### Vedic Math - Multiplication of numbers whose last digits add to 10 and first digits are same and vice-versa.

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We had discussed this method in our previous articles for 2-digit numbers and today we shall explain the same method for 3-digit numbers.

A. Numbers whose last digits add to 10 and the remaining first digits are the same
Case 1: When sum of last two digits number is 100
Example: 392 x 308
Here we can see that right digits sum is 100 i.e.(92 + 8) and left side digits are same. Here we can now apply the same method, which we discussed earlier for 2-digit number. But this time we must expect to have four figures on the right-hand side.

• First, multiply the right side numbers(92 x 08) and the result is 0736.
• Second, multiply 3 by the number that follows it, i.e.4, so the result of (3 x 4) is 12.
• And now the final output is 120736.

Example: 795 x 705
Here 95 + 05 = 100 and left side digits are same i.e. '7'. Hence it qualifies for this case.
In calculation, we shall multiple the last two digits and the left digit i.e. '7; with its next number '8'. So the calculation is:
795 x 705 = 7 x 8 | 95 x 05
= 56 | 0475
= 560475

Example: 866 X 834
Here 66 + 34 = 100 and left side digit is 8 and its next number is 9. So the calculation is:
848 x 852 = 8 x 9 | 66 x 34            (Note: For 66 x 34, we shall discuss in our upcoming articles)
= 72 | 2244
= 722244

Case 2: When sum of whose last digits is 10  and the remaining first digits are the same
Example: 241 x 249
Here we can see that right digits sum is 10 i.e.(9 + 1) and left side digits are same i.e. 24. So we can now apply the same method as described above.
241 x 249 = 24 x 25 | 1 x 9
= 0600 | 09
= 60009

### Vedic Math - Multiplication of numbers with a series of 1's

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In the previous article, we learnt the technique of "how to multiply numbers with a series of 9’s". In this article, we shall learn the technique of "how to multiply numbers with a series of 1’s". So, we shall multiply the numbers with 1, 11, 111,..... etc.

In this technique, we use "vertically and crosswise" vedic sutra. Take example of ab x uv, and apply the sutra as follows:

a         b
u         v
---------------------
a x u | av + ub | b x v
---------------------

(Here '|'  is used just as separator)

Here we are splitting the answer in three parts as following:
• vertically                         =(b x v)
• crosswise multiplication and add   =(a x v) + (b x u)
• vertically                         =(a x u)
During multplication with 11, u=1 and v=1, means:

a               b
1               1
--------------------
a  |  a + b  |  b
--------------------

Example:  Multiply 53 by 11

### Vedic Math - Multiplication of numbers with a series of 9’s

Note: Vedic Math Blog has been moved to http://vedicmath.vedantatree.com/. Please bookmark the new address for new and existing blogs.

Another special case of multiplication is, multiplication with numbers like 9, or 99, or 999, or 9999.....so on. It feels like if multiplier is a big number, the calculation will be tough. But, with the help of vedic math formulae, the multiplication is much easier for all '9' digits multiplier. By using the method given below, we can multiply any number with 99,999,9999, etc. very quickly.

Please note that the methods or the vedic formulae, that we use in this calculation, are "By one less than the one before"  and "All from 9 and the last from 10".

There are three cases for the multiplication of numbers with a series of 9's.
• Case 1: Multiplying a number with a multiplier having equal number of 9’s digits                                              (like 587 x 999)
• Case 2: Multiplying a number with a multiplier having more number of 9’s digits                                             (like 4678 x 999999)
• Case 3: Multiplying a number with a multiplier having lesser number of 9’s digits                                             (like 1628 x 99)

The method to solve 'Case 1' and 'Case 2' is the same, but for 'Case 3', the method is different. Let us start with 'Case 1'.

Case 1: Multiplying a number with a multiplier having equal number of 9’s digits

Multiply 587 by 999

587
x  999
------------
586 413

Solution is,
•  Let us first do the calculation by conventional method to understand the solution. Result will be 586413.
• Split the answer in two parts i.e. '586' and '413'.
• Let's see the first part of the result, i.e. 586. It is reduced by 1 from the number being multiplied i.e. 587 - 1 = 586. {Vedic sutra "By one less than the one before"}
• Now see the last part, i.e. 413. Subtract the multiplicand i.e. 587 from 1000 (multiplier + 1). Vedic Sutra applied here is "All from 9 and the last from 10", and hence we substract 587 from 1000. So the outcome will be (9 -5 = 4, 9 - 8 = 1, 10 - 7 = 3) , and result is 413. Refer to image below for more clarity:

### Vedic Math - Squaring of numbers near '50'

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In this article, we pick another special case of squaring i.e. squaring numbers which are near 50. It can have two cases, which are:
• Case 1: Numbers greater than 50.
• Case 2: Numbers lesser than 50
In both the cases, we need to take 50 as the base value. First let us take 'Case 1' i.e. 'Numbers greater than 50'.

Case 1 -
Take an example: Say, 542=2916
Consider this answer in two parts: 29 (first part) and 16(second part). Now let us study, how Vedic Math can help us to achieve this answers or both of these parts.

As we are taking '50' as base, so the number presentation will be like 50 + 4. So the first part is 50, and second part is 4.

For the first part of the answer:
1. Pick the first part i.e. 50.
2. Pick the first digit i.e. 5
3. Square this digit i.e. 52 > 25
4. Add 4(second part) to it
5. And we get our first part of the answer i.e. 29 (25 + 4).
For the second part of answer, following are the steps:
1. Pick second part i.e. 4
2. Square this digit i.e. 42 > 16
3. And we get our second part of the answer i.e. 16

Let us understand it with another example to make it more clear. Say, 612=3721
As we are taking '50' as base, so the number presentation will be like 50 + 11. So the first part is 50, and second part is 11.

For the first part of the answer:
1. Pick the first part i.e. 50.
2. Pick the first digit i.e. 5
3. Square this digit i.e. 52 > 25
4. Add 11(second part) to it
5. And we get our first part i.e. 36 (25 + 11).
For the second part of answer, following are the steps:
1. Pick second part i.e. 11
2. Square this digit i.e. 112 > 121
3. Now the result is of three digit. So the first digit (1) will be added to the first part i.e. 1 + 36 = 37
4. So first part becomes 37 now
5. And second part of answer will be 21
So the answer is 3721. Refer to image below for visual representation.

### Vedic Math - Squaring Of Numbers Ending with '5'

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In this article, we shall discuss a very common and interesting trick to square those numbers quickly which are having '5' as last digit. For example, what is the result of 652, 852, 1252 ?

Let us start with an example:- 35 x 35. How will you multiply?

The conventional approach is-

35
x 35
-------
175
105
--------
1225
--------

In above problem, we followed the following steps:
1. In first step, we multiply 5 by 35, get 175 and wrote it below the line.
2. In second step, we multiply 3 by 35, get 105, wrote it below the first step and leave one space from right.
3. In last, we add results from both the steps and get 1225 as answer.
Now here is the magical trick or quicker way to do this calculation using Vedic Math (to square any number with a 5 on the end). Let us have a look on the same example once again, following 'Vedic Math' steps to solve it.
1. In 35, the last digit is 5 and other number is 3.
2. Add 1 to the top left digit 3 to make it 4 (i.e. 3+1=4) (See the image below).
3. Then multiply original number '3' with increased number i.e. '4'. Like 3 x 4, and we get 12.
4. Now you can see that this is the left hand side of the answer.
5. Next, we multiply the last digits, i.e 5 x 5 and write down 25 to the right of 12.
6. And here we come up with a desired answer, 1225
7. Visual representation is given below.

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Our exploration for above three tools is:

### What is Vedic Math?

Note: Vedic Math Blog has been moved to http://vedicmath.vedantatree.com/. Please bookmark the new address for new and existing blogs.

We are going to start a new and very interesting section and that is 'VEDIC MATH'. Many of us are interested in increasing our productivity with calculations. This is where ‘Vedic Math’ helps us. It teaches us many ways to do the calculations quickly and if practiced correctly then all the calculations can be done in mind. Hence it helps us not only in our work, but routine works also. Vedic Math is also very useful for students to get rid of math phobia and improve grades. With these techniques one could be able to solve the mathematical problems 15 times faster. It improves mental calculations, concentration and confidence. Isn’t this great!

Once you are aware of the basics of Vedic Math, you can practice and make yourself a human calculator. Vedic Mathematics is magical. Let us take a simple example of multiplication to feel what Vedic Math is and what it can do.

So, let’s try 14 times 11.
•     Split the 14 apart, like:
•     1    4
•     Add these two digits together
•     1 + 4 = 5
•     Place the result, 5 in between the 14 to have 154
•     And the result is
•     14 X 11 = 154

This is a very basic example to show the magical power of Vedic Math. Once you learned all the techniques, you will be able to do various complex calculations very fast as mentioned above. Before we proceed towards the different techniques of Vedic mathematics in detail, we first give you brief background of Vedic Mathematics history.

'Vedic Mathematics' is the name given to the ancient system of mathematics derived from ancient treasure of knowledge called ‘Veda’. ‘Veda’ means knowledge. Vedic Mathematics believes to be a part of ‘Atharva Veda’. It a unique technique of calculations based on simple rules and principles, using which any mathematical problem related to arithmetic, algebra, geometry or trigonometry can be solved quickly and possibly orally (once you master it).

Vedic Mathematics was devised probably thousands of years back; however it was rediscovered again from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to his research all of mathematics is based on sixteen Sutras or word-formulae. For example, 'Vertically and Crosswise` is one of these Sutras. These formulae describe the way the mind naturally works and are therefore a great help in solving the problems.