MENTAL MATH - PERFORMING THE MULTIPLICATION OF NUMBERS MENTALLY, EASILY AND QUICKLY

Please study

Vedic Mathematics before Mental Math.

That knowledge is a prerequisite here.

There we applied the "the Urdhva Tiryak Sutra"
(meaning : Vertically and cross-wise) to perform
multiplication of two digit numbers mentally,
easily and speedily.

Here, we extend the method to perform
multiplication of three digit numbers.

There and Here we deal with the principles
appled to Multiplication. For the application
to other topics of mental math, go to
Vedic Maths eBook.
for more details, see
near the bottom of the page.













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
three digit numbers.

Let us say the three digits of the first number be
'a'(Hundreds' digit) and 'b'(tens' digit) and 'c'(units' digit)

And those of the second number be
'p'(Hundreds' digit) and 'q'(tens' digit) and 'r'(units' digit).

Write the digits of the two numbers
one below the other as follows.

          a  b  c
          p  q  r

The product of these numbers has five parts
which are given below seperated by '/'.

          a  b  c
          p  q  r
          ---------
          ap/(aq+pb)/(ar+pc+bq)/(br+qc)/cr
          ---------

'ap' is the ten thousands' part which is
the vertical product of the first column.

(aq+pb) is the Thousands' part which is the
sum of the cross-wise products 'aq' and 'pb'.

(ar+pc+bq) is the Hundreds' part which is the
sum of the cross-wise products 'ar' and 'pc'
and the vertical product 'bq'.

(br+qc) is the Tens' part which is the
sum of the cross-wise products 'br' and 'qc'.

'cr' is the units' part which is
the vertical product of the last column.

Let us see the method by examples.

Example 1 0f Mental Math

To find 211 x 142

          211                  
          142
          -----
          2x1/2x4+1x1/2x2+1x1+1x4/1x2+4x1/1x2
          = 2/9/9/6/2
          211 x 142 = 29962 

The product has five parts and each part
is seperated by slash (/).

First part of the product
= vertical product of first column
= 2x1 = 2

Second part of the product
= sum of the cross-wise products of the
first two columns = 2x4+1x1 = 8+1 = 9

Third part of the product
= sum of the cross-wise products of the
first and last columns + vertical product
of middle column = 2x2+1x1+1x4 = 4+1+4 = 9

Fourth part of the product
= sum of the cross-wise products of the
last two columns = 2x4+1x1 = 8+1 = 9

Fifth part of the product
= vertical product of last column
= 1x2 = 2

Thus 211 x 142 = 29962.

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.

          211                  
          142
          -----
          2/9/9/6/2
          -----
          211 x 142 = 29962

Example 2 0f Mental Math

To find 421 x 513

          421
          513
          ----
          4x5/4x1+5x2/4x3+5x1+2x1/2x3+1x1/1x3
          = 20/4+10/19/7/3
          = 20/4+10+1/9/7/3
          = 20/15/9/7/3
          = 20+1/5/9/7/3
          = 21/5/9/7/3
          ----
          421x513=215973

Here, the five parts are obtained as in previous
example. But some parts have more than one digit.
In such cases, units's digit is retained and
the other digits (shown in small letters) are
carried over to the immediate left place as
explained in the case of multiplication of
two digit numbers in Vedic mathematics.

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.

          421
          513
          ----
          20/14/19/7/3
          =21/5/9/7/3  
     (worked out from right)
          421 x 513 = 215973

Example 3 0f Mental Math

To find 456 x 789

          456
          789
          ----
          4x7/4x8+7x5/4x9+7x6+5x8/5x9+8x6/6x9
          = 28/32+35/36+42+40/45+48/54
          = 28/67/118/93+5/4
          = 28/67/118/98/4
          = 28/67/118+9/8/4
          = 28/67/127/8/4
          = 28/67+12/7/8/4
          = 28/79/7/8/4
          = 28+7/9/7/8/4
          = 35/9/7/8/4
          456x789=359784

Here, obtaining the five parts and doing the
carry over method is similar to the previous
example. Here, the carry over is to be done
in all the parts.

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.

          456
          789
          ----
          28/67/118/93/54
          =35/9/7/8/4 
      (worked out from right)
          456 x 789 = 359,784.

Example 4 0f Mental Math

To find 567 x 89

          567
          089
          ----
          0/40/93/110/63
          =5/0/4/6/3 
   (worked out from right)
          567 x 89 = 50463

Example 5 0f Mental Math

To find 987 x 21

987
021
----
0/18/25/22/7
=2/0/7/2/7 (worked out from right)
987 x 21 = 20,727

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Proof of the method adopted

We know
(ay² + by + c)(py² + qy + r)
= y⁴(ap) + y³(aq + pb) + y²(ar + pc + bq) + y(br + qc) + cr

y4 term	y3 term	y2 term	y term	constant term
ap	(aq+pb)	(ar+pc+bq)	(br+qc)	cr

Multiplying three digit numbers is similar with y = 10

10⁴term and units term are vertical products
and 10³ term, 10² term, 10 term are
cross-wise products' sum.

[a(10²) + b(10) + c][p(10²) + q(10) + r]
= 10⁴(ap) + 10³(aq+pb) + 10²(ar+pc+bq) + 10(br+qc) + cr

Ten Thousands' term Thousands' term Hundreds' term Tens'term Units' term

ap aq + pb ar + pc + bq br + qc cr

Ten Thousands'	Thousands'	Hundreds'	Tens'	Units'
ap	(aq+pb)	(ar+pc+bq)	(br+qc)	cr

The digits of the two numbers :

          a  b  c
          p  q  r

The five parts of the product :

ap/(aq+pb)/(ar+pc+bq)/(br+qc)/cr (proved.)

To find the mental math for products
of four digit numbers, go to

Math Magic.

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Exercise on Mental Math

You may take any three digit numbers
and apply the above method for multiplying
and verify your answers with calculator.

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