We deal with a method which is totally unconventional but help in Carrying out multiplication easily, speedily and executing mentally. The method, based on a sutra (Formula) of Ancient Indian Vedic Mathematics, called "The Urdhva Tiryak Sutra" which means Vertically and cross-wise is applied to multiplication of numbers. Here, we will apply it to multiplication of two digit numbers. We provide lucid explanation of the method with a number of Examples.
There are methods, which are totally unconventional but help in Carry
ng out tedious and cumbersome arithmetical operations, easily, speedily and in some cases executing them mentally.
People who are deeply rooted in the conventional methods, may find it difficult, at first reading, to understand the methods.
But, these methods, based on the Sutras (aphorisms or Formulas) of Ancient Indian Vedic Mathematics are simple and easy to understand, remember and apply, even by little children.
Here, we will apply a sutra of Vedic Math as applied to multiplication of two digit numbers.
The Urdhva Tiryak Sutra (meaning : Vertically and cross-wise) :
This is a general Formula applicable to all cases of multiplication.
Using this principle, we can find the product of two numbers easily.
Multiply vertically and crosswise to get the digits of the product.
Examples will clarify the method.
Before seeing the examples, let us see the formula for finding the product of two digit numbers.
Let us say the two digits of the first number be 'a' (tens' digit) and 'b'(units' digit).
And those of the second number be 'p' (tens' digit) and 'q'(units' digit).
Write the digits of the two numbers one below the other as follows.
a b
p q
The product of these numbers has three parts which are given below seperated by '/'.
a b
p q
------
ap/(aq+pb)/bq
------
'ap' is the Hundreds' part which is the vertical product of the first column.
(aq+pb) is the Tens' part which is the sum of the cross-wise products 'aq' and 'pb'.
'bq' is the units' part which is the vertical product of the second column.
Let us see the method by examples.
Example 1 :
To find 21 x 13
21
13
-----
2x1/2x3+1x1/1x3
= 2/7/3
21 x 13 = 273
The product has three parts and each part is seperated by slash (/).
First part of the product = product of first vertical digits = 2x1 = 2
Second part of the product = sum of the cross-wise products = 2x3+1x1 = 6+1 = 7
Third part of the product = product of second vertical digits = 1x3 = 3
Thus 21 x 13 = 273.
Example 2 :
To find 41 x 51
41
51
----
4x5/4x1+5x1/1x1
=20/9/1
41x51 = 2091
The product has three parts and each part is seperated by slash (/).
First part of the product = product of first vertical digits = 4x5 = 20
Second part of the product = sum of the cross-wise products = 4x1+5x1 = 4+5 = 9
Third part of the product = product of second vertical digits = 1x1 = 1
Thus 41 x 51 = 2091.
Example 3 :
To find 23 x 42
23
42
----
2x4/2x2+4x3/3x2
=8/16/6
=8+1/6/6
=9/6/6
23x43 = 966
The product has three parts and each part is seperated by slash (/).
First part of the product = product of first vertical digits = 2x4 = 8
Second part of the product = sum of the cross-wise products = 2x2+4x3 = 4+12 = 16
This is a two digit number.
Units digit (6) is retained and tens' digit (1) is carried over to the immediate left place.
See that in 16, 1 is written in small letters indicating its carry over to the immediate left place and 6 is written normally indicating it is retained in its place.
Third part of the product = product of second vertical digits = 3x2 = 6
Thus 23 x 42 = 966.
To explain the procedure clearly, so many steps are shown.
In practice we can do the calculations mentally and write the answer in fewer steps as follows.
23
42
----
8/16/6
=9/6/6 (worked out from right)
23 x 42 = 966.
Example 4 :
To find 37 x 29
37
29
----
3x2/3x9+2x7/7x9
=6/27+14/63
=6/27+14+6/3
=6/47/3
=6+4/7/3
=10/7/3
37x29=1073
The product has three parts and each part is seperated by slash (/).
First part of the product (before carry over from second part) = product of first vertical digits = 3x2 = 6
Second part of the product (before carry over from third part) = sum of the cross-wise products = 3x9+2x7 = 27+14
Third part of the product = product of second vertical digits = 7x9 = 63
This is a two digit number.
Units digit (3) is retained and tens' digit (6) is carried over to the immediate left place.
See that in 63, 6 is written in small letters indicating its carry over to the immediate left place and 3 is written normally indicating it is retained in its place.
Now,
Second part of the product (after carry over from third part) = 27+14+6 = 47
This is a two digit number.
Units digit (7) is retained and tens' digit (4) is carried over to the immediate left place.
See that in 47, 4 is written in small letters indicating its carry over to the immediate left place and 7 is written normally indicating it is retained in its place.
Now,
First part of the product (after carry over from second part) = 6+4 = 10
Thus 37 x 29 = 1073.
To explain the procedure clearly, so many steps are shown.
In practice we can do the calculations mentally and write the answer in fewer steps as follows.
37
29
----
6/41/63
=10/7/3 (worked out from right)
37 x 29 = 1073.
Example 5 :
To find 68 x 79
68
79
----
6x7/6x9+7x8/8x9
=42/54+56/72
=42/110+7/2
=42/117/2
=42+11/7/2
=53/7/2
68x79=5372
By this time, you are familiar with the procedure and you can understand the above problem.
Thus 68 x 79 = 5372.
To explain the procedure clearly, so many steps are shown.
In practice we can do the calculations mentally and write the answer in fewer steps as follows.
68
79
----
42/110/72
=53/7/2 (worked out from right)
68 x 79 = 5372.
Proof of the method adopted :
We know,
(ay + b)(py + q) = y2(ap) + y(aq + bp) + bq
Multiplying two digit numbers is similar with y = 10
10^2 term and units term are vertical products and 10 term is cross-wise products' sum.
[a(10) + b][p(10) + q] = 102(ap) + 10(aq + bp) + bq
The digits of the two numbers :
a b
p q
The three parts of the product :
ap/(aq + pb)/bq (proved.)
For More Vedic Math Problems , go to
http://www.math-help-ace.com/Mental-Math.html
About the author
Name: KVLN
Age: 47 years old
Qualifications: B.Tech., M.S. (from IIT, Madras) Has 14 years of teaching experience. Love for math and love for teaching makes him feel more than happy to help. For First-Rate Math Help, go to the author's web site http://www.math-help-ace.com/ It Aims to help to make every one an ace (expert) in math. Explains lucidly math topics for kids and teens with solved examples and exercises. Highlights the salient points and formulas. Helps to develop confidence and desire to continue. Helps to perceive the work as less demanding. Helps to to integrate the current problem with existing knowledge and ideas. Helps every student succeed in math by making the journey a pleasant one.
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