = 25

now add 1 to 3 thus 3 + 1 = 4

multiply 4 x 3 = 12

thus answer 1225
 
Trick number 2 tells you how to multiply a two digit number by 11 but what if you have a number like 12345678 . Well that is very easy if you our trick as given below . Read it carefully.

Let the number be 12345678 __ thus 12345678 x 11

Write down the number as 012345678 ( Add a 0 in the beginning)

Now starting from the units digit write down the numbers after adding the number to the right

So the answer will be 135802458

This one is simple if you think over it properly all you got to do is to add the number on the right . If you are getting a carry over then add that to the number on the left. So I will tell you how I got the answer . Read carefully.

The number was 12345678 ___ I put a 0 before the number ____ so the new number 012345678

Now I wrote ___ 012345678

Then for the answer

8 + 0 = 8

7 + 8 = 15 (1 gets carry carried over)

6+1+7 = 14 ( 1 gets carried over)

5 + 1 + 6 = 12 ( 1 gets carried over)

4 + 1 + 5 = 10 ( 1 gets carried over)

3 + 1 + 4 = 8

2 + 3 = 5

1 + 2 = 3

0 + 1 = 1

Thus the answer = 135802458
 
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