ແບບຮຽນຄະນິດສາດ ມ1

 

0D r;doadIPolk,kf.-hg7jnv’\kpspa’]ts;kj ’ D c]t I F]ts;kj ’ I c]t `ສະຫງສ.ວວນ.ລິສຂະສິດ
c]t ]ts;jk’ D c]t ` D
!WD L c,oj d5j,rqo]tg,nv’xtgmf]k;F V dj5,rqo]tg,nv’ot7vos^;’;P’9aoF
S c,jod5j, rqo]tg,nv’c0;’lts;aootg0fF X d5,j rqo]tg,nv’0v’c0;’-P’
0;k’D
dD { 4hk;kj a c,oj 7qoot7vos^;’;P’9aoX a∈V ?F gIqk9tg;qhk.fhco;.f]ts;kj ’

a c]t L L gIqk0Powfhco;.f]ts;kj ’ V c]t L L
{ g-aoj fP;daoF 4hk m∈S F gIqk0Powfhco;.fS ]ts;jk’ m c]t L.............DDDDDDDDS
]ts;jk’S c]t L D
{ gIqklk,kf0Powfhco;.fF ]ts;jk’ X c]t L L
0D .shc8h,I6[d5j,c[[ Venn le]a[d5j, L c]t lk,vto5d5j, V; S; X.
7D V ∩ S∩ X = ................................. L
!!D gryoj .shd5j, U = {2, 6,8, 7,1,5} ; E = {1, 2, 6} ; F = {8}

G = {1, 2} ; H = {7, 5, 6, 2,1}; K = {9, 2, 3}

dD .o[aofkd5j,  E,F,G,H,K ,ud5j,.fcfjgxaovto5dj5,0v’ U D
0D 4hk[+c,oj vto5d5j,0v’ U 9q’j .shgsfzqoD
!@D dj5,  A  wfh4nddeoqfc[[0tspkpfa’j oUG
                A = {a,b, c, m, n,u} S B = {a, d, m}S C = {b, c, n,u}

dD 8k8t]k’]5,j oU gryjo\kp 1 le]a[ a ≠ C . .sh8nj,.l[j vj os;j k’.sh4nd8hv’D
0D .sh0Povq’xtdv[0v’dj5,  A,B c]t C .oczo;kf0v’ Venn c]t czo
;kf0v’ Caroll .

AB C
a1 0
b

c

d

m

n

u

197

[qfmu @* dko8af0v’d5j,{dko3I,0v’dj5,

!D dyf9tde

dD dko8af0v’d5j,

dyf9tdemu !

.shoadIPola’gdfg[y’j czo;kfc;o mu1j 60j hk’0;koUD E 
 c,jovq’xtdv[0v’d5j, A [+ L c]t c,oj

vq’xtdv[0v’dj5, B [+ L

gIqklk,kfg;qhkwfh;jk c,jovq’xtdv[Ij;,0v’ 7  A 
A c]t B D a  B 
ovd9kd ,uvq’xtdv[8q;vjno[+mugj xaovq’xtdv[
Ij;,0v’ A c]t B L 0 
3  x 

 gIqk,uvq’xtdv[0v’ A c8[j +c,jovq’xtdv[0v’ ສະຫງສ.ວວນ.ລິສຂະສິດ

B g-aoj a c]t ,uvq’xtdv[0v’ B [+c,jovq’xt

dv[0v’ A g-ajo DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

 gIqk,uvq’xtdv[muj[+c,oj vq’xtdv[0v’ A c]t [+c,oj vq’xtdv[0v’ B g-aoj

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

 faj’oAo d5j, { , U, 3 } c,jodj5,vq’xtdv[I;j ,0v’d5j, A c]t B gvUo;jkG

rkdlj;o8afdao0v’dj5, A c]t dj5, B laopt]ad A ∩ B = { , U, 3 }

dyf9tdemu @
.shoadIPola’gdfg[y’j czo;kf0v’ c;oXVenn ? djP;da[l;j o8af0v’dj5,

A  B  F   A  B  F   A  B  F  

XI6[!? XI6[@? XI6[#?
.sh.ljlulj;o8afdao0v’ .sh.llj ul;j o8afdao0v’ .sh.ljlul;j o8afdao0v’

198

A c]t  B .oI6[! A c]t  B .oI6[@ A c]t  B .oI6[#

A ∩ B = .................. A ∩ B = .................. A ∩ B = ..................

.97;k,

vq’xtdv[0v’rkdl;j o8afdao0v’dj5, A c]t B  c,oj dj5,0v’vq’xtdv[Ij;,
daoma’\qf0v’ A c]t  B . laopt]ad A ∩B vkj o;jk A 8af B D \kp;jk
a∈A ∩ B c,oj a∈A c]t a∈B

dyf9tdemu #
.shoadIPola’gdfg[y’j czo;kf0v’ d5j, A, B, C C

9 35 0 ສະຫງສ.ວວນ.ລິສຂະສິດ
1 2 68
B
7
A4

 .sh0Podj5, A, B, C c[[0tspkp
A = {1, 2,........................................}

B = { 0,............................................}

C = { ................................................}
 .sh-vdskvq’xtdv[I;j ,0v’ A c]t B c]h;0Po A ∩B = { ...............}
c]t vq’xtdv[Ij;,0v’ B c]t A c]h;0Po B∩ A = { ...............}
gIqkg;qhkwfhco;.f]ts;kj ’ A ∩ B c]t B∩ A ltoAo A ∩ B..................B∩A
 .shd5j, M = A ∩BF.sh-vdskl;j o8afdao0v’ M c]t C s^n

M ∩ C = { ............}
ltoAo M ∩ C = ( A ∩ B ) ∩C = { ........................}
 .shdj5, P = B∩C = { ..................}F c]t A ∩ P = { ..................}D
ltoAo A ∩ P = A ∩ ( B∩C ) = { ...........................................}

199

 la’gdfgsao;jk ( A ∩ B )∩C ........... A ∩( B∩C )
 9kddko7yfw]0j hk’gmy’r;dgIqklts^5[wfh;jkG

75o]adltot

!D m5dM dj5, A, B S A ∩B = B∩A dko8af0v’d5j,gxaodko7eo;omuj
la[xjPo[jvo wfhD

@D m5dM d5j, A, B, C S A ∩( B∩C ) = ( A ∩ B )∩C = A ∩ B∩ C dko8af
dao0v’d5j,gxao dko7eo;omuj 3I,\j6 wfhD

0D dko3I,0v’dj5,

dyf9tdemu $

 .shoadIPola’gdfg[y’j czo;kfc;o c]t czo < A 3I, B <

7nrkdl;j ooEg[uDDDDDDDDD
;kf0v’dk3io 0hk’g[Nv’0;koUDglAolufe[yd0uf ສະຫງສ.ວວນ.ລິສຂະສິດ

vhv,d5j, A c]t B -bj’xtdv[gxaodj5,.\mj ujwfh,k9kd DDDDDDDDDDDDDDDDDDDDDDDDDDDDD

dj5, A 3I,d5,j B D

 vq’xtdv[0v’d5j,.\j A 3I,da[ B wfh,k9kd 213
lk,rkdl;j o7nG
rkdlj;omuj,uvq’xtdv[Ij;,0v’ A c]t B c,jo

A∩B 7nrkdlj;ooEg[uDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD A
rkdlj;omu,j uvq’xtdv[0v’ A c8[j +,uDDDDDDDDD0v’ B

7nrkdl;j ooEg[uDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD B     

rkdlj;omuj,uvq’xtdv[0v’ B c8[j +,uDDDDDDDDDD0v’ A

7nrkdlj;ooEg[uDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

d5j,.\j A 3I, B laopt]adG ''A∪B''gvUo;kj G E
 le]a[1j6.oczo;kf0v’dk3io A∪B c,jolj;o.f0v’I6[ L

200

.97;k, { vq’xtdv[Ij;,0v’( A∩ B )

vq’xtdv[0v’d5j, A∪B c,jo { vq’xtdv[0v’d5j, A c8j[+c,oj vq’xt
dv[0v’dj5, B

{ vq’xtdv[0v’d5j, B c8[j +c,jovq’xt
dv[ 0v’ dj5, A

dyf9tdemu &
gryoj .shdj5, A = {1, 4, 6, 8 }; B = { 5, 6, 7, 8, 9 }
{ .sh0Poc[[0tspkp A ∩B = { ......................}
{ .sh0Povq’xtdv[0v’ A c8[j +c,jovq’xtdv[0v’ B : ....................................
{ .sh0Povq’xtdv[0v’ B c8j[+c,oj vq’xtdv[0v’ A : ....................................
{ ltoAo A ∪B = { ........................} c]t B∪A = { .........................}
{ la’gdfwfhco;.f]ts;jk’dj5, A ∪B........................B∪A

75o]adltot
ສະຫງສ.ວວນ.ລິສຂະສິດ
m5dMd5j, A, B S A ∪ B = B∪A dko3I,d5j,c,oj dko7eo;o la[xjPo[vj o wfhD

dyf9tdemu *

.shoadIPo0Poc[[0tspkpdj5, A, B, C faj’czo;kf0v’c;o]j5,oUD

{ A = { m, c..............................} A  a 5  c  4  2  B 
C 
B = { .....................................} 9  m n  8 
7 
C = { .....................................}
{ E = A ∪ B = { a,....................} d 
{S = E ∪ C = { a,....................}
{ F = B ∪ C = { a,....................}

{ P = A ∪ F = { a,....................}

201

{ .shlq,mP[d5j, P da[d5j, S S P.............S gIqkwfh A ∪ F.......E ∪C
{ oadIPola’gdfco;.f]ts;kj ’ ( A ∪B )∪C...............A ∪( B∪C )
75o]adltot

m5dMd5j, A, B, C S A ∪( B∪C ) = ( A ∪B )∪C = A ∪B∪C dko3I,
dj5, c,jodko7eo;omuj 3I,\6j wfhD

@D [qfg/ydsaf

!D .shoadIPo0uf0hklj;omuj3I,dao0v’dj5, M c]t N 8k,c8j]tI6[D

M  N F  M  N  F  M  N  F 

0uf0hklj;o M∪ N 0uf0hklj;o M∪ N ສະຫງສ.ວວນ.ລິສຂະສິດ
0uf0hkl;j o M∪ N

c]t M ∪ N = .................

@D oadIPo 237qo 0v’shv’,! lk,kfs^Uody]kwfh1jk’|hvp|bj’xtgrf.olv’xt
grf7n [kog8t c]t [kolq’j D Ih6;kj 18 7qo s^Uody]kg8t[ko c]t 157qo s^Uo[ko
lqj’D 4k,;jk ,uoad IPo9ad7qo musj ^Uowfhma’dy]kg8t[ko c]t [kolqj’D
#D 3fpoe.-hs^addkomuj;kj dko8afdao0v’d5j,c,oj la[xPj o[vj owfh c]t 3I,\j6wfhD
9q’j -Uc9h’;kj G ( A ∩ B )∩C = ( A ∩C )∩B D

$D gryjo.shdj5, A rtpao-tot]k;S d5j, B vadlvol6’S dj5, C vadlvodk’ c]t d5j,
D vad lvo8OD

{ .sh0Podj5, A S d5j,S B Sdj5, C c]t D c[[0tspkpD
{ .sh8jn,d5j,.l[j jvo9Eg,af]5,j oUG
dD A ∪ B = ................... ; A ∪ A = ...................
0D B∩C = ................... ; C∩ D = ...................

202

ສະຫງສ.ວວນ.ລິສຂະສິດ7D D ∩ A = ................... ; A ∩ C = ...................
’D B∪C∪ D = ................... ; B∪ D = ...................
&D .shd5j, E = { 3, , 7, a,U, 2, x } c]t vto5d5j, C ={3, , 2} F D ={U, 7}D
dD .shc8h,czo;kf0v’ carroll dj5, E c]t C, D
0D .sh-vdskd5j, C ∩ D = ....................; C ∪ D = { .......................}
*D dD I6h;jk B⊂ A gIqkwfh A ∪B = A
0D 9kdzqoIa[0hk’gmy’Fc]h;gvqk,koe.-hdko7yfw]8j +wxoUG

A∪A = A; A∪φ = A; φ ∪φ = φ

(D3fpoe.-hs^addkomuj;kj <dko3I,d5j,<0v’dj5,c,oj dko7eo;omujla[xjPo[jvowfh
c]t 3I,\6jwfhF.shryl6f;kj G
( A∪B )∪C = ( A∪C )∪B L
)D oadIPoshv’,! csj’|b’j F wfhrkdaowxmvj ’mPj ;8k,[jvo8jk’MF .olk,[vj ofaj’
oUS zklkfsuo;afr6F oE8qd8kf7v’rtgra’ c]t 8kfgspbv’D r;dg0qkg9qhkwfh
c['j daofaj’oUG 9 7qowxmjP;;afr6rP’cs’j fP;F37qowxmPj ;;afr6 c]t 8kfgpbv’F
5 7qo wxmPj ;;afr6 c]t 7vortgra'F7 7qowxmPj ;8kfgpnv’rP’cs’j fP;F 6 7qo
wxmjP;7vortgra’ c]t 8kfgpbv’D oadIPowx7vortgra’ma’\qf17 7qo .ooAo,u
4 7qo wx7vortgra’rP’cs’j fP;D
dD .sh0Pogxaoczo;kf0v’c;o dj6, P 9eo;ooadIPowxmjP;;afr6

d5j, C 9eo;ooadIPowxmjP;7vortgra’
d5j, M 9eo;ooadIPowxmjP;8kfgpbv’
0D ,uoadIPo9ad7qomuj,u3vdkfwfhwxmPj ;\qfma’lk,[jvo L ,uoadIPo9ad7qomuwj fhwx
mjP;rP’c8j 2 [jvogmqjkoAo L
7D oadIPomujwxmPj ;;afr6,u9ad7qo L wxmPj ;8kfgpbv’,u9ad7qo L c]t oadIPoma’
\qf0v’shv’ ,! csj’oaho,u9ad7qo L
_D czo;kf0v’ mugj saooU c,oj d5j,
E:oadIPo0v’3I’IPocsj’|b’j c]t vto5d5j,
A : dj5,oadIPomujIPorklkva’dyf
B: d5j,oadIPomujIPorklk/ia’
C:d5j,oadIPomuIj Porklk9uo

203

dD .sh.llj ucf’dj5,oadIPomuIj Porklkva’dyf c]t B  A 
rklk9uoc8j[+wfhIPorklk/ia’
0D le]a[oadIPomu\j kpfh;p m F gIqklk,kfg;qhk n h  m 
wfh;jk oadIPo m IPorklkva’dyfrP’rklk d  c 
fP;D gIqklk,kfg;qhkwfhco;.fle]a[oadIPo i 
mju\kpfh;p n, c, h, d, j, i
j 

C 

7D .shc8h,.\jle]a[c8]j t7e4k,]j5,oUG
{ .shmklud5j,0v’oadIPomujIPorklk9uo c8j[+wfhIPorklkva’dyf
{ .shmklud5j,0v’oadIPomujIPoma’lk,rklk
{ .shmkludj5,0v’oadIPomu[j +wfhIPorklkva’dyf c]t [+wfhIPorklk/ia’D

!WD .shdj5, E = { a, b } X gvUo;jk d5j,76j ?D

dD .sh-vdsk 4 vto5dj5,0v’dj5, E : P = {φ , { a },.............................}

0D 8q;1jk’ φ ∩ { a } = φ ; { a }∩{ a, b } = { a }
{ a }∩ { b } = φ c]t 8+MwxD .sh3I,gvqk7e8v[8kj ’M g0qhk.o8k8t
]k’ dko8afdao0v’d5j,fa’j ]5j,oUG
ສະຫງສ.ວວນ.ລິສຂະສິດ

∩ φ {a} {b} E

φφ

{a} φ φ {a}

{b} φ

E {a}

204

7D 8q;1jk’ φ ∪ { a } = { a } ; { a }∪{ a, b } = { a, b }
{ a }∪ { b } = E c]t 8+wxoU .sh3I,gvqk7e8v[8kj ’M g0qhk.o8k8t
]k’dko3I,dao0v’d5j,]5j,oUD
∪ φ {a} {b} E

φ {a}

{a} {a} EE

{b} E

EE

’D 8q;1jk’ ltg\uzqo { a }∩ X = φ g[yj’8k,8k8t]k’dko8afdao0v’dj5,r;dສະຫງສ.ວວນ.ລິສຂະສິດ
gIqk9trq[

{ a }∩ φ = φ S gIqkwfh7kj X = φ
{ a }∩ { b } = φ gIqkwfh7jk X = { b }
faj’oAo X ,ulv’7jk7n φ s^n { b }
.sh-vdsk7kj 0v’ X .oltg\uzqo8+wxoUG
 { b }∩ X = { a } ; gIqkwfh X = ..........................
 E ∩X = E gIqkwfh X = ..........................
 X ∩{a} = {a} gIqkwfh X = ................; X = ................

 X ∪{b} = E gIqkwfh X = ................; X = ................

 X ∪{a} = {a} gIqkwfh X = ................; X = ................

 E ∪ X = E gIqkwfh X = ................; X = ................

X = ................; X = ................

 X ∪{b} = {b} gIqkwfh X = ................; X = ................

205

[qfmu @( vaf8kl;j o

!D dyf9tde

dyf9tdemu !

 dv’\kdxulv’dv’F dv’mu|b’j ,u 7 |;j p c]tdv’mulv’,u12|;j p 4k,;jk
dv’.f|hvpd;j kdao L DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
4hkgvqkdv’mu|b’j Xmuj,u9eo;o\kdxu 7 |;j p?mP[.lj dv’mulv’ Xmu,j u\kdxu12
|j;p? \kp7;k,;jk 7yfsk7jk0v’dv’mu\b’j 3fpgvqkdv’mulv’gxaosq;\;j pD
oadIPo 7yf ;kj zqoIa[9ts^kpd;j k1s^n |hvpdj;k1 L 8A’[A’g]dc[[.f L DDDDDDDDDDDDD
ltoAo vaf8kl;j o9eo;o0v’dv’muobj’mP[.l9j eo;o0v’dv’mulv’gxaog]d
lj;oc,jo DDDDDDDDDDDDDDDDDDDD

 4hkgvqkdv’mulv’mP[.ljdv’mu|b’j \kp7;k,;jkG 7yfsk7kj 0v’dv’mulv’
3fpgvqkdv’mu|b’j gxaosq;|;j pD
oadIPo7yf;kj zqoIa[9ts^kpdj;k1s^n |hvpdj;k1 L 8A’[A’g]dc[[.f L
ltoAo vaf8klj;o9eo;o0v’dv’mulv’mP[.l9j eo;o0v’dv’mu|b’j gxaog]d
lj;oc,joG DDDDDDDDDDDD
ສະຫງສ.ວວນ.ລິສຂະສິດ
dyf9tdemu @

 vaf8kl;j o9eo;ooadIPopy’ mP[.lj 9eo;ooadIPo-kp.o3I’IPocsj’|b’j

gmqkj da[ 3
5

 \kp7;k,;kj G 9eo;ooadIPopy’ sko.sh DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

9tgmqkj da[3 ÷ 5

 s^n 0Pogxaog]dlj;oG 9eo;ooda IPopy' = DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
9eo;ooda IPo-kp

4hk;kj 9eo;ooadIPo-kp,u 1057qoF 4k,;jkG 9eo;ooadIPopy’,ugmqkj .f L

206

;ymucdh
9eo;ooadIPopy’cmofh;p a c]t 9eo;ooadIPo-kpcmofh;pb D

a = 3 b cmofh;p105 gIqkwfh a = 3
b 5 105 5

a ×105 = 3 ×..............; a× 105 = 3×........ = ..............
105 5 105 5

faj’oAo a = ..................9eo;ooadIPopy’c,oj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

.97;k,

gryjogvUo;jkG vaf8klj;o0v’xt]y,ko a mP[.lxj t]y,ko b -b’j b ≠ 0

c,jozqosko4nd0v’ a .sh b s^n ( a ÷ b ) s^n gxaog]dl;j o a  g-y’j
b

a gvUo;jkG rqfmu|b’j c]t b c,oj rqfmulv’D

\kpgsf ສະຫງສ.ວວນ.ລິສຂະສິດ

vaf8klj;o0v’lv’xt]y,ko-toyffP;daoF c,oj 9eo;o|b’j mujltcf’;kj G xt]y

,kooYUmq[9adgmnjvxt0v’xt]y,kooAo Xs^n xt]y,kooUgmqjk9adgmnjv0v’xt]y,kooAo?D

8q;1jk’G vaf8kl;j o0v’lv’9eo;o15 c]t 3 c,jozqosko4nd0v’15.sh 3 g-y’j

c,oj |bj’7n 5 ×3 = 15 D

15 = 5 grkt;kj G 15 = 5×3
3

dyf9tdemu #

 gryoj .shzqosko s^n Xg]dlj;o? 5 ; 15 ; 1 .shoadIPo0Pog]dl;j o8+wxoU gxao
20 60 4
9eo;omqfltoupq,D gIqkg;qhkwfhco;.f]ts;kj ’g]dlj;ofaj’djk; L

9qj’.sh0=hla’gdf]ts;jk’zqoskoma’lk, s^n Xg]dlj;oma’lk,? 0hk’gmy’G

 fa’j oaho

×...........   ÷ .......... 1
5 ........... ...........
20 = =
×...........   60
÷ ..........

207

dyf9tdemu $

9qj’0Po9eo;o8nj,.l[j vj o9Eg,afD

15 ÷ .......... ........... × 10 ...........
27 ........... ...........
= =

÷3 × ...........

75o]adltot

 8k,dyf9tde 3 ; 4 0hk’gmy’gIqklts^5[wfh;kj G

a = a × n = a : m X g-yj’ b , n , m , ≠ 0 ?
b b × n b : m

 zqoIa[0v’lv’9eo;o a -bj’ b ≠ 0  [+xjPocx’G
b
{ 4hkgryjo76or6fc]t9eo;or6fda[9eo;o8jk’l6ovaofP;dao

{ s^n 4hkgryjoskor6fc]t9el;or6f.sh9eo;o8kj ’l6ovaofP;dao

@D [qfg/ydsaf ສະຫງສ.ວວນ.ລິສຂະສິດ

!D gryjoc[j’fyox6d/a’mq’j |bj’gxaolk,rkdl;j o7nG rkdlj;omu|b’j c,jox6dg0qhkF rkd

l;j omulv’c,jox6d,aofhk’ c]t rkdl;j omulk,c,oj x6d\kd4q;j fyoF 8k,dko

9af laofaj’oUG vaf8klj;ofyox6dg0qhk8+fyox6d,aofhk’c,jo 6 S vaf8kl;j ofyox6d,ao
5

fhk’8+ fyox6d\kd4qj;fyoc,jo 5 c]t vaf8kl;j ofyox6d\kd4q;j fyo 8+fyox6d/a’
4

ma’\qf c,oj 4 D
15
dD goNvmujx6d/a’rkdl;j o.fs^kpd;j k\6j c]t goNvmujx6d/a’rkdlj;o.f|hvpd;j k\5jL

X.shvtmy[kp?

0D .sh7yfw]sj kgoNvmujfyox6d/a’c8]j trkdlj;og,nvj Ih6;kj goNvmujfyox6d/a’ma’\qfc,oj

525 ha D

@D 1j6mtok7kocsj’|b’j ,uma’7qowx/kdg’yo c]t dh61n,g’yoD Ih6;jkvaf8klj;o0v’zh6

wx1n,g’yo8+9eo;ozh6/kdg’yoc,jo 2 c]t Ih6;kj 9eo;o7qozh6muwj x/kdg’yo,u
7
180 7qoD 4k,;jk 9eo;o7qomujwxdh61n,g’yo,ugmqkj .f L 9eo;o7qowx/kdg’yo c]t

dh61n,g’yo,u ma’\qfgmqjk.f L

208

#D 9qj’8nj,.llj tg\uzqo8+wxoUG

dD 1 = .......... 0D .......... = 15 7D .......... = 15
4 80 50 25 5 25

’D 60 = 3 9D 35 = .......... lD 242 = 11
.......... 5 50 50 .......... 5

$D vaf8klj;o 0v’oadIPofugfaoj mhkpxu 8+oadIPoma’\qf0v’shv’,! 0v’3I’IPo

csj’ |bj’c,jo 1 Xs^n g;qhk;kj G oadIPofugfajomhkpxu0v’shv’,! ,u 1  0v’oadIPo
4 4
ma’ \qf .oshv’oAo?D 4k,;jk oadIPoma’\qf.oshv’oAo,u9ad7qo L

I6h;kj oadIPofugfajo0v’shv’oaho ,u 9 7qoD

&D goNvmugj fujog8t[ko0v’3I’IPocsj’|bj’,u 10800m2 F ovdoAogryo9afgxaogoNv

mu0j v’dy]kvnjoM mP[.lj goNvmujma’\qf0v’3I’IPoc,oj 2 c]t goNvmuvj njoM mP[
9

.ljgoNvmuj0v’gfuoj g8t[koc,oj 5 D Ih6;kj goNvmuj0v’3I’IPoma’\qfc,jo 2  gmnjv
9
ສະຫງສ.ວວນ.ລິສຂະສິດ
gonhvmuj0v’gfujog8t[koD 4k,;jk goNvmuj0v’dy]kvnoj M c]t goNvmujvnoj M,u9ad m2 L

*D 9qj’0Po <vaf8klj;o< 9eo;o0v’I6[g]0k7toyf|bj’ mP[.lj 9eo;o0v’I6[

g]0k 7toyf8jk’M faj’ltcf’1j6.oI6[]5,j oUG

dD 9eo;oI6[;q’,qo mP[.lj 9eo;oI6[9t85]af c,oj G DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
0D 9eo;oI6[9t85]af mP[.lj 9eo;oI6[;q’,qo c,jo G DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
7D 9eo;oI6[fvd9ao mP[.lj 9eo;oI6[lk,c9 c,jo G DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
’D 9eo;oI6[lk,c9 mP[.lj 9eo;oI6[fvd9ao c,oj G DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
9D 9eo;oI6[lujc9lkd mP[.lj 9eo;oI6[fvd9ao c,oj G DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
lD 9eo;oI6[fvd9ao mP[.lj 9eo;oI6[7k’\6lkd c,jo G DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
-D 9eo;oI6[lucj 9lkd mP[.lj 9eo;oI6[7k’\6lkd c,jo G DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

209

pD 9eo;oI6[7k’\6lkd mP[.lj 9eo;oI6[lk,c9 c,oj G DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
(D .oc8]j td=]tou0v’I6[fa’j ]5j,oUF.shoadIPomklu8k,vaf8kl;j omu.j sh0v’goNvmumj a’
\qf16jI6[]5j,oUD

15 2

26 3

 rkdlj;omujmkluc8]j tI6[,u9ad8kdk3i L

1 0v’I6[ = ........................... 8kdk3i s^n 1 × 24 ...................
2 2

5 0v’ 24 8kdk3i =.............. 8kdk3i s^n 24 × 5 ...................
6 6

2 0v’ 24 8kdk3i = .............. 8kdk3i s^n 24 × 2 ...................
3 ສະຫງສ.ວວນ.ລິສຂະສິດ3
)D .shdkj p7noc]h;8jn,.l.j sh4nd8hv’D

dD 76 = ......... = 19 7D 7 = 28 = .........
12 6 ......... 5 ......... 100

0D 50 = ......... = 1 ’D 1, 5 = ......... = 0, 3
100 10 ......... 2, 5 25 .........

7D 16 = ......... = 1, 6 9D 16 = 4 = .........
100 25 ......... 4 .........

_D gryoj 4vdoE.l4j a’wfhrP’ 3 0v’[=]y,kfma’\qfD ,aopa’[+r= 27 A grnvj .shoEg8a,
4
4a’ r=fuD oadIPolk,7qowfhc8h,|j;p9vdc]tmklufa’j ]5j,oU

I6[! I6[@ I6[#

27A 27A 27A

dD c,oj I6[.f.olk,I6[mujmklu4ndda[39f0=hoU L
0D 4k,;kj 4a’oU[ao95wfh9ad]yf X A ? L

210

[qfmu @) vaf8kl;j orq;rao

!D dyf9tde 0, 35 105

dyf9tdemu !
];’pk;gxao ( km ) G 1, 24

.shxjPogxao ( m ) G 1240 ... ...

9qj’la’gdfg[yj’zqowfhIa[0v’c4;lv’ c,oj .fh,kc[[.f L

dyf9tdemu @

xt95[aooU ok’ 8yd ,uvkp5 18xu c]t c,j0v’]k;,uvkp5 48 xuF g,jnvvkp50v’ok’
8yd,u 25 xuF c,j0v’]k;9t,uvkp5 DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
vkp50v’c,j]k;c,oj wfh,kfh;p;ymuDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
ສະຫງສ.ວວນ.ລິສຂະສິດ
dyf9tdemu #

.sh8nj,.l8j k8t]k’]5j,oUD

0hk’0v’I6[9t85]af( cm ) 8 4 10 1
];’Iv[0v’I6[9t95]af( cm ) ........... 16 ........... 4

® 9eo;o8jk’M0v’c4;mulv’wfh,k3fpDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

® 9qj’lq,mP[zqosko Xg]dl;j o? 8+wxoUG 32 ; 16 ; 40 ; 4 D
8 4 10 1

dyf9tdemu $

.sh8jn,.lj[vj o9Eg,af0v’8k8t]k’]5,j oUG

0hk’0v’I6[9t85]af( cm ) 8 4 10 2
........... 16 ........... 4
goNvmuj0v’I6[9t95]af( cm2 )

9eo;o8jk’M0v’c4;mulv’wfh,k3fp,u]adltot7hkp7ndaoda[.odyf9tdemu3s^n [+L

211

lq,mP[zqosko Xg]dl;j o? 8+wxoUG 64 ; 16 ; 100 ; 4
8 4 10 2

dyf9tdemu &

9qj’8nj,.lj[jvo9Eg,af0v’8k8t]k’]5,j oUG

[=]y,kfoE,ao 12 .......... ......... 15
30 250 60 500 ×...
÷.... cvaf-a’ X]yf?
9eo;og’yo8hv’ .........
6050 .......
9kj p Xdu[?

Â-vdsk9eo;o8jk’M .oc4;mulv’0v’8k8t]k’ 3fpvu’.lj9eo;o.oc4;

mu|bj’0v’8k8t]k’ wfhfh;p;ymuDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
Â-vdsk9eo;o8kj ’M .oc4;mu|b’j 0v’8k8t]k’ 3fpvu’.lj9eo;o.oc4;mu

lv’0v’8k8t]k’wfhfh;p;ymu.f LDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
Â9qj’lq,mP[zqosko0v’9eo;o.oc4;mulv’.sh9eo;o.oc4;mu|b’j G
ສະຫງສ.ວວນ.ລິສຂະສິດ
6050 ; ...... ; 30 250 ; 60 500 ; ......
1 2 ........ ......... 15

gsao;kj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

0=hla’gdfG
Â8k,dyf9tdemu ! ];’pk;mujgxao ( m ) c]t ];’pk;mujgxao km gxaovaf8kl;j o

rq;raoD
 8k,dyf9tdemu # ];’Iv[0v’I6[9t85]af c]t ];’pk;0v’0hk’gxaovaf8k

lj;orq;raodaoD
 8k,dyf9tdemu & 9eo;og’yo8hv’9kj pgxaovaf8kl;j orq;raoda[[=]y,kfoE,aoD
 l;j odyf9tdemu @F$ lv’xt]y,komujrq;raodao.oc8j]tdyf9tde[+gxaovaf8k
l;j orq;raowfhD

212

dyf9tde *D

9qj’la’gdfg[y’j lv’8k8t]k’0v’vaf8kl;j orq;rao]j5,oU c]h;.sh7evtmy[kp9e
o;o0v’c4;mulv’wfh,k9kd9eo;o0v’c4;muob’j fh;p;ymu.f L
9eo;o0v’c4;muobj’wfh,k9kd9eo;o0v’c4;mulv’fh;p;ymu.f L
8k8t]k’ !G

5 6 7 8 9 10
÷..... ×.....

15 18 21 24 27 30

8k8t]k’ @G

5 6789 10
÷..... 1, 2 1, 4 1,6 1,8 ×.....

1 2
ສະຫງສ.ວວນ.ລິສຂະສິດ
lts^5[
9eo;o0v’c4;mu|b’j ma’lv’8k8t]k’gruj,0BoXs^ns^5f]q’?1,2,...gmnjv c]t 9eo;o
0v’c4;mulv’d=gru,j 0bho Xs^ns^5f]q’?1,2,...gmjnv g-ajofP;daoD

@D .97;k,

lv’xt]y,koX7n a′,b′,c′.....c]t a,b,c.....?mugj xao<vaf8klj;orq;rao< c,oj lv’
xt]y,komugj ruoj gvqk7jk0v’xt]y,komu|b’j X7n a,b,c,....?76oda[9eo;ovaofP;  k ,
grnjv.shwfh7kj 0v’xt]y,komulv’X7n a′,b′,c′,....?D s^n gvqk7kj 0v’xt]y,komulv’
X a′,b′,c′,....? sko.sh9eo;ofP; k  grnjv.shwfh7kj 0v’xt]y,komu|bj’X a,b,c.....?D
9eo;o k fa’j djk;gvUo;jk <8q;lextlyf0v’vaf8kl;j orq;rao<

213

\kpgsfD

xt]y,ko a′,b′,c′.....gxaovaf8klj;orq;raoda[ a,b,c..... d+8+g,njv;kj G

a′ = b′ = c′ = k
a b c

8q;1kj ’G

!D ]k7k0v’l=fec,jogxaovaflj;oda[9eo;o0v’l=feD .sh8jn,.lj8k8t]k’8+wxoU

c]t .sh-vd sk8q;lextlyf0v’vaf8kl;j orq;raofaj’dkj ;D

9eo;o0v’l=fe 62 7 .........
......... 12 000
÷..... XglAo? ×.....
]k7k
Xdu[? 4800 .........

ສະຫງສ.ວວນ.ລິສຂະສິດ
@D 7qo’kozh6|b’j wfhIa[g[hp]hP’336000 8+ 14,ND 4k,;jk ]k;9twfhIa[g[hp]hP’gmqjk
.f4hk]k;gIaf;PdIvf 21,N c]t grnjvIvf 30,N gf L
X0=hcotoe.shlhk’8k8t]k’vaf8kl;j orq;rao c]t -vdsk8q;lextlyf0v’vaf
8klj;orq;rao ?

9eo;o,N 14 21 30 ×.....
336 000 ......... .........
g[hp]hP’muwj fhIa[
Xdu[?

#D .odj5,0v’oadIPocs’j |b’j F wfhc]dxjPodao]ts;jk’g0qhk|q, c]t \kd[u8k,
g’njvow0faj’8k8t]k’]5,j oUG
g0qhk|q, Xdhvo? 6 18 15 3 24
\kd[u X|j;p? 4 12 10 2 16

9eo;o\kd[uda[9eo;og0qhk|q, c,oj gxaovaf8kl;j orq;raodao s^n [+ L 4hkc,joF
.sh-vdsk8q;lextlyf0v’vaf8klj;orq;raoD

214

ສະຫງສ.ວວນ.ລິສຂະສິດ8q;1jk’ Xdkocdhg]d39f0v’vaf8klj;orq;rao?D
!D dko7yfw]3j fpoe.-hdko7yfw]sq;|j;pdvj oG
czoj cr 20m ,u]k7k100000 du[D 4hkczoj cr 7 m 9t,u]k7kgmqjk.f L
 -vdsklv’xt]y,komudj jP;0hv’ .o[qf39foUc,oj G
{ 7;k,pk;0v’czjocrS ]k7kczoj crD
 I6h;jk lv’xt]y,kogxaovaf8klj;orq;raodao L
{ ]k7kczjocrgxaovaf8klj;orq;raoda[];’pk;0v’czjocrD
 -vdskczoj cr1m 9tgxao]k7kgmqjk.f L
{ ]k7kczjocr1m c,oj 100000 ÷ 20 = ................... du[D
 -vdsk]k7kczoj cr7m D
{ ]k7kczoj cr 7 m c,jo 5000×.............= ..............du[
@D dko7yfw]3j fpoe.-hdko[;d c]t dko76oG
]qf7ao|b’j c]oj X3fp.-h7;k,w;ltg\u?wfh165km F3fp.-hg;]k3-qj;3,’D 4k,;jk
4hk.-hg;]k 9 -qj;3,’ s^n 12 -q;j 3,’ ]qf7aofaj’dkj ;9tc]oj wfh9ad km L
 -vdskxt]y,komudj Pj ;0hv’.o[qf39foUc,oj G
{ w]ptmk’wxwfhFg;]kD
 lv’xt]y,kofa’j djk;gxaovaf8klj;orq;raodaoD
{ w]ptmk’ c]t g;]kgxaovaf8kl;j orq;raodaogrkt]qfc]oj fh;p7;k,w;lt

g\uD
 la’gdfg[y’j dkorq;rao]ts;kj ’g;]kmu.j -h.o[qfg]dD
{ 9 -qj;3,’c,jo DDDDDDDDDDDDDDDDDDgmnjv 0v’3-qj;3,’D fa’j oAo ]qf7aofa’j djk;.-hg;]k

9 -q;j 3,’9tc]oj wfh165km × ............... = .................-b’j gxaow]ptmk’DDDDDDDDDD
DDDDDDDgmnjv 0v’w]ptmk’muj.-h.og;]k 3 -q;j 3,’D
{ 12 -qj;3,’ c,jozqo[;d0v’9-qj;3,’ + ...-qj;3,’D faj’oAo ]qf7ao fa’j dkj ;9t

c]jowfh 495km + 165km = ...

0=h7;ogvqk.9.lDj

dkooe.-h8k8t]k’grjnv7yfw]j
 .shla’gdfg[yj’8k8t]k’]5,j oU mugj vqk0=h,6o,k9kd8q;1kj ’ 2 0hk’gmy’D

215

8k8t]k’ ! × ..... 8k8t]k’ @ +

g;]kX-q;j 3,’? 3 9 g;]k 3 9 .......
w]ptmk’ ( km ) 165 ....... w]ptmk’ 165 ....... .......

× ..... +

8k8t]k’ # −

g;]kX-qj;3,’ 12 9 .......
w]ptmk’( km ) ....... ....... .......

 9kd8k8t]k’0hk’gmy’gIqkwfhG −

• 165 = 165 × 3 s^n 165 = 495 X9kd8k8t]k' !?
3 3×3 3 9 X9kd8k8t]k' @?
X9kd8k8t]k' #?
• 165 = 495 = 165 + 495 s^n 165 = 495 = 660ສະຫງສ.ວວນ.ລິສຂະສິດ
3 9 3 + 9 3 9 12

• 660 = 495 = 660 − 495 s^n 660 = 495 = 165
12 9 12 −9 12 9 3

 3fpmq;j wxD

a = c = a+c = a−c ; b,d ≠ 0
b d b+d b−d

#D dko7yfw]3fpoe.-h8q;lextlyfD
16jIhko7hk0kpxN,csj’|bj’ gryjo0kpxN,c[[IPog]d.o]k7k 4500 du[ 8+|bj’sq;D
.sh8jn,.lj8k8t]k’]5j,oUG

9eo;oxN, 1 5 ......... ×.....
Xsq;? 4 500 31500 45 000

]k7k
Xdu[?

216

 -vdskxt]y,komujdjP;0hv’.o[qf39foUc,joG
{ 9eo;oxN,F ]k7kD
 lv’xt]y,kofaj’djk;gxaovaf8klj;orq;rao -b’j 9eo;o8kj ’M.oc4;mulv’wfh
,k9kd9eo;o8kj ’M c4;mu|b’j 76oda[DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD c]t 9eo;ofaj’djk;
gvUo ;kj lextlyf 0v’vaf8kl;j orq;raoD
 faj’oAo xN, 5 sq; 9b’j ,u]k7k 5 × ................. = ....................
{ gIqkgsao;kj ............. ×4500 = 45000 faj’oAo lk,kf-NxN,wfhDDDDDDDDDDDDDDDDDsq;

#D [qfg/ydsaf

!D dD mhk; [5ol;o 8hv’dkox6dk3ishv’oE0v’]k;F ]k;7yfw]j;jk 9tg]nvd-Ndk
3imuj,u]k7k 25000 du[8+1m2D dvj o]k;9tla’j -NF ]k;wfhlbdlkg[y’j g’yomuj]k;8hv’
9jkp8k,9eo;odk3imu]j k;8hv’dkoF .shoadIPo7yfw]-j j;p]k; 3fp8jn,8q;g]d.lj
shv’8k8t]k’musj ;jk’]j5,oUG
8k8t]k’ !D
ສະຫງສ.ວວນ.ລິສຂະສິດ
9eo;o;af45 1 10 20 25 35 40 45

X m2 ?

9eo;og’yo

muj8hv’9jkp 25000 ...... ...... ...... ...... ...... ......

Xdu[?

0D phvo;jk mhk; [5ol;o gxao]6d7hkmujfuF zh60kp.fhltg\u0kp.sh]k;faj’oUG
 4hk]k;-Ndk3ic8j1m2 sk 20m2 ]k;9twfh9jkp 25000 du[ 8+ 1m2
 4hk]k;-N9eo;odk3ic8j 21m2 sk 40m2 ]k;9twfh9jkp 20000 du[ 8+

1m2 D
 4hk]k;-N9eo;odk3is^kpd;kj 40m2 ]k;9twfh9jkp15000 du[ 8+ 1m2 D
.shoadIPo7yfw]-j ;j p]k;vud7A’|b’j 3fp8nj,8q;g]d.ljshv’8k8t]k’ muj,ug,af9E
]j5,oUG

217

8k8t]k’ @D

9eo;odk3iX m2 ? 1 10 20 25 35 40 45

9eo;og’yo 25000 ........ ........ 600000 ........ 900000 ........
muj8hv’9jkp Xdu[? 25000 25000 25000 24000 ........ 25000 ........

]k7k8+X1m2 ?

7D .shoadIPola’gdfg[yj’8k8t]k’! 0hk’gmy’9trq[;kj G

 .oc8]j t4aoF9eo;o0v’c4;mulv’wfh,kfh;p DDDDDDDDDDDDDDDDDDDDD 9eo;o0v’

c4; DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD da[9eo;o7q’7kj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

Â.oc8]j t4aoF9eo;o0v’c4;mu|bj’wfh,kfh;pDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD9eo;o

0v’c4; DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD .sh9eo;o7q’7kj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

 8q;lextlyf0v’vaf8klj;orq;rao.o8k8t]k’! c,oj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
ສະຫງສ.ວວນ.ລິສຂະສິດ
’D .shoadIPo la’gdfg[yj’8k8t]k’ @ 0hk’gmy’F dko7yfw]jc,jo7hkp7ndaoda[8k8t

]k’ ! s^n [+ L phvospa’ LDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

@D I6[lujc99t85]afF-b’j 0hk’cmdwfh5cm D ];’Iv[0v’I6[c,oj 5cm×... = ...cm

c]t goNvmucj ,jo 5cm×... = ...cm2 D

dD.shoadIPo7yfw]sj k];’Iv[ c]t goNvmu0j v’I6[9t85]affaj’[vd.o8k8t]k’]5j,

oUG

8k8t]k’ #D

0hk’X cm ? 23 4 576 1 0,5

];’Iv[X cm ? 8 ...... ...... ...... ...... ...... ...... ......

gonhvmju X cm2 ? 4 ...... ...... ...... ...... ...... ...... ......
 9q’j la’gdfg[y’j .oc8]j t4aoF9eo;o0v’c4;mulv’wfh,kfh;pDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

9eo;o0v’c4;mu|b’j da[DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

 s^n 9eo;o8kj ’M 0v’c4;mu|bj’wfh,kfh;p DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 9eo;o

8kj ’M 0v’c4;mulv’.shDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

 .oc8]j t4ao9eo;o0v’c4;mulk, c,jowfh,k7hkp7ndko7yfw]0j v’c4;mulv’

218

s^n [+ L phvospa’ L DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

#D .shoadIPo8jn,.lj8k8t]k’ 4 sk7 ]5,j oU grjnv.shlv’9eo;o s^n Xlv’xt]y,ko?

gxao vaf8kl;j orq;raoD

8k8t]k’ $D 8k8t]k’ &D

16 8 5 0,3 ×9 3 .... .... 4 ×....
.... .... .... .... .... 0 15 6

8k8t]k’ *D 8k8t]k’ (D

0, 2 1000 .... 0,003 ×.... 2 0,1 .... ....

×0,1

.... 2000 40 .... .... .... 2 0,1

$D 8k8t]k’8 Ivf11 c,jogxaovaf8kl;j orq;rao s^n [+ L 4hkc,jo.sh[vdlext
ສະຫງສ.ວວນ.ລິສຂະສິດ
lyf0v’vaf8klj;orq;rao0v’c8]j t8k8t]k’]5,j oU Xrhv,ma’vtmy[kp?G

8k8t]k’ )D 8k8t]k’ _D

5 0,5 0 7 1234

35 3,5 0 49 3 6 9 14

8k8t]k’ !WD 8k8t]k’ !!D

10 20 100 5 2345

5 10 50 2,5 3456

&D 8k8t]k’]5j,oU c,oj dPj ;da[D w]ptmk’mu]j qfc]oj 8k,g;]kD

mn 3 6 7 ×....
km 4,5 9 10,5

219

ສະຫງສ.ວວນ.ລິສຂະສິດdD w]ptmk’mu]j qfc]oj gxaovaf8klj;orq;raoda[g;]k s^n [+ L
0D .sh-vdsk7;k,w;0v’]qf7aofa’j dkj ;gxaosq;|;j p km mn s^n km h
*D [=]ylafmvj ’mPj ;csj’|b’j wfh;k’]k7kdkomjv’mPj ;gxaod5,j faj’oUG
 4hkmPj ;16jxtgmf]k; 11;ao9twfh9jkp 1045000 du[8+ 1 7qo
 4hkmPj ;1j6,Po,k 18;ao9twfh9jkp 1940000 du[8+ 1 7qo
 4hkmPj ;1j6s;Pfok, 16;ao9twfh9kj p 1360000 du[8+ 1 7qo
 4hkmjP;1j6dex6g9p 21;ao9twfh9jkp 1785000 du[8+ 1 7qo
dD dko.-h9kj p.odkomvj ’mjP;gxaovaf8kl;j orq;raoda[g;]kmjv’mPj ;s^n[+ L

X.shd;f7no?D
0D gryoj gsao;kj dko.-h9kj p dkomjv’mjP;1j6,Po,k,u]k7kl6’s^kpD gryjo9yj’wfh8af
]kpdkoouhvvd c]t wfhxa[dko.-h9kj pdkomvj ’mjP; 11;aomuxj tgmf]k; 3fp
.shdko.-h9kj p.odkomjv’mPj ; gxaovaf8kl;j orq;raoda[g;]kmvj ’mjP; 9tgxao
g’yogmqkj .f le]a[dkomvj ’mjP; 11;ao .oxtgmf]k; L X0=hcotoeG 7e8v[
0=h d s^n 0=h 0 7;ogIaf8k8t]k’]ts;kj ’dko.-h9kj pc]tg;]kmjv’mPj ;?D
(D ]6’,u8hv’dkox6dk3i shv’ovolv’shv’ shv’mu|bj’gxaoI6[lucj 9lkd ];’pk;
cmdwfh 4,50m c]t];’d;hk’3m c]tshv’mulv’gxaoI6[9t85]af mu,j u0hk’cmd
wfh 5m D grjnvx6dk3ishv’mu|b’j ]k;8hv’wfh.-hg’yo-Ndk3i 675000 du[D

dD 4k,;jk ]k7k-Ndk3i8+1m2 gxao]k7kgmqkj .f L
0D ]k;9t8hv’.-h9kj pg’yoma’|qfgmqjk.f grnjvx6dk3ilv’shv’fa’j dkj ; L
)D mhk; vvHf c]t ok’ 8yd lv’vhkpohv’,uvkp5 12xu c]t 7 xu 8k,]e fa[D
c,jg0qk g9qhkma’lv’wfhgvqkg’yo.sh9eo;o 10545du[ grnjvc[j’xaodaozh6vhkpwfh
g;qhk;jkG r;d gIqk7;oc[’j xaog’yo9eo;ooU8k,vaf8kl;j oda[vkp5c8]j t7qoD
4k,;kj c8j]t7qo 9twfhg’yozh6]tgmqkj .f L
_D lk,lPj ;wfh]q’mnogIafm5]tdyfohvpvao|b’j zh6mu19kj p 25000 du[F zh6mu 2 9kj p
15000 du[ c]t zh6mu39jkp10000 du[Ds^a’9kdoAo |bj’xug0qkg9qhkwfhdew] 20000
du[D ma’ lk, 7qowfh8qd]q’c[j’xaodew]faj’dkj ; 8k,vaf8kl;j orq;raoda[9eo;o
g’yoc8j]t7qomuwj fh]q’mnowxc8j8QoD 4k,;kj c8]j t7qo9twfhdew]gmqjk.f L
!WD r;dgryjowfhoe.-h[ks^vfcmdvj5ostr6,gxaolv’c[[F c[[mu|bj’ c,josq;|j;p
gxaovq’lkg-aolyf Xcelsius : c ?vudc[[|b’j c,oj sq;|;j pgxaovq’lk2kgiaowI

220

X Fahrenheit : F ? ma’lv’c[[wfh\kp0ufgxaovq’lkfa’j I6[]5j,oUG

100Dc oE2qf 212DF

x 104D F

0Dc oEdhk, 32D F

dD 4hk;jkvj5ostr6,sq;|;j p2kgiaowIXDF ?]q[32 vvd zqoIa[9twfhgxaovaf8kl;j o
rq;raoda[sq;\j;pg-aolyfXDc ?.sh8jn,.lj[jvo9Eg,af0v’8k8t]k’]5,j oU

vj5ostr6,  Dc 0 ....... 100 Dc ສະຫງສ.ວວນ.ລິສຂະສິດ
×....
( )vj5ostr,6 D F −32
....... 72 180 .......

0D .sh0Pol6fdkorq;raodao]ts;kj ’sq;|;j p DF da[ Dc
!!D Ih6;jk];’Iv[X P ?0v’glAo;q’,qoc,oj zqo76o0v’glahozjk.9dk’ X D ? c]t
9eo;o π D Ih6;jk glahozkj .9dk’0v’glaho;q’,ao,u DDDDD gmjnv0v’glaho]aflt\u X R ?D
dD .sh8jn,.l[j vj o9Eg,af0v’8k8t]k’]5j,oU.sh4nd8hv’D

glAo]aflt\u R ( cm )  2, 5 3 79 ×....   ×....
glAozjk.9dk’ D ( cm ) ....... ....... ....... .......

];’Iv[ P ( cm ) ....... ....... ....... ....... ×....  

0D ];’Iv[X P ? gxaovaf8klj;orq;raoda[glAo]aflt\uX R ? s^n [+ L DDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
c]t 8q;lextlyf0v’vaf8klj;orq;raoc,joDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

221

!@D Ihkovaflegoqkgvdtlkocsj’|bj’wfh7yfw]7j kj vafgvdtlkofaj’oUG
 1|hksk 200 |hk 7yfw]7j jkvaf110du[ 8+1|hkD
 201|hksk300|hk 7yfw]7j kj vafrP’c8j80 du[ 8+1|hkD
dD .shlhk’8k8t]k’ dko7yfw]jsk7jkvafgvdtlko8k,9eo;o|hk]5,j oUG 15|h
30|hkF 45 |hkF 75|hkF90|hkF180|hkD
 ]k7kc,jogxaovaf8klj;orq;rao8+9eo;o\hk s^n [+ L DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

c]t 8q;lextlyfc,jo DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
0D .shlhk’8k8t]k’dko7yfw]jsk7kj vafgvdtlko8k,9eo;o|hk]5,j oUD

30 |hkF 75|hkF180|hkF 201|hkF 225 |hkD
 ]k7kc,jogxaovaf8kl;j orq;rao8+9eo;o\hk s^n [+L phvospa’ L
!#D .sh-vdsk7kj 0v’ x c]t y 8k,8k8t]k’]5j,oU gxaolv’;ymuG

9eo;o\kddhP’  2 459
X kg ?
ສະຫງສ.ວວນ.ລິສຂະສິດ
]k7k 3600 x 9000 y
Xdu[?

dD fh;p;ymu-vdsk8q;lextlyf0v’vaf8kl;j orq;rao L
0D fh;p;ymu dko[;dF dko76oD

222

[qfmu @_ -vdskrqfmu[j +maoI6h0v’vaf8klj;orq;rao

!D dyf9tde

dyf9tdemu !

,;olko0v’cxh’ 15 22
( kg )

,;olko0v’g0qhk9ju 21 x
( kg )

{ ,;olko0v’g0qhk9uj gxaovaf8kl;j orq;rao8+,;olko0v’g0qhkcxh’D gIqklk,kf

0Poltg\uzqo0v’zqoskofa’j oUG 21 = x D 21, 15, x c]t 22 gvUo;kj rqf 0v’
15 22
lv’vaf8klj;orq;raoD
ສະຫງສ.ວວນ.ລິສຂະສິດ
{ -vdsk x rqfmulugj ,jnvIh6lk,rqfD
15 kg 0v’g0qhkcxh’lk,kf.shg0qhk9ujwfh DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

1 kg 0v’g0qhkcxh’lk,kf.shg0qhk9uwj fh 21÷15 s^n 21 kg = ...................
15

22 kg 0v’g0qhkcxh’lk,kf.shg0qhk9uwj fh 22 × 21 = ( 22 × 21 ) ÷ 15 = ............
15

21 = ..... gIqkwfh 22 × 21 = ............. ⎫ lk, kf0PoG 22 × 21 = ............×15
15 22 .....× 15 = ............. ⎬
⎭

{ 0h=la’gdf 21 = ..... mP[gmqjk 21 × 22 = 15 × ...............zqo76ow0;j gmqjkdaoD
15 22

dyf9tdemu @

gryoj mqf]v’x6dg0qhk.ljok8vo|b’j ,ugoNvmuj 3500m2 lk,kfzt]yfg0qhkwfh 2,45 t D

.og’njvow0vaofP;dao gryjowfhmqf]v’x6d.ljok8vomuj.sjpdj;k g-yj’ ,ugoNv muj 1,5ha D

4k, ;kj zqo zt]yf9twfh9ad38o L

223

8k8t]k’

goNvmuoj k • xPj o 2, 45t = .....................kg

( )m2 3500 15000 • xPj o1,5ha = ........................m2

zqozt]yf x ×... • 8q;lextlyf0v’vaf8klj;orq;rao]t
( kg ) 2450 s;kj 'zqozt]yfda[goNvmujokc,jo9e
o;omu7j 6oda[3500 c]h; gmqjkda[ 2450  

8q;lextlyffaj’djk;c,jo 2 450 = ...........................
3500

x = 15000 × ........................ s^n x = 15000 × 2 450 = ........................
3500

okgoNvmu1,5ha lk,kfzt]yfg0qhkwfh ...................kg s^n .......................Τ

0=hla’gdf 2 450 = 15 x mP[gmqkj zqo76ow0;j 3500 × x = ...........×15000
3500 000
ສະຫງສ.ວວນ.ລິສຂະສິດ
x = 15 000 ×........... = ..............
...............

dyf9tdemu #

]k7k-Uo\6 600g gxaog’yo 15000 du[F 4hk8hv’dko-N-Uo\6900g F 9t8hv’9jkp

g’yogmqkj .fL

9q’j -vdskrqfmulujg,nvj Ih6lk,rqf ,;olko0v’-Uo(g) 600 900
djvo g’yomuj8hv’9jkp(k) 15000 x

;ymucdhmu ! fh;pdkoskoc]tdko76o fh;pdko[;d

÷2 × 3 +

600 ....... 900 600 ....... 900

15000 ....... x 15000 ....... x

÷2 × 3  
+ 

224

;ymucdhmu @
3fpdko-vdsklextlyf0v’vaf8kl;j orq;raoD

600 900 • lextlyf0v’vaf8kl;j orq;rao
×.....
c,oj 15 000 : 600 = 15 000 = ...
600
15000 x

• 900× 15000 = 900 × ... = ... ltoaho]k7k-Uo\6900g gxao9eo;o
600
g’yoDDDDDDDDDDDDDDDDDDDDD la’gdfgsao;kj G 600 × ... = 15000 × 900 zqo76ow0j; gmqkj daoD

;ymucdhmu #
fh;pzqo76ow0;j gmqkj dao

600 × x = 900 × 15000

x ( 900 ) 600 15 000 ສະຫງສ.ວວນ.ລິສຂະສິດ
= × 15 000 ÷ = 900 × 600 = ...................................

]k7k-U\6 900g c,oj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

@D .97;k,

-vdskrqfmulu0j v’vaf8kl;j orq;raoF c,jodko7yfw]j-vdsk9eo;omu[j +maoIh6.o8k
8t]k’0v’vaf8kl;j orq;raomudj eoqffh;plu9j eo;o Xg-y’j I6hlk,9eo;odvj oc]h;?D
8q;1jk’G

12 20 • -vdsk7kj 0v’ x 3fp.-hzqo76ow0j;gmqjkdaoD
36 x
• 36 = x gIqkwfh 1 2 × x = 3 6 × 2 0
12 20

• 12 x = 36× 20

• x = ( 36 ÷ 12 ) × 20

• x = 3× 20 = 60

rqfmuluj x 0v’vaf8klj;orq;raoc,oj 60 D

225

3fpmqj;wxG

gryjo.shlv’vaf8kl;j omugj xaovaf8kl;j orq;rao a = c mP[gmqkj da[ a×d = b×c
b d
Xzqo76ow0;jgmqjkdao? a,b,c,d gvUo;jkG rqf0v’vaf8klj;orq;rao c]t b,d ≠ 0

#D [qfg/ydsaf

!D .sh-vdskrqfmuluj .o8k8t]k’0v’vaf8klj;orq;rao]5,j oU 3fpoe.-hzqo76ow0;j D
dD

35 21 24 y 152 95

25 x 32 64 z 55

0D 19 z 56 34 y
x 72 84 136 6
12 ສະຫງສ.ວວນ.ລິສຂະສິດ
72

@D .sh-vdsk7jk x, y .o8k8t]k’vaf8kl;j orq;rao]5,j oUD

3 x 78
5 20 y

dD grnjv-vdsk7jk x, y .shcpd8k8t]k’gmy’oUgxaolv’8k8t]k’ 9kdoAo .sh
8jn,9eo;o.lj[vj os;kj ’]5j,oUD

x y

0D c]h;.sh7yfw]-j vdsk7kj 0v’ x c]t y L

226

#D grjnvc8h,czoza’0v’gInvoFgryjo8hv’sp=hl;j o0t|kf0v’gInvo]q’F0t|kfsp=h

l;j o 0v’gInvo gxaovaf8kl;j orq;raoda[ 0t|kf8q;9y’ 0v’gInvoD

dD lhk’8k8t]k’ c]h;.sh-vdsk7jk0v’ a,b .o8k8t]k’]5j,oU X3fpoe.-h

zqo76ow0;j ?D

0t|kf8q;9y’ 2 400 1600 a
6
÷x s^n × 1 ( cm ) ×x
x
0t|kflj;osp=h
12 b
( cm )

gIqk,u 12 = b mP[gmqkj da[ 2 400×b =............×............
2 400 1600

b = 12×.......... = ..................... gIqkwfh7jk b =........................
2 400

c8j 12 = 6 mP[gmqjk 12×a = 2 400×........................
2 400 a
ສະຫງສ.ວວນ.ລິສຂະສິດ
a = 2 400×.......... = ..................... gIqkwfh7kj a = .....................
12

0D -vdsk8q;lextlyf x 0v’vaf8kl;j orq;raoF x 76oda[9eo;o.oc4;mu|b’j
grnjv.shwfh DDDDDDDDDDDDDDDDDDDDDDDDDDDDDF x gxaog]dlj;o g-y’j c,oj ,kf8kl;j ospa’ L

÷

2 400× x = ..................... gIqkwfh x = ......... = 1
2 400 .........
÷

7D -vdsk,kf8kl;j osp=hc,oj G -vdsk 8q;lextlyf0v’vaf8klj;orq;rao0v’0t
|kfsp=hl;j oda[0t|kf8q;9y’

,kf8kl;j osp=h = DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

0t | kf8q;9y'

’D l6f7yfw]-j vdsk0t|kflj;osp=hc,oj G
0t|kfl;j osp=h = ,kf8kl;j osp=h×............ 

227

9D zqosko( 12÷ x ) c,oj 2400 s^n 12 = 2 400
x

12×1 = 2 400 s^n 12× 1 = 2 400 X 1 c,jo9eo;oxUo0v’ x s^n x c,jo9eo;oxUo
x x x

0v’ 1 ?D
x

l6f7yfw]-j vdsk0t|kf8q;9y’c,oj G
0t|kf8q;9y’ =X9eo;oxUo0v’,kf8kl;j o? ×DDDDDDDDDDDDDDDDDDDDDDDDDD

$D .o8k8t]k’8+wxoUF gryoj .sh9eo;o0v’c4;mulv’ c,oj gxaovaf8klj;oda[9e
o;o0v’c4;mu|bj'D

dD .sh8jn,.l[j vj os;jk’0v’8k8t]k’]j5,oU g-y’j dko7yfw]fj h;p ;ymudko76ogmqkj oAo
c]t .sh[vd8q;lextlyfD

3 9 18 54 24 27 48 99 0, 6
×.....

2 ...... ...... ...... ...... ...... ...... ...... ......
ສະຫງສ.ວວນ.ລິສຂະສິດ
0D .sh8jn,.l[j vj os;kj ’0v’8k8t]k’]5,j oU g-yj’dko7yfw]fj h;p;ymudko[;d s^n dko
]q[ gmqjkoaho c]t .sh-vd8q;lextlyfD

0,5 2 3 5 7 8 10 9,5 4,5
×.....

3 12 18 ...... ...... ...... ...... ...... ......

&D ]qf3fplko7ao|b’j c]jowfh 260km F \qfoE,aocvaf-a’13A D
dD 4hk]qf7aofja’dkj ;9tc]oj .shwfh550km F 9t.-hoE,aocvaf-a’\qf9ad]yf L
0D ]qf7aofaj’djk;9tc]oj wfh9ad km 4hk;kj oE,aocvaf-a’\qf 85A L
X0=hcotoeG .shlhk’8k8t]k’F .sh7yfw]jgxaolk,;ymu7nG fh;pdko76oF fh;p8q;lext
lyf c]t fh;pzqo76ow0j; ?D

228

*D lv’glaholtcf’mugj xaolv’glaho-nj w]ptmk’Xkm? 6
d1c]t d2 D c8j]tglaho-njgxao8q;-U
[vdg4y’w]ptmk’mucj ]jo0v’]qf
lv’7ao3fpvu’.ljg;]kmjuc]jo

5 d 2 d1
4

3

2

1

0 1 2 3 4 g;]k (mn)

dD djkp7no9eo;o9kdI6[ltcf’F c]h;8nj,.l8j k8t]k’]5,j oUG

8k8t]k’ d1 8k8t]k’ d2
g;]k   g;]k
(mn) 0 1 2 3 (mn) 0 1 2 3
ສະຫງສ.ວວນ.ລິສຂະສິດ
w]mk’   w]mk’
(km) .... 2 .... 6
(km) 0 1,5 .... ....

0D .shd;f7noF 8k8t]k’ d1c]t8k8t]k’ d2 ;kj gxaovaf8kl;j orq;rao 9q’j -vd
sk8q;lextlyf0v’vaf8kl;j orq;rao0v’r;d,aoD

7D oadIPorq[lextlyf0v’vaf8kl;j orq;rao0hk’gmy’ 3fpla’gdf9kd glaholt

cf’16j95f.f L

(D .sh-vdsk a,b,c 0v’8k8t]k’8+wxoUF grnvj .shgxao8k8t]k’0v’vaf8klj;orq;

rao 0v’9eo;o.oc4;mulv’da[9eo;o0v’c4;muobj’D .sh-vdsk7kj x, y.

 x c,jolextlyf0v’vaflj;orq;

rao0v’9eo;oc4;mulv’da[

9eo;oc4;mu|bj’D × y 7 91 21 c × x
4 a b 51
7× x = 4 ; x = 4
......
 y  c,oj lextlyf0v’vaf8kl;j orq;rao 0v’9eo;oc4;mu|b’j da[9eo;o0v’

229

c4;mulv’

4× y = 7 ; y = .......
4

ltoAo a = 91× 4 = .......................
7

b = 21×.......................

c = 51×.......................

)D .shoadIPovtmy[kpc8]j t8k8t]k’]5,j oU ;jkc,jo8k8t]k’vaf8klj;orq;rao
s^n[+ L

dD fh;p;ymu7yfw]zj qo76ow0;j D
0D fh;p;ymuvjno|b’j ;ymuD

9, 6 12 14 16 18 24 12 20
12 15 13 15 12 16 75 125
ສະຫງສ.ວວນ.ລິສຂະສິດ
_D .shoadIPola’gdfg[yj’I6[lv’I6[]5j,oUG

I6[X@?

B

I6[X!?

B

A C A F C
E D E G D

F
G

dD .shoadIPo.-hw,h[aomafcmd];’pk;0v’mjvo-nj [AB]; [AC] ; [BC] ;  [CD] ;

230

               [EF] ; [FG] 0v’I6[X!?c]tI6[X@? c]t8n,j .l[j vj os;jk’0v’8k8t]k’]5,j oUD

AB AC BC CD EF FG

];’pk;.oI6[ ! (cm)
];’pk;.oI6[ @ (cm)

0D oadIPola’gdfgsaoco;.fF le]a[8k8t]k’0hk’gmy’ L
7D .shoadIPo.-h[aomafcmd,5,grjnvcmd,6,8jk’M 0v’I6[ X!? c]tI6[ X@? oadIPo

la’gdfgsaoco;.f L
’D I6[ X@? c,oj I6[0tspkp9kdI6[ X!? c,jo s^n [+ L 4hkc,oj .sh[vd,kf8klj;o

0tspkpD
9D I6[ X!? c,oj I6[sp=h9kdI6[ X@? c,jo s^n [+ L 4hkc,jo.sh[vd,kf8kl;j osp=hD

!WD 8k8t]k’]j5,oU c,oj dko-Uogxnv’oE,aocvaf-a’ 8+w]ptmk’muj]qf7ao|bj’c]oj
wfh3fp.-h7;k,w;ltg\u7n 90km h D
ສະຫງສ.ວວນ.ລິສຂະສິດ
w]ptmk’ 100 250 300 450

×y ( km ) ×x
oE,aocvaf-a’
(A) 8 20 24 36

dD .shd;f7no;kj G 8 = 20 = 24 = 36 = ...........................
100 250 300 450

9eo;ooE,aocvaf-a’gxaovaf8kl;j orq;raoda[w]ptmk’ s^n [+ L .sh-vdsk7jk

0v’ x  8q;lextlyf0v’vaf8klj;orq;raoD 7jk0v’ x  ,u7;k,\kpspa’ L

0D .shd;f7no;jkG 100 = 250 = 300 = 450 = ...........................
8 20 24 36

w]ptmk’mucj ]joc,oj gxaovaf8kl;j orq;rao8+oE,aocvaf-a’muj\qfwxs^n[+ L

.sh-vdsk7jk0v’ y . 7kj 0v’ y ,u7;k,\kpspa’ L
!!D ,;olko0v’3]stmugj xaomv’ c,jogxaovaf8klj;orq;raoda[[=]y,kf0v’,aoD

231

[=]y,kf0v’mv’ 8 12 17 50

×y ( cm3) ×x

,;olko0v’ 447

mv’ ( g)

dD 7yfw]7j kj 0v’ x , lextlyf0v’vafl;j orq;rao0v’,;olkomv’da[[=]y,kf0v’
,aoD 7kj 0v’ x  ,u7;k,\kpspa’ L -vd8nj,9eo;o.l[j vj os;jk’3fpoe.-hzqo
76ow0;jD

0D 7yfw]j7kj 0v’ y lextlyf0v’vaf8klj;orq;rao 0v’[=]y,kfmv’8+,;olko0v’
mv’D 7jk0v’ y  ,u7;k,\kpspa’ L

!@D .sh-vdsk7jk0v’ x  ]5j,oU g,jnvIh6;kj lv’vaf8kl;j o]5,j oUgxaovaf8kl;j orq;
raodaoD

dD x = 8 0D 108÷9 = 42 ÷ xສະຫງສ.ວວນ.ລິສຂະສິດ
15 24

7D 36 = 54         ’D 8÷12 = x ÷3
x 3

9D 75 = x lD x ÷54 = 12 ÷36
35 14

-D 156 = 26         pD 180 ÷ x = 8 ÷5  
288 x

232

[qfmu #W dko7yfw]lj j;oIhvp

!D dyf9tde

dyf9tdemu !

g9hplts^kdmuj8yf16jdhvogoupc0’ -U[vdw0,ao 45 : D 9q’j -vdsk,;olko0v’w0,ao

.ogoupc0’ 280gF c]h;0Po8n,j .lj[vj o9Eg,afD

\kp7;k,;jkG

Ð .o 100g 0v’goupc0’,uw0,ao16j 45g

Ð .o 1g 0v’goupc0’,uw0,ao16j 45 ÷ ............ = 45 = ............
......

Ð .o 280g 0v’goupc0’,uw0,ao1j6 280× 45 = 280 × 45 = ............
......... ............
\kpgsfG

45 : vjko;jk 45 gxug-ao ສະຫງສ.ວວນ.ລິສຂະສິດ

45 : 0Pogxaog]dlj;oc,oj 45
100
dyf9tdemu @

Ihko0kpg7jnv’o5’j sqj,csj’|b’j wfhxtdkfs^5f]k7k0kpg7jnv’o5’j 8kj ’M ]q’15 :

dD 9qj’djkp7nolhk’8k8t]k’ c]t 8jn,.lj[jvo9Eg,afD

3lh’ glNv dt3x’j glNvdao|k;

]k7kgfu, 24000 32000 28000 41000

Xdu[? ......... ×.....
l;j os^5f15 : ×.....

Xdu[? 3600 ......... .........

]k7kmujs^5fc]h; .........
Xdu[? 20 400 ......... .........

0D lj;os^5f0v’c8]j t]kpdko c,oj gxaovaf8kl;j orq;raoda[]k7kgfu,[+L 9qj’vt

my[kp c]t -vdsk8q;lextlyf rhv,ma’0Pogxaog]dl;j oIhvpD

7D ]k7kmujs^5fc]h; c,oj gxaovaf8kl;j orq;raoda[]k7kgfu,[+L 9q’j vtmy[kp c]t

-vdsk8q;lextlyf rhv,ma’0Pogxaog]dlj;oIhvpD

233

@D .97;k,

grnjv7yfw]j x: 0v’9eo;o|bj’F gryoj 76o9eo;ooAoda[ x  
100

8q;1jk’G

7yfw]j30 : 0v’ 250 c,oj 250× 30 = 250×30 = 25×3 = 75
100 100
faj’oAo 30 : 0v’ 250 c,oj 75

#D [qfg/ydsaf

!D 9qj’7yfw]j c]h;0Po8n,j .l[j vj o9Eg,afD

25 : 0v’ 520 kg c,jo 520 × 25 = ........... S
100

 Ð 10 : 0v’ 200 du[ = ................ Ð 30 : 0v’ 100 7qo = ................

Ð 50 : 0v’ 24 7qo = ................ Ð 90 : 0v’ 1000 m = ................
ສະຫງສ.ວວນ.ລິສຂະສິດ
Ð 15 : 0v’ 40 kg = ................ Ð 125 : 0v’ 200g = ................

@D .og]dlj;o8+wxoU c,oj g]dlj;o.fmP[gmqkj da[ 75:G

50 ; 75 ; 7 ; 4 ; 3
200 100 5 3 4

#D 7yfw]jg]dlj;ogxaog]dl;j oIhvpD

9qj’-vdsk[=]y,kf0v’oEmuj16j.oc8]j ts^vfcdh;F 7yfw]j.shgxaog]dlj;oIhvpD

XI6[!? XI6[@? XI6[#?

8q;1jk’G -vdsk[=]y,kf0v’oE.os^vfcdh; XI6[@?D

;ymucdhmu !

oE.os^vfcdh; XI6[@? ,u 1 0v’[=]y,kfs^vfcdh; mP[gmqjkda[ x :D
4

234

ltoAo 1 = x s^n 4× x = 1 × 100 gIqkwfh x = (1×100) ÷ 4 = ...................
4 100

faj’oAo 1 = 25 s^n 1 mP[gmqjkda[ 25 :
4 100 4

;ymucdhmu @

×.....
1 ........ 1 ........ 1
4= 100 s^n 4 = 100 fa’j oaho 4 mP[gmqjkda[ 25 :

  ×.....

;ymu-vdskg]dlj;oIhvp0v’oE.os^vfcdh; XI6[! c]t I6[#? c,jo7yfw]j7hkp7nda[

0hk’gmy’D

$D 7yfw]gj ]dl;j oIhvp.shgxaog]dl;j o c]t .shgxaog]dmqfltoypq,D

9q’j 8nj,.l[j jvo9Eg,af.o8k8t]k’]5,j oUD

g]dlj;o g]dlj;o 9eo;omqf ;ymucdhG ສະຫງສ.ວວນ.ລິສຂະສິດ

Ihvp ltoypq, 35 : c,oj 35
35 : 100
50 : 7 0, 35
125 : 20 .......... ÷5
.......... ..........
35 = 7
.......... ........
100 ÷....

 

fa’j oAo 35 = 7 c]t
100 ........

25 : .......... .......... 35 = 35 ÷100 = 0, 35
100

&D .sh8jn,.lj[vj o9Eg,af8+wxoU.sh4nd8hv’D

dD DDDDDDDDDDDDDDDDDDDDDDDDDD : 0v’350 gmqkj da[ 280,
0D DDDDDDDDDDDDDDDDDDDDDDDDDD : 0v’ 91 gmqjkda[ 9,1,
7D DDDDDDDDDDDDDDDDDDDDDDDDDD : 0v’ 200 gmqjkda[ 600.

235

ສະຫງສ.ວວນ.ລິສຂະສິດ*D xt3spd]5j,oU 4nd s^n zyf c]t 9q’j d;f7no4hkzyfD
dD !W : 0v' $ WWW du[ c,jo ! WWW du[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
0D 7yfw]jsk @& : 9eo;o|b’j c,jo gvqk9eo;ooAo sko.sh $ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
7D 7yfw]jsk ! : 9eo;o|bj’ c,oj gvqk9eo;ooAo sko.sh !WW DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
’D 7yfw]jsk & : 9eo;o|b’j c,jo gvqk9eo;ooAo sko.sh !W c]t zqoIa[sko

.sh @ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
9D 7yfw]jsk _W : 9eo;o|bj’ c,jo gvqk9eo;ooAo 76oda[ WF_ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
lD!WW : 0v' & WWW du[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

(D .shoadIPola’gdfg[y’j 0=h,6o.o8k8t]k’]5j,oU

dyf9tdemujoadIPo9eo;o 40 7qo0v’ ,D! wfhxt8y[af
dy]k0v’3I’IPo s^yhovyog8ugof fqo8u{Ihv’gra’

60 : 30 : 25 :
dD 9qj’-vdsk9eo;ooadIPo muxj t8y[afc8j]tdyf9tdeD
0D 8q;g]d0hk’gmy’X.o0=hd? gxaowxwfh s^n [+ L phvospa’ L

)D
lt4u8ydkolv[4k,7;k,gsao0v’zh6g[y’j 3m]trk[9e
o;o 1 500 7qo wfhzqolts^5[faj’oU
,adg[yj’gxaoxt9e ,adg[y’j gxao7ah’7k; [+,adg[y’j
39 : 51: 10 :

dD 4k,;kj ,u9ad7qo,adg[y’j gxaoxt9eF ,adg[y’j gxao7A’7k; c]t [+,adg[yj’D
0D zqoIa[0hk’gmy’gxaowxwfh s^n [+ L .shgsfzqoD

_D dD g7njv’.-h-toyf|bj’ ,u]k7k.og,jnvdjvo 12000 du[F c8xj t95[ao,u]k7k
10000 du[D |kp7;k,;kj ]k7kwfhs^5f]q’ 20 :F 4nd s^n zyfL 9qj’d;f7noD

0D g7njv’.-h,u]k7k.og,jnvdvj o 9 000 du[F c8jxt95[ao,u]k7k10 000 du[D |kp
7;k,;jk ]k7kwfhgry,j 0Bo10:F 4nd s^n zyf L 9qj’d;f7noD

236

!WD "g0qhk|q,dhvo|b’j [ao95oE8ko 12: " D 9q’j vtmy[kpxt3spdfa’j djk;D
dD 9q’j -vdsk,;olko0v’oE8komuj1j6.og0qhk|q,g-y’j ,uoE|ad 100gF 150gF
200g c]t 145gD
0D grnjv-vdsk,;olko0v’oE8ko mu[j ao9516j.og0qhk|q, g-yj’,uoE|ad 265gF
gryjo8hv’76o 265×.....................

!!D.opk,radc]h’F gryjola’gdfgsao;jkoadIPowfhwxmafltotlbdlkfja’oUG 24 : wx
8k,sh;p|v’7v’[b’F 15: wx8k,r6 c]t -qoot[qf16: wx8k,my;mafxjkw,h
c]t oE8qd8kf 8kj ’MF ovdoAoc,jo[+wfhwxmjP;D
dD 9q’j c8h,7no.\Ij 6[8kdk3i0hk’oU rhv,
ma’mklu8kj ’dao8k,9eo;o X:? 0v’
oadIPomujwxmafltotlbdlkc8j]t[vj oD
0D ,uoadIPo9adgxug-aoX:?mu[j +wfhwxmPj ;D

!@D c8j]tI6[]5j,oUF goNvmuj lufe,u9adgxug-ao X:? 0v’goNvmujma’\qf0v’c8]j tI6[D
ສະຫງສ.ວວນ.ລິສຂະສິດ

XI6[ !? XI6[ @? XI6[ #?

!#D 9qj’c8h,7noI6[lucj 9lkd]5j,oUD 60 mm
dD9qj’-vdsk];’Iv[0v’I6[D

0D9qj’c8h,I6[.\j -b’j ];’pk; 25 mm
0v’c8j]t0hk’,u7;k,pk;

80 : 0v’I6[gfu,c]h;-vdsk

];’Iv[0v’I6[lujc9lkd.\jD

7D 4k,;kj ];’Iv[0v’I6[lujc9lkd.\9j t,u9adgxug-ao0v’];’Iv[gfu,0v’

I6[lucj 9lkd L

237

ສະຫງສ.ວວນ.ລິສຂະສິດ!$D \=hs6’g0qhk|;j p|bj’,u]k7k 200000 du[F 8Qogfnvo,a’dvo]k7kwfhgrY,0Bo10:D
8qhogfnvod5,rk]k7kwfhs^5f]q’ 10: 0v’]k7k.ogfnvo,a’dvoF 8Qogfnvo,uok
]k7k wfhgrY,0Bo10: X3fpvu’.l]j k7k0v’gfnvo d5,rk?D c]t s^5f]q’F
grY,0Boc[[fP;daooU9qoIvfgfnvo,u45okD 4k,;jk 8Qogfnvo,u45ok ]k7k
\=hs6’g0qhk|;j poAo 9tgxaog’yogmqjk.f L
X0=hcotoeG 9q’j lhk’8k8t]k’dkorq;rao]ts;jk’]k7k8qhoF l;j os^5fF l;j o
gruh, c]t ]k7kmusj ^5fc]h; { gryh,c]h;?D

!&D goNvmuj0v’fyo.o3]doU,u 125 000 000km2 D 28 : 0v’goNvmufj yoma’\qf gxao
mtg]-kp c]t50 : 0v’mtg]-kpc,jo8A’1j6mt;u[vk2yiddkD 4k,;kj
goNvmumj tg]-kp 16jmt;u[vk2yiddk,ugmqkj .f L

!*D mhk; ;P’ 1kdwfhdew] 72450 du[F le]a[g7jnv’o5’j muj-N,k.o]k7k345000du[D
dD 9q’j 7yfw]sj kgxug-aodew]3fpvu’.lj]k7k-ND
0D .sh-vdsk]k7k0kp c]t gxug-aodew]3fpvu’.l]j k7k0kpD

!(D oE|ad6511g 0v’3]stg’yo mujxqo16j.oczjo3]stxtlq,]ts;kj ’g’yo c]t
mv’D -b’j oE\ad0v’3]stg’yofa’j djk;,u 68: 0v’oE|ad0v’3]stxtlq,D
4k,;jk 9t8hv’ 8nj,3]stmv’9adg grjnv.shwfhczoj 3]stxtlq,fa’j djk; L

238

[qfmu #! vaf8kl;j orq;raoxUo

!D dyf9tde

dyf9tdemu !
dD 9q’j 0Po8nj,.lj[vj o9Eg,afD
Ð oadIPo37qo Ij;,dao05f|kozad|bj’8vo .shc]h;rkp.o 4 ,ND
Ð 4hkoadIPo 17qo 9t05f|kozad8vooAo.shc]h; 9t8hv’.-hg;]kgryj,0Bo s^n .-hg;

]ks^5f]q’ L X4hkmP[.lj37qo?DDDDDDDDDDDDDDDD 4hk.-hg;]kgryj,0Bo9tgry,j 0Bo9adgmnjv LDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD c]t.-hg;]k9ad,N L DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
Ð 4hkoadIPo6 7qo 9t05f|kozad8vooAo.shc]h; 9t8hv’.-hg;]ks^5f]q’ s^n .-hg;
]kgry,j 0Bo L X4hkmP[.lj 17qo? DDDDDDDDDDD 4hk.-hg;]ks^5f]q’9ts^5f]q’9adgmnjv LDDDDDDDDD
c]t .-hg;]k9ad,N L DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

0D 9qj’lhk’8k8t]k’dkorq;rao]ts;kj ’9eo;ooadIPoda[9eo;o,NF grnjv05f
|kozad|bj’8vo c]h;8nj,.l[j vj o9Eg,afD
ສະຫງສ.ວວນ.ລິສຂະສິດ
÷ .... ×.....
9eo;ooad 316 .og'jnvow0G
IPoX7qo? 4 ..... .....
{ 05f|kozad8vofP;daoD
9eo;o,N { oadIPom5dM7qogIaf;Pdwfh
X,N?
gmqjkdaoD

× ..... ÷ .....

8k8t]k’0hk’gmy’oU c,jo8k8t]k’vaf8kl;j orq;rao s^n [+ L oadIPola’gdfgsao
co;.f ]ts;jk’dkoxPj ocx’0v’9eo;o,N c]t dkoxPj ocx’0v’9eo;ooad
IPo L

239

9q’j 7yfw]j G

zqo76o0v’lv’9eo;o.o4aomu 1 3 × 4 = ........................

zqo76o0v’lv’9eo;o.o4aomu 2 1 × 12 = ........................

zqo76o0v’lv’9eo;o.o4aomu 3 6 × 2 = ........................

gIqkla’gdfgsao;jk zqo76o0v’lv’9eo;o.o4aomu1 F 4aomu 2 c]t 4aomu3 DDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

dyf9tdemu @
dD 9qj’0Po 8nj,.lj[vj o9Eg,afD

de,tdvo3 7qo 05fIvj ’oEwfh 12m rkp.o 1 ,ND
{ rkp.o !,Nde,tdvo17qo 9t05fIjv’oEwfhs^5f]q’ DDDDDDDDDDDDDDD gmjnv s^n 05fIvj ’oE
wfh DDDDDDDDD m X4hkmP[.lj37qo?D
{ rkp.o ! ,N4hk8hv’dko9t05f.shwfh 24m 9t8hv’.-hde,tdvo9ad7qo L
.shoadIPolhk’8k8t]k’dkorq;rao ]ts;jk’9eo;ode,tdvoda[];’pk;
0v’Ivj ’muj05fwfhc]h;D
ສະຫງສ.ວວນ.ລິສຂະສິດ
÷ 3 ×.....
× .........
de,tdvo
X7qo? 3 1 6

7;k,pk;0v’
Ijv’ X m ? 12 ..... .....

÷ 3 × .....

{ 8k8t]k’0hk’gmy’ c,jogxaovaf8kl;j orq;rao s^n [+ L DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
4hkc,jo.sh-vdsk8q;lextlyf DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
{ oadIPola’gdfgsaogxaoco;.f ]ts;jk’8q;g]d0v’9eo;o];’pk;c,af c]t
8q;g]d0v’9eo;ode,tdvoLDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

240

dyf9tdemu #
oEg8a,4a’F 4hk4vd.ljd;fcdh;0t|kf[ao95 75cA .shg8a,r=fu9twfh 79 |j;pD
4hkgryjoxPj o,k4vd.ljd;fcdh;0t|kf[ao95 25cA 9t.-hd;fcdh;9ad|j;pL DDDDDDDD

8k8t]k’

0t|kf[ao95 75 25 Ð .og’jnvow0oEg8a, 14a’F
X cA ? 79 Ð 8k8t]k’0hk’-hkpoUc,oj
L
9eo;od;fcdh; gxao8k8t]k’ vaf8kl;j o
X|;j p? rq;rao [+ L

Ð 4hk4vd.ljd;fcdh;0t|kf[ao95 75cA 9twfh.-hd;fcdh; DDDDDDDDDDDDDDDDDDDDDDDDDDD|j;pDສະຫງສ.ວວນ.ລິສຂະສິດ
Ð 4hk4vd.ljd;fcdh;0t|kf[ao95 1cA 9twfh.-hd;fcdh;gry,j 0BoDDDDDDDDDDDDDDDDDDDDDgmjnvF

ltoAo 9t8hv’wfh.-hd;fcdh; 75×............ = ............ |;j pD
Ð 4hk4vd.ldj ;fcdh;0t|kf[ao95 25cA 9twfh.-hd;fcdh;|hvp]q’DDDDDDDDDDDDDDDgmjnvF

ltoAo 9t8hv’wfh.-hd;fcdh; ( 75×............ ) ÷25 = ............ |;j pD

0=hla’gdfG
{ 8k,dyf9tdemu !F 9eo;o,N grjnv05f|kozadda[9eo;ooadIPo gxaovaf8k
l;j orq;raoxUoD
{ 8k,dyf9tdemu #F 0t|kf[ao950v’d;fcdh;da[9eo;od;fcdh; gxaovaf
8klj;orq;raoxUoD
{ 8k,dyf9tdemu @F 7;k,pk;0v’Ivj ’da[9eo;ode,tdvo gxaovaf8kl;j o
rq;raoD

@D .97;k,

lv’xt]y,ko7n a,b,c, ...c]t a′,b′,c′, ... gxaovaf8kl;j orq;raoxUoF c,jolv’xt
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2,3, ...gmnjvD

241

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8q;1kj ’G
!D .og’njvow0muj8hv’dkowx.shwfh 36kmF 9t.-hg;]k9adokmu4hkskd;kj 7;k,w;

0v’]qfc]jowfh 30km h c]t 48km h 8k,]efa[D

[qfcdhG
Ð 7;k,w; 30km h 9t.-hg;]kDDDDDDDDDDDDDDokmu grnjvwx.shwfh 36km L

\kp7;k,;kj G w]ptmk’ 30km .-hg;]k DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDokmu
w]ptmk’ 1km .-hg;]k 60÷........... = ............... okmu
w]ptmk’ 36km .-hg;]k( 60÷....... )×36 = ........ okmu

faj’oAo 7;k,w; 30km h .-hg;]k 72 okmu grjnvwx.shwfh 36kmD
ສະຫງສ.ວວນ.ລິສຂະສິດ
Ð 7;k,w; 48km h 9t.-hg;]kDDDDDDDDDDDDDDokmu grjnvwx.shwfh 36km L

\kp7;k,;jkG w]ptmk’ 48km .-hg;]k DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDokmu

w]ptmk’ 1km .-hg;]k 60÷........... = ............... okmu

w]ptmk’ 36km .-hg;]k( 60÷....... )×36 = ........ okmu

fa’j oAo 7;k,w; 48km h .-hg;]k 45 okmu grnjvwx.shwfh 36kmD

8k8t]k’ ×1, 6

7;k,w;X km h ? 30 48

g;]kXokmu? 72 45

0=hla’gdf ! ÷ .....

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45 okmu gIqkg;qhkwfh;jkG c,oj 8k8t]k’vaf8kl;j orq;raoxUo0v’g;]kda[7;k,w;D

242

0=hla’gdf @

.sh7yfw]j a F 72 ÷ 1 = a s^n 72 = 1 × a
30 30

72 = 1× a s^n 72 = a s^n 72×30 = 1×a = a Xzqo76ow0;jgmqkj dao?
1 30 1 30

.sh7yfw]jb F 45 ÷ 1 = b s^n 45 = 1 ×b
48 48

45 = 1×b s^n 45 = b s^n 45×48 = 1×b = b Xzqo76ow0;gj mqkj dao?
1 48 1 48
gsao;kj 72×30 = 45×48 s^n a = b D

faj’oAoG 72 ÷ 1 = 45 ÷ 1 s^n 72 = 45 s^n 72×30 = 45×48
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0Pogxao 72 = 45 s^n 72×30 = 45×48 ສະຫງສ.ວວນ.ລິສຂະສິດ
1 1

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;ymucdhmu!

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÷ 2 ,kgxao 30 7qoD

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lv’9eo;o0v’4aomu@F 30×10 = 300 D ltoAo 15×20 = 30×10

lts^5[G g;]kgxaovaf8kl;j orq;raoxUoda[9eo;ode,tdvoD

243

ສະຫງສ.ວວນ.ລິສຂະສິດ#D [qfg/ydsaf

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dyosphk.sh\qf5 ,af rkp.o10,N L

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244