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By cross – multiplication,n ( ax + b) = m (cx + d)nax + nb = mcx + mdnax - mcx = md – nbx( na – mc ) = md – nb

md - nbx = ________

na - mc.

Now look at th problm onc a!ain

ax + b m _____ = __cx+ d n

para"artya !i"s md - nb, na - mc andmd - nb

x = _______na - mc

#xampl $%   &x + $ $& _______ = ___

'x + & $

md - nb $& (&) - $($) & - $ *

x = ______ = ____________ = _______ = __na- mc $ (&) - $&(') -

= '

#xampl %   'x +  ________ = __&x + $&

() ($&) -()()

x = _______________()(') - ()(&)

($) - '* ($ - *) $$ $= __________ = _________ = ______ = __

& – $ & – $ / $$

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0yp (i") % 1onsidr th problms o2 th typ m n _____ + ____ = *

x + a x + b

0ak 3.1.4 and procd.

m(x+b) + n (x+a) ______________ = *

(x + a) (x +b)

mx + mb + nx + na ________________ = *

x + a)(x + b)

(m + n)x + mb + na = * (m + n)x = - mb - na

-mb - nax = ________

(m + n)

0hus th problm m n ____ + ____ = *, by para"artya procssx + a x + b

!i"s dirctly -mb - nax = ________

(m + n)

#xampl $ %   & ' ____ + ____ = *x + ' x – 5

!i"s -mb - nax = ________ Not that m = &, n = ', a = ', b = - 5(m + n)

-(&)(-5) – (') (') $ - $5 = _______________ = ______ = __

( & + ')

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#xampl %   5

 ____ + _____ = *x + $ x – $

!i"s -() (-$) - (5) ($) $* - 5 x = ________________ = ______ = __ =

+ 5 $$ $$

6 . 7ol" th 2ollowin! problms usin! th sutra 8ara"artya – yo9ayt.

$) &x + = x – & 5) (x + $) ( x + ) = ( x – &) (x – ')

) (x&) + $=x - $ ) (x – ) (x – )= (x – &) (x – )

&) x + ) (x + ) (x + )= (x + & ) (x + $) ______ = __

&x-

') x + $ & _______ = $

&x - $

)

 ____ + ____ = *x + & x – ' 

66)$.7how that 2or th typ o2 :uations

m n p ____ + ____ + ____ = *, th solution isx + a x + b x + c

- mbc – nca – pabx = ________________________ , i2 m + n + p =*.

m(b + c) + n(c+a) + p(a + b)

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. ;pply th abo" 2ormula to st th solution 2or th problm

8roblm &  ____ + ____ - ____ = *x + ' x + 5 x +  

som mor simpl solutions %  

m n m + n ____ + ____ = _____x + a x + b x + c

Now this can b writtn as,

m n m n ____ + ____ = _____ + _____x + a x + b x + c x + c

m m n n ____ - ____ = _____ - _____x + a x + c x + c x + b

m(x +c) – m(x + a) n(x + b) – n(x + c) ________________ = ________________

(x + a) (x + c) (x + c) (x + b)

mx + mc – mx – ma nx + nb – nx – nc ________________ = _______________

(x + a) (x + c) (x +c ) (x + b)

m (c – a) n (b –c)

 ____________ = ___________x +a x + b

m (c - a).x + m (c - a).b = n (b - c). x + n(b - c).ax < m(c - a)- n(b - c) = na(b - c) – mb (c - a)

or x < m(c - a) + n(c - b) = na(b - c) + mb (a - c)

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0hus mb(a - c) + na (b - c)x = ___________________

m(c-a) + n(c-b).

By para"artya rul w can asily rmmbr th 2ormula.

#xampl $ %   sol" &

. 7unyam 7amya 7amuccay  

0h 7utra >7unyam 7amyasamuccay> says th >7amuccaya is th sam, that7amuccaya is ?ro.> i.., it should b :uatd to @ro. 0h trm >7amuccaya>has s"ral manin!s undr di22rnt contxts.

i) A intrprt, >7amuccaya> as a trm which occurs as a common 2actor in allth trms concrnd and procd as 2ollows.

#xampl $%  0h :uation x + &x = 'x + x has th sam 2actor x in allits trms. Cnc by th sutra it is @ro,i.., x = *.

Dthrwis w ha" to work lik this%

x + &x = 'x + x$*x = x

$*x – x = *x = *

0his is applicabl not only 2or xE but also any such unknown :uantity as2ollows.

#xampl % (x+$) = &(x+$)

No nd to procd in th usual procdur lik

x + = &x + &x – &x = & –

x = - or x = - F = -$

7imply think o2 th contxtual manin! o2 >7amuccaya>

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Now 7amuccaya is ( x + $)x + $ = * !i"s x = -$

ii) Now w intrprt >7amuccaya> as product o2 indpndnt trms inxprssions lik (x+a) (x+b)

#xampl &%   ( x + & ) ( x + ') = ( x – ) ( x – 5 )

Cr 7amuccaya is & x ' = $ = - x -57inc it is sam , w dri" x = *

0his xampl, w ha" alrady dalt in typ ( ii ) o2 8ara"artya in sol"in!simpl :uations.

iii) A intrprt > 7amuccaya >as th sum o2 th dnominators o2 two 2ractionsha"in! th sam numrical numrator.

1onsidr th xampl.  

$ $ ____ + ____ = *&x- x-$

2or this w procd by takin!3.1.4.

(x-$)+(&x–) ____________ = *(&x–)(x–$)

x–& __________ = *(&x–)(x–$)

x – & = * x = &

&x = __

6nstad o2 this, w can dirctly put th 7amuccaya i.., sum o2 thdnominators

i.., &x – + x - $ = x - & = *

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!i"in! x = & x = &

6t is tru and applicabl 2or all problms o2 th typ

m m ____ + _____ = *ax+b cx+d

7amuccaya is ax+b+cx+d and solution is ( m G * )

- ( b + d )x = _________

( a + c )

iii) A now intrprt >7amuccaya> as combination or total.

62 th sum o2 th numrators and th sum o2 th dnominators b thsam, thn that sum = *.

1onsidr xampls o2 typ  

ax+ b ax + c _____ = ______ax+ c ax + b

6n this cas, (ax+b) (ax+b) =(ax+c) (ax+c)ax + abx + b = ax + acx +c 

abx – acx = c – b x ( ab – ac ) = c – b 

c–b  (c+b)(c-b) -(c+b)x = ______ = _________ = _____

a(b-c) a(b-c) a

;s pr 7amuccaya (ax+b) + (ax+c) = *

ax+b+c = *ax = -b-c

-(c+b)x = ______

a Cnc th statmnt.

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#xampl '%  &x+ ' &x +

 ______ = ______&x+ &x + '

7inc N$ + N = &x + ' + &x + = 5x + ,;nd H$ + H = &x + ' + &x + = 5x + A ha"N$ + N = H$ + H = 5x +

Cnc 2rom 7unya 7amuccaya w !t 5x + = *

5x = -

- -&x = __ = __

5

#xampl %  x + x + $

 _____ = _______x+$ x +

Cnc N$ + N = x + + x + $ = $*x + $

;nd H$ + H = x + $ + x + = $*x + $N$ + N = H$ + H !i"s $*x + $ = *$*x = -$

-$x = ____

$*

1onsidr th xampls o2 th typ, whr N$ + N = I (H$ + H ), whr I isa numrical constant, thn also by rmo"in! th numrical constant I, wcan procd as abo".

#xampl 5%  

x + & x + $ _____ = ______'x + x + &

Cr N$ + N = x + & + x + $ = &x + '

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H$ + H = 'x + + x + & = 5x + = ( &x + ' )

Jmo"in! th numrical 2actor , w !t &x + ' on both sids.

&x + ' = * &x = -' x = - ' &.

") >7amuccaya> with th sam manin! as abo", i.., cas (i"), w sol" thproblms ladin! to :uadratic :uations. 6n this contxt, w tak thproblms as 2ollowsK

62 N$ + N = H$ + H and also th di22rncsN$ L H$ = N L H thn both th thin!s ar :uatd to @ro, th

solution !i"s th two "alus 2or x.

#xampl %  

&x + x +  _____ = ______x + &x +

6n th con"ntional txt book mthod, w work as 2ollows %

&x + x +  _____ = ______x + &x +

( &x + ) ( &x + ) = ( x + ) ( x + )x + $x + ' = 'x + *x +

x + $x + ' - 'x - *x – = *x – x – $ = *

x – $x + x – $ = *x ( x – & ) + ( x – & ) = *

(x – & ) ( x + ) = *x – & = * or x + = *

x = & or -

Now 7amuccayaE sutra coms to hlp us in a bauti2ul way as 2ollows %Dbsr" N$ + N = &x + + x + = x +

H$ + H = x + + &x + = x +

Murthr N$ L H$ = ( &x + ) – ( x + ) = x – &N L H = ( x + ) – ( &x + ) = - x + & = - ( x – & )

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Cnc x + = * , x – & = *x = - , x = &

i.., x = - , x = &

Not that all ths can b asily calculatd by mr obsr"ation.

#xampl %  

&x + ' x + 5 ______ = _____5x + x + &

Dbsr" thatN$ + N = &x + ' + x + 5 = x + $*

andH$ + H = 5x + + x + & = x + $*

Murthr N$ LH$ = (&x + ') – (5x + )= &x + ' – 5x – = -&x – & = -& ( x + $ )

N L H  = (x + 5) – (x + &) = &x + & = &( x + $)

By 7unyam 7amuccayE w ha"

x + $* = * &( x + $ ) = *x = -$* x + $ = *

x = - $* x = -$

= - '"i)7amuccayaE with th sam sns but with a di22rnt contxt andapplication .

#xampl %  

$ $ $ $ ____ + _____ = ____ + ____x - ' x – 5 x - x -

sually w procd as 2ollows.x–5+x-' x–+x-

 ___________ = ___________(x–') (x–5) (x–) (x-)

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x-$* x-$* _________ = _________x–$*x+' x–$*x+$5

( x – $* ) ( x – $*x + $5 ) = ( x – $* ) ( x – $*x + ')x&–*x+&x–$*x+$**x–$5* = x&–*x+'x–$*x+$**x-'*

x& – &*x + $&x – $5* = x& – &*x + $'x – '*$&x – $5* = $'x – '*

$&x – $'x = $5* – '*– $5x = - *

x = - * - $5 =

Now 7amuccayaE sutra, tll us that, i2 othr lmnts bin! :ual, th sum-total o2 th dnominators on th 3.C.7. and thir total on th J.C.7. b thsam, that total is @ro.

Now H$ + H = x – ' + x – 5 = x – $*, andH& + H' = x – + x – = x – $*

By 7amuccaya, x – $* !i"s x = $*

$*x = __ =

#xampl $*%  

$ $ $ $ ____ + ____ = ____ + _____

x - x – x - x – $

H$ +H = x – + x – = x – $, andH& +H' = x – + x –$ = x – $

Now x – $ = * !i"s x = $

$x = __ = O

#xampl $$%  

$ $ $ $ ____ - _____ = ____ - _____

x + x + $* x +5 x +

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0his is not in th xpctd 2orm. But a littl work r!ardin! transpositionmaks th abo" as 2ollows.

$ $ $ $ ____ + ____ = ____ + _____

x + x + x +5 x + $*

Now 7amuccayaE sutra applis

H$ +H = x + + x + = x + $5, andH& +H' = x + 5 + x + $* = x + $5

7olution is !i"n by x + $5 = * i.., x = - $5.x = - $5 = - .

7ol" th 2ollowin! problms usin! 7unyam 7amya-7amuccayprocss.

$. ( x + ) + & ( x + ) = 5 ( x + ) + ( x + )

. ( x + 5 ) ( x + & ) = ( x – ) ( x – )

&. ( x - $ ) ( x + $' ) = ( x + ) ( x – )

$ $'. ______ + ____ = *

' x - & x –

' '. _____ + _____ = *

&x + $ x +

x + $$ x+5. ______ = _____x+ x+$$

&x + ' x + $. ______ = _____

5x + x + &

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'x - & x+ '. ______ = _____

x+ & &x -

$ $ $ $. ____ + ____ = ____ + _____

x - x - x - & x - ' 

$ $ $ $$*. ____ - ____ = _____ - _____

x - x - 5 x - $* x -  

7unyam 7amya 7amuccay in 1rtain 1ubs%  

1onsidr th problm ( x – ' ) & + ( x – 5 ) & = ( x – ) &. Mor th solutionby th traditional mthod w 2ollow th stps as !i"n blow%

( x – ' )& + ( x – 5 )& = ( x – )& x& – $x + 'x – 5' + x& – $x + $*x – $5

= ( x

&

 – $x

 + x – $ )x& – &*x + $5x – * = x& – &*x + $*x – *$5x – * = $*x – *

$5x – $*x = * – *5x = &*x = &* 5 =

But onc a!ain obsr" th problm in th "dic sns

A ha" ( x – ' ) + ( x – 5 ) = x – $*. 0akin! out th numrical 2actor ,w ha" ( x – ) = *, which is th 2actor undr th cub on J.C.7. 6n such a

cas P7unyam samya 7amuccayQ 2ormula !i"s that x – = *. Cnc x = 0hink o2 sol"in! th problm (x–') & + (x+') & = (x–$) & 

0h traditional mthod will b horribl "n to think o2.

But ( x – ' ) + ( x + ' ) = x – = ( x – $ ). ;nd x – $. on J.C.7.

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cub, it is nou!h to stat that x – $ = * by th sutraE.

x = $ is th solution. No cubin! or any othr mathmatical oprations.

;l!braic 8roo2 %  

1onsidr ( x – a)& + ( x – b )& = ( x – a – b )& it is clar thatx – a + x – b = x – a – b

= ( x – a – b )

Now th xprssion,

x& -5xa + $xa – a& + x& – 5xb +$xb – b&  =(x&–&xa–&xb+&xa+&xb+5axb–a&–&ab–&ab–b&) 

= x&–5xa–5xb+5xa+5xb+$xab–a&–5ab–5ab–b& 

cancl th common trms on both sids

$xa+$xb–a&–b& = 5xa+5xb+$xab–a&–5ab–5ab–b& 5xa + 5xb – $xab = 5a& + 5b& –5ab – 5ab 5x ( a + b – ab ) = 5 < a& + b& – ab ( a + b )

x ( a – b ) = < ( a + b ) ( a + b –ab ) – ( a + b )ab= ( a + b ) ( a + b – ab )= ( a + b ) ( a – b ) 

x = a + b

7ol" th 2ollowin! usin! P7unyam 7amuccayQ procss %

$. ( x – & ) & + ( x – ) & = ( x – 5 ) & 

. ( x + ' ) & + ( x – $* ) & = ( x – & ) & 

&. ( x + a + b – c ) & + ( x + b + c – a ) & = ( x + b ) & 

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#xampl %  

(x + )& x + $ ______ = _____(x + &)& x + '

with th txt book procdurs w procd as 2ollows

x& + 5x + $x + x + $ _______________ = _____x& + x + x + x + '

Now by cross multiplication,

( x + ' ) ( x& +5x + $x + ) = ( x + $ ) ( x& + x + x + )x' + 5x& + $x+ x + 'x& + 'x + 'x + & =

x' + x& + x + x + x& + x + x + x' +$*x& + &5x + 5x + & = x' + $*x& +&5x + 'x +

5x + & = 'x + 5x – 'x = – &

x = - x = -

Dbsr" that ( N$ + H$ ) with in th cubs on

3.C.7. is x + + x + & = x + andN + H on th ri!ht hand sid

is x + $ + x + ' = x + .

By "dic 2ormula w ha" x + = * x = - .

7ol" th 2ollowin! by usin! "dic mthod %

$. (x + &)&  x+$ ______ = ____(x + )&  x+

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.(x - )&  x - &

 ______ = ____(x - )&  x -  

5. ;nurupy - 7unyamanyat  

0h 7utra ;nurupy 7unyamanyat says % >62 on is in ratio, th othr on is@ro>.

A us this 7utra in sol"in! a spcial typ o2 simultanous simpl :uations

in which th co22icints o2 >on> "ariabl ar in th sam ratio to ach othras th indpndnt trms ar to ach othr. 6n such a contxt th 7utra saysth >othr> "ariabl is @ro 2rom which w !t two simpl :uations in th 2irst"ariabl (alrady considrd) and o2 cours !i" th sam "alu 2or th"ariabl.

#xampl $%  &x + y = 'x + $y = 5

Dbsr" that th y-co22icints ar in th ratio % $ i.., $ % &, which is

sam as th ratio o2 indpndnt trms i.., % 5 i.., $ % &. Cnc th othr"ariabl x = * and y = or $y = 5 !i"s y =

#xampl %  &&x + $'y = $5$5x + &$y = ''

0h "ry apparanc o2 th problm is 2ri!htnin!. But 9ust an obsr"ationand anurupy sunyamanyat !i" th solution x = , bcaus co22icint o2 xratio is

&& % 5 = $ % & and constant trms ratio is $5$ % '' = $ % &.

y = * and && x = $5$ or 5 x = '' !i"s x = .

7ol" th 2ollowin! by anurupy sunyamanyat.

$. $x + y = $ . &x + y = '

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$5x + 5y =$5 $x + y = 5

&. 'x – 5y = ' '. ax + by = bmx – y = &5 cx + dy = dm  

6n sol"in! simultanous :uadratic :uations, also w can tak th hlp o2 th sutraE in th 2ollowin! way%

#xampl & %  7ol" 2or x and y

x + 'y = $*x + xy + 'y + 'x - y = *

x + xy + 'y + 'x - y = * can b writtn as( x + y ) ( x + 'y ) + 'x – y = *

$* ( x + y ) + 'x – y = * ( 7inc x + 'y = $* )$*x + $*y + 'x – y = *

$'x + y = *

Now x + 'y = $*$'x + y = * and ' % %% $* % *

2rom th 7utra, x = * and 'y = $*, i..,, y= * y = $*' = O0hus x = * and y = O  is th solution.

. 7ankalana - Rya"akalanabhyam  

0his 7utra mans >by addition and by subtraction>. 6t can b applid in sol"in!a spcial typ o2 simultanous :uations whr th x - co22icints and th y- co22icints ar 2ound intrchan!d.

#xampl $%  'x – &y = $$&

&x – 'y = $6n th con"ntional mthod w ha" to mak :ual ithr th co22icint o2 xor co22icint o2 y in both th :uations. Mor that w ha" to multiply :uation( $ ) by ' and :uation ( ) by & and subtract to !t th "alu o2 x andthn substitut th "alu o2 x in on o2 th :uations to !t th "alu o2 y orw ha" to multiply :uation ( $ ) by & and :uation ( ) by ' and thnsubtract to !t "alu o2 y and thn substitut th "alu o2 y in on o2 th

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:uations, to !t th "alu o2 x. 6t is di22icult procss to think o2.

Mrom 7ankalana – "ya"akalanabhyam

add thm,  i.., ( 'x – &y ) + ( &x – 'y ) = $$& + $i.., 5x – 5y = *' x – y = &

subtract on 2rom othr,  i.., ( 'x – &y ) – ( &x – 'y ) = $$& – $i.., x + y = x + y = $

and rpat th sam sutra, w !t x = and y = - $Rry simpl addition and subtraction ar nou!h, how"r bi! th

co22icints may b.

#xampl %  $x – '5y = '

'5x – $y = -'$&

Dh S what a problm S ;nd still

 9ust add, '&$( x – y ) = - '&$ x – y = -$

subtract, $' ( x + y ) = & x + y =

onc a!ain add, x = ' x = subtract - y = - 5 y = &

7ol" th 2ollowin! problms usin!7ankalana – Rya"akalanabhyam.

$. &x + y = $x + &y = $

. x – $y = 5$x – y = 5

&. 5x + 5y = '$55x + 5y = &

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. 8uranapuranabhyam  

0h 7utra can b takn as 8urana - ;puranabhyam which mans by thcompltion or non - compltion. 8urana is wll known in th prsnt

systm. A can s its application in sol"in! th roots 2or !nral 2orm o2:uadratic :uation.

A ha" % ax 2 + bx + c = *

x + (ba)x + ca = * ( di"idin! by a )

x + (ba)x = - ca

compltin! th s:uar ( i..,, purana ) on th 3.C.7.

x

+ (ba)x + (b

 'a

) = -ca + (b

 'a

)

[x + (ba)] = (b - 'ac) 'a 

 ________- b T U b – 'ac

8rocdin! in this way w 2inally !t x = _______________a

Now w apply purana to sol" problms.

#xampl $.   x& + 5x + $$ x + 5 = *.

7inc (x + )& = x& + 5x + $x + ;dd ( x + ) to both sids

A !t x& + 5x + $$x + 5 + x + = x + i..,, x& + 5x + $x + = x + i..,, ( x + )& = ( x + )

this is o2 th 2orm y& = y 2or y = x + solution y = *, y = $, y = - $

i..,, x + = *,$,-$which !i"s x = -,-$,-&

#xampl %   x&

 + x

 + $x + $* = *

A know ( x + &)& = x& + x + x + 7o addin! on th both sids, th trm (x + $*x + $ ), w !t

x& + x + $x + x + $*x + $ = x + $*x + $i..,, x& + x + x + = x + 5x + + 'x + i..,, ( x + & )& = ( x + & ) + ' ( x + & ) – '

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y& = y + 'y – ' 2or y = x + &y = $, , -.

Cnc x = -, -$, -

0hus purana is hlp2ul in 2actori@ation.Murthr purana can b applid in sol"in! Bi:uadratic :uations also.

7ol" th 2ollowin! usin! purana – apuranabhyam.

$. x& – 5x  + $$x – 5 = *. x& + x  + &x + $ = *&. x + x – & = *'. x' + 'x & + 5x  + 'x – $ = *  

. 1alana - Ialanabhyam  

6n th book on Rdic 4athmatics  7ri Bharati Irishna 0irtha9i mntiondth 7utra >1alana - Ialanabhyam> at only two placs. 0h 7utra mans>7:untial motion>.

i)  6n th 2irst instanc it is usd to 2ind th roots o2 a :uadratic :uationx –$$x – = *. 7wami9i calld th sutra as calculus 2ormula. 6ts application at

that point is as 2ollows.Now by calculus 2ormula w say% $'x–$$ = TU&$

; Not 2ollows sayin! "ry Vuadratic can thus b brokn down into twobinomial 2actors. ;n xplanation in trms o2 2irst di22rntial, discriminant withsu22icint numbr o2 xampls ar !i"n undr th chaptr Vuadratic#:uationsE.

ii)  ;t th 7cond instanc undr th chaptr Mactori@ation and Hi22rntial1alculusE 2or 2actori@in! xprssions o2 &rd, 'th and th d!r, th procduris mntiond as>Rdic 7utras rlatin! to 1alana – Ialana – Hi22rntial1alculus>.

Murthr othr 7utras $* to $5 mntiond blow ar also usd to !t thr:uird rsults. Cnc th sutra and its "arious applications will b takn upat a latr sta! 2or discussion.

But sutra – $' is discussd immdiatly a2tr this itm.

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Now th rmainin! sutras %

$*. WXR;HYN;4 ( 0h d2icincy )$$. RW;Z[67;4;Z[6C ( Ahol as on and on as whol )$. \#ZXNW;] I#N; 1;J;4#^; ( Jmaindr by th last di!it )$&. 7D8XN0W;HR;W;4;N0W;4 ( ltimat and twic th pnultimat )$. ^60;7;411;W;C ( 0h whol product is th sam )$5. ^;I; 7;411;W;C ( 1ollcti"ity o2 multiplirs )

0h 7utras ha" thir applications in sol"in! di22rnt problms in di22rntcontxts. Murthr thy ar usd alon! with othr 7utras. 7o it is a bit o2incon"ninc to dal ach 7utra undr a sparat hadin! xclusi"ly andalso indpndntly. D2 cours thy will b mntiond and also b applid insol"in! th problms in th 2orth comin! chaptr whr"r ncssary. 0hisdcision has bn takn bcaus up to now, w ha" tratd ach 7utraindpndntly and ha" not continud with any othr 7utra "n i2 it isncssary. Ahn th nd 2or combinin! 7utras 2or 2illin! th !aps in thprocss ariss, w may opt 2or it. Now w shall dal th 2ourtnth 7utra, th7utra l2t so 2ar untouchd.

$*. #kanyunna 8ur"na 

0h 7utra #kanyunna pur"na coms as a 7ub-sutra to Nikhilam which !i"s

th manin! >Dn lss than th pr"ious> or >Dn lss than th on b2or>.

$) 0h us o2 this sutra in cas o2 multiplication by ,,.. is as 2ollows .

4thod %  

a) 0h l2t hand sid di!it (di!its) is ( ar) obtaind by applyin! th kanyunnapur"na i.. by dduction $ 2rom th l2t sid di!it (di!its) .

.!. ( i ) x K – $ = 5 ( 3.C.7. di!it )

b) 0h ri!ht hand sid di!it is th complmnt or di22rnc btwn thmultiplir and th l2t hand sid di!it (di!its) . i.. / J.C.7 is - 5 = &.

c) 0h two numbrs !i" th answrK i.. / = 5&.

#xampl $%   x 7tp ( a )  !i"s – $ = ( 3.C.7. Hi!it )7tp ( b )  !i"s – = ( J.C.7. Hi!it )7tp ( c )  !i"s th answr

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#xampl %   $ x 7tp ( a ) %  $ – $ = $'7tp ( b ) %   – $' = ( or $** – $ )7tp ( c ) %  $ x = $'

#xampl &%   ' x

;nswr %

#xampl '%   &5 x ;nswr %

#xampl %   x ;nswr %

Not th procss % 0h multiplicand has to b rducd by $ to obtain th 3C7and th ri!htsid is mchanically obtaind by th subtraction o2 th 3.C.7 2romth multiplir which is practically a dirct application o2 Nikhilam 7utra.

Now by Nikhilam

' – $ = & 3.C.7.x – & = 5 J.C.7. ($**–')

 _____________________________

& 5 = &5

Jconsidr th #xampl '%  

&5 – $ = & 3.C.7.x – & = 5'' J.C.7.

 ________________________& 5'' = &5''

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and in #xampl %   x w writ

* – $ = 3.C.7.x – = $ J.C.7.

 __________________________

$ = $

;l!braic proo2 %  

;s any two di!it numbr is o2 th 2orm ( $*x + y ), w procd( $*x + y ) x

= ( $*x + y ) x ( $** – $ )= $*x . $* – $*x + $* .y – y= x . $*& + y . $* – ( $*x + y )= x . $*& + ( y – $ ) . $* + < $* – ( $*x + y )

0hus th answr is a 2our di!it numbr whos $***th

 plac is x,$**th

 plac is( y - $ ) and th two di!it numbr which maks up th $*th and unit plac is thnumbr obtaind by subtractin! th multiplicand 2rom $**.(or apply nikhilam).

0hus in & / . 0h $***th plac is x i.. &

$**th plac is ( y - $ ) i.. ( - $ ) = 5

Numbr in th last two placs $**-&=5&.

Cnc answr is &55&.

;pply #kanyunna pur"na to 2ind out th products

$. 5' x . & x &. &$ x

'. '& x . 5 x 5. $ x  

A ha" dalt th cass

i) Ahn th multiplicand and multiplir both ha" th sam numbr o2 di!itsii) Ahn th multiplir has mor numbr o2 di!its than th multiplicand.

6n both th cass th sam rul applis. But what happns whn th multiplirhas lssr di!its`

i.. 2or problms lik ' / , $' / , 5& / tc.,

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Mor this lt us ha" a r-look in to th procss 2or propr undrstandin!.

4ultiplication tabl o2 .  

a b

x = $ & x = ' x = & 5- - - - - - - - - - x = x = $

$* x = *

Dbsr" th l2t hand sid o2 th answr is always on lss than thmultiplicand (hr multiplir is ) as rad 2rom 1olumn (a) and th ri!ht handsid o2 th answr is th complmnt o2 th l2t hand sid di!it 2rom as rad

2rom 1olumn (b)

4ultiplication tabl whn both multiplicand and multiplir ar o2 di!its.  

a b$$ x = $* = ($$–$) – ($$–$) = $*$ x = $$ = ($–$) – ($–$) = $$$& x = $ = ($&–$) – ($&–$) = $

-------------------------------------------------$ x = $ ----------------------------

$ x = $ $* x = $ * = (*–$) – (*–$) = $*

0h rul mntiond in th cas o2 abo" tabl also holds !ood hr

Murthr w can stat that th rul applis to all cass, whr th multiplicandand th multiplir ha" th sam numbr o2 di!its.

1onsidr th 2ollowin! 0abls .

(i)  

a b$$ x = $ x = $* $& x = $$

----------------------$ x = $5 $ x = $ $* x = $ *

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(ii)  $ x = $ x = $ & x = *

-----------------------

x = x = 5 $&* x = *

(iii)  & x = &$ '5 x = '$ '& x = ' 5 x = 5* &

-------------------------so on.

Mrom th abo" tabls th 2ollowin! points can b obsr"d%

$) 0abl (i)  has th multiplicands with $ as 2irst di!it xcpt th last on. Cr3.C.7 o2 products ar uni2ormly lss than th multiplicands. 7o also with* x

) 0abl (ii)  has th sam pattrn. Cr 3.C.7 o2 products ar uni2ormly & lssthan th multiplicands.

&) 0abl (iii)  is o2 mixd xampl and yt th sam rsult i.. i2 & is 2irst di!ito2 th multiplicand thn 3.C.7 o2 product is ' lss than th multiplicandK i2 ' is2irst di!it o2 th multiplicand thn, 3.C.7 o2 th product is lss than th

multiplicand and so on.

') 0h ri!ht hand sid o2 th product in all th tabls and cass is obtaind bysubtractin! th J.C.7. part o2 th multiplicand by Nikhilam.

Ipin! ths points in "iw w sol" th problms%

#xampl$ %  ' /

i) Hi"id th multiplicand (') o2 by a Rrtical lin or by th7i!n % into a ri!hthand portion consistin! o2 as many di!its as th multiplir.

i.. ' has to b writtn as '  or '% 

ii) 7ubtract 2rom th multiplicand on mor than th whol xcss portion onth l2t. i.. l2t portion o2 multiplicand is '.

on mor than it ' + $ = .

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A ha" to subtract this 2rom multiplicandi.. writ it as

' % %-

---------------

& %

0his !i"s th 3.C.7 part o2 th product.

0his stp can b intrprtd as tak th kanyunna and sub tract 2rom thpr"ious i.. th xcss portion on th l2t.

iii) 7ubtract th J.C.7. part o2 th multiplicand by nikhilam procss.i.. J.C.7 o2 multiplicand is

its nikhilam is

6t !i"s th J.C.7 o2 th product

i.. answr is & % % = &.

0hus ' / can b rprsntd as

' % %- %

------------------& % % = &.

#xampl %   $' /

Cr 4ultiplir has on di!it only .

A writ $ % '

Now stp (ii),$ + $ = $&

i.. $ % '-$ % &

------------7tp ( iii ) J.C.7. o2 multiplicand is '. 6ts Nikhilam is 5

$' x is $ % '-$ % & % 5

-----------------$$ % $ % 5 = $$$5

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0h procss can also b rprsntd as$' x = < $' – ( $ + $ ) % ( $* – ' ) = ( $' – $& ) % 5 = $$$5

#xampl &%   $5& x

7inc th multiplir has di!its, th answr is7implicity > in doin! problms is absnt.

#xampl $% '5 / '&  

;s pr th pr"ious mthods, i2 w slct $** as bas w !t

'5 -' 0his is much mor di22icult and o2 no us.'& -

 

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Now by anurupynaE w considr a workin! bas 6n thr ways. A can sol"th problm.

4thod $%  0ak th narst hi!hr multipl o2 $*. 6n this cas it is *.

0rat it as $** = *. Now th stps ar as 2ollows%

i) 1hoos th workin! bas nar to th numbrs undr considration.i.., workin! bas is $** = *

ii) Arit th numbrs on blow th othr

i.. ' 5' &

 

iii) Arit th di22rncs o2 th two numbrs rspcti"ly 2rom * a!ainst achnumbr on ri!ht sid

i.. '5 -*''& -*

 

i") Arit cross-subtraction or cross- addition as th cas may b undr th lindrawn.

") 4ultiply th di22rncs and writ th product in th l2t sid o2 th answr.

'5 -*''& -*

 ____________& -' x –

=

"i) 7inc bas is $** = * , & in th answr rprsnts &/*.

Cnc di"id & by bcaus * = $**

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0hus & F !i"s $O whr $ is :uotint and $ is rmaindr . 0his $ asJmindr !i"s on * makin! th 3.C.7 o2 th answr + * = (orJmaindr O x $** + )

i.. J.C.7 $ and 3.C.7 to!thr !i" th answr$ A rprsnt it as

'5 -*''& -*

 ) &

 $O

= $ = $

#xampl %   ' / '.

Aith $** = * as workin! bas, th problm is as 2ollows%

' -*' -*

 ) '* $5 

* $5

' x ' = *$5

4thod %  Mor th xampl $%  '5/'&. A tak th sam workin! bas *. Atrat it as *=/$*. i.. w oprat with $* but not with $** as in mthod

now

($ + ) = $

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Now in th xampl %  ' x ' w can carry as 2ollows by tratin! * = x $*

4thod &%  A tak th narst lowr multipl o2 $* sinc th numbrs ar '5and '& as in th 2irst xampl, A considr '* as workin! bas and trat it as '/ $*.

7inc $* is in opration $ is carrid out di!it in $.

7inc ' / $* is workin! bas w considr ' / ' on 3.C.7 o2 answr i.. $5and $ carrid o"r th l2t sid, !i"in! 3.C.7. o2 answr as $. Cnc thanswr is $.

A procd in th sam mthod 2or ' / '

3t us s th all th thr mthods 2or a problm at a !lanc

#xampl &%  ' / &

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4thod - $%   Aorkin! bas = $** = *

' *'& *&

 ) $    $ = =

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4thod %  workin! bas = x $** = **

4thod - &.  7inc '** can also b takn as workin! bas, trat '** = ' / $** as

workin! bas.

0hus

No nd to rpat that practic in ths mthods 2inally taks us to workout all ths mntally and !ttin! th answrs strai!ht away in a sin!l lin.

#xampl %   & / '

Aorkin! bas = $**** = ***

& -$**' -***

 ) &5 **' sinc $*,*** is in opration

$ **' = $**'

or takin! workin! bas = x $*** = ,*** and

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Ahat happns i2 w tak '*** i.. ' / $*** as workin! bas` _____

& ***' * 7inc $*** is an opration

 

'5 $5  ___ ___;s $*** is in opration, $5 has to b writtn as $5 and '*** as bas,

th 3.C.7 portion *** has to b multiplid by '. i. . th answr is

; simplr xampl 2or bttr undrstandin!.

#xampl 5%   x '

Aorkin! bas * = x $* !i"s

7inc $* is in opration.

s anurupyna by slctin! appropriat workin! bas and mthod.

Mind th 2ollowin! product.

$. '5 x '5 . x &. ' x ''. $ x $ . 5 x ' 5. x &*

. ' x 5 . 5 x 5 . 'x'

$*. & x $  

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$. ;dyamadynantya – mantyna

0h 7utra > adyamadynantya-mantyna> mans >th 2irst by th 2irst and thlast by th last>.

7uppos w ar askd to 2ind out th ara o2 a rctan!ular card board whosln!th and bradth ar rspcti"ly 52t . ' inchs and 2t. inchs. nrallyw continu th problm lik this.

;ra = 3n!th / Brath

= 5E ' / E 7inc $E = $, con"rsion

= ( 5 / $ + ') ( / $ + ) in to sin!l unit

= 5 5 = $5 7:. inchs.

7inc $ s:. 2t. =$ / $ = $''s:.inchs w ha" ara

$5 = $'') $5 (& 

$'' '& 

'* i.., & 7:. 2t $ 7:. inchs

 $

By Rdic principls w procd in th way th 2irst by 2irst and th last by last

i.. 5E ' can b tratd as 5x + ' and E as x + ,

Ahr x= $2t. = $ inKx is s:. 2t.

Now ( 5x + ' )(x + )

= &*x + 5..x + '..x + &= &*x + 'x + *x + &

= &*x + 5. x + &= &*x + ( x + ). x + & Aritin! 5 = x $ + = &x + . x + &= & 7:. 2t. + x $ 7:. in + & 7:. in= & 7:. 2t. + 5 7:. in + & 7:. in= & 7:. 2t. + $ 7:. in

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6t is intrstin! to know that a mathmatically untraind and "n unducatdcarpntr simply works in this way by mntal ar!umntation. 6t !os in hismind lik this

5E '

E

Mirst by 2irst i.. 5E / E = &* s:. 2t.

3ast by last i.. ' / = & s:. in.

Now cross wis 5 / + x ' = ' +* = 5.

;d9ust as many >$> s as possibl towards l2t as >units> i.. 5 = / $ + , twl">s as s:uar 2t mak th 2irst &*+ = & s:. 2t K l2t bcoms x

$ s:uar inchs and !o towards ri!ht i.. x $ = 5 s:. in. towards ri!ht i"s5+& = $s:.in.

0hus h !ot ara in som sort o2 & s:uints and anothr sort o2 $ s:. units.i.. & s:. 2t $ s:. in

;nothr #xampl%

Now $ + = $', $* x $ + ' = $* + ' = $''

0hus ' 5 x & ' = $' 7:. 2t. $'' 7:. inchs.

7inc $'' s:. in = $ / $ = $ s:. 2t 0h answr is $ s:. 2t.

A can xtnd th sam principl to 2ind "olums o2 paralllpipd also.

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6. Mind th ara o2 th rctan!ls in ach o2 th 2ollowin! situations.

$). l = &E , b = E ' ). l = $E , b = E

&). l = ' yard & 2t. b = yards 2t.($yard =&2t)

'). l = 5 yard 5 2t. b = yards 2t.

66. Mind th ara o2 th trap@ium in ach o2 th 2ollowin! cass.Jcall ara =O h (a + b) whr a, b ar paralll sids and h is thdistanc btwn thm.

$). a = &E , b = E ', h = $E

). a = E 5, b = 'E ', h = &E

&). a = E ', b = 'E 5, h = E $.  

Mactori@ation o2 :uadratics%  

0h usual procdur o2 2actori@in! a :uadratic is as 2ollows%

&x + x + '= &x + 5x + x + '= &x ( x + ) + ( x + )= ( x + ) ( &x + )

But by mntal procss, w can !t th rsult immdiatly. 0h stps ar as2ollows.

i) . 7plit th middl co22icint in to two such parts that th ratio o2 th 2irstco22icint to th 2irst part is th sam as th ratio o2 th scond part to th lastco22icint. 0hus w split th co22icint o2 middl trm o2 &x + x + ' i.. in

to two such parts 5 and such that th ratio o2 th 2irst co22icint to th 2irstpart o2 th middl co22icint i.. &%5 and th ratio o2 th scond pat to th lastco22icint, i.. % ' ar th sam. 6t is clar that &%5 = %'. Cnc such split is"alid. Now th ratio &% 5 = % ' = $% !i"s on 2actor x+.

ii) . 7cond 2actor is obtaind by di"idin! th 2irst co22icint o2 th :uadratic byth 2ist co22icint o2 th 2actor alrady 2ound and th last co22icint o2 th:uadratic by th last co22icint o2 th 2actor.

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i.. th scond 2actor is

&x ' ____ + ___ = &x +

x

Cnc &x + x + ' = ( x + ) ( &x + )

#!.$%  'x + $x +

i) 7plit $ into and $* so that as pr rul ' % = $* % = % $i..,, x + $ is2irst 2actor.

ii) Now'x 

 ___ + __ = x + is scond 2actor.

x $

#!.%  $x – $'xy – y 

i) 7plit –$' into –*, 5 so that $ % - * = & % - ' and 5 % - = & % - '. Both arsam.i.., ( &x – 'y ) is on 2actor.

ii) Now$x  y 

 ____ + ___ = x + y is scond 2actor.&x -'y

0hus $x – $'xy – y = ( &x – 'y ) ( x + y ).

6t is "idnt that w ha" applid two sub-sutras anurupynaE   i..proportionalityE and adyamadynantyamantynaE   i.. th 2irst by th2irst and th last by th lastE to obtain th abo" rsults.

Mactoris th 2ollowin! :uadratics applyin! appropriat "dic mathssutras%

$). &x + $'x + $

). 5x – &x +

&). x – x +

'). $x  – &xy + $*y

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$&. Wa"adunam 0a"adunikrtya Rar!anca Wo9ayt

0h manin! o2 th 7utra is >what "r th d2icincy subtract that d2icit 2romth numbr and writ alon! sid th s:uar o2 that d2icit>.

0his 7utra can b applicabl to obtain s:uars o2 numbrs clos to bass o2powrs o2 $*.

4thod-$ %  Numbrs nar and lss than th bass o2 powrs o2 $*.

#! $%    Cr bas is $*.

0h answr is sparatd in to two parts by aEE

Not that d2icit is $* - = $

4ultiply th d2icit by itsl2 or s:uar it

$ = $. ;s th d2icincy is $, subtract it 2rom th numbr i.., –$ = .

Now put on th l2t and $ on th ri!ht sid o2 th "rtical lin or slashi.., $.

Cnc $ is answr.

#!. %  5 Cr bas is $**.

7inc d2icit is $**-5=' and s:uar o2 it is $5 and th d2icincysubtractd 2rom th numbr 5 !i"s 5-' = , w !t th answr $50hus 5 = $5.

#!. &%  ' Bas is $***

H2icit is $*** - ' = 5. 7:uar o2 it is &5.

H2icincy subtractd 2rom ' !i"s ' - 5 =

;nswr is *&5

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;nswr is 5 *$''

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= $*** ( * ) +

= *.

4thod. %  Numbrs nar and !ratr than th bass o2 powrs o2 $*.

#!.($)%   $& .

6nstad o2 subtractin! th d2icincy 2rom th numbr w add and procdas in 4thod-$.

2or $& , bas is $*, surplus is &.

7urplus addd to th numbr = $& + & = $5.

7:uar o2 surplus = &

 =

;nswr is $5 = $5.

#!.()%   $$

Bas = $**, 7urplus = $,

7:uar o2 surplus = $ = $''

add surplus to numbr = $$ + $ = $'.

;nswr is $' $'' = $''

Dr think o2 idntity a = (a + b) (a – b) + b  2or a = $$, b = $%

$$ = ($$ + $) ($$ – $) + $ = $' ($**) + $''= $'** + $''= $''.

(x + y) =x + xy + y 

= x ( x + y ) + y = x ( x + y + y ) + y 

= Bas ( Numbr + surplus ) + ( surplus) 

!i"s$$ =$** ( $$ + $ ) +$ 

= $** ( $' ) + $''

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= $'** + $''= $''.

#!. &%  $** 

= ( $** + )

 

= $*** *5 < sinc bas is $*,***

= $****5.

4thod - &%  0his is applicabl to numbrs which ar nar to multipls o2 $*,$**, $*** .... tc. Mor this w combin th upa-7utra >anurupyna> and>ya"adunam ta"adunikritya "ar!anca yo9ayt> to!thr.

#xampl $%   &  Narst bas = '**.

A trat '** as ' x $**. ;s th numbr is lss than th bas w procdas 2ollows

Numbr &, d2icit = '** - & = $

7inc it is lss than bas, dduct th d2icit

i.. & - $ = &5.

multiply this rsult by ' sinc bas is ' / $** = '**.

&5 x ' = $*'

7:uar o2 d2icit = $ = $''.

Cnc answr is $*' $'' = $*''

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#xampl &%   5 Narst bas = *

#xampl '%   '$5 Narst ( lowr ) bas = '**

Cr surplus = $5 and '** = ' x $**

#xampl %   *$ Narst lowr bas is *** = x $***

7urplus = $

;pply ya"adunam to 2ind th 2ollowin! s:uars.

$.   .   &.   '. $' 

. $$5  5. $*$   . $  . ' 

. 5  $*. $*   $$.   $. 5*$' . 

7o 2ar w ha" obsr"d th application o2 ya"adunam in 2indin! th s:uars o2numbr. Now with a sli!ht modi2ication ya"adunam can also b applid 2or