ແບບຮຽນວິຊາຄະນິດສາດ ປ4

17[qfmu dkooe.-h[aomaf c]t [aomaflkd

I. dyf9tdeD
!D 9qj’lhk’[aomaflkdfh;pdkora[g9hp8k,0Ao8vo]5j,oU G

!D g9hpczoj |b’j 8k,.9D @D ra[8k,.9];’0;k’D #D ra[.shc7,
ra[7A’mu ! g8a’daoD

@D 9qj’oe.-h[aomaflkdmujlhk’.o0=hmu ! ,kcmd,5,0v’I6[]5j,oU D

dd d
W
0W 0
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I6[ ! I6[ @ I6[ # ສະຫງສ.ວວນ.ລິສຂະສິດ

d d d
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{ I6[.f,uglAo-n8j A’lkddao c]t ,u9ad,5,lkd X _W = ? .oc8j]tI6[D

#D 9q’j oe.-h[aomafgrnjvc8h,I6[]5,j oU7no D

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48

II. .97;k,D
gryoj oe.-h[aomaflkdg0Qk.odko;afcmd,5, grnvj 1kdI6h,5,oAo;kj wfhlkds^n[+ L

gIqklk,kflhk’[aomaflkdfh;pdkora[g9hp

g9hp8k,.9 ra[gmnjvmu !

ra[gmnvj mu @

lv’glAo-nmj usj kj ’ltg\udaoc,oj lv’glAo-nuoj Ao0t|kodaoD 4hk0uf8+wxd+9t[+
8afdao9adgmnvj D
ສະຫງສ.ວວນ.ລິສຂະສິດ
II. [qfg/udsafD
! 9qj’c8h,I6[]5j,oUc]h;oe.-h[aomaflkdgrnjv;afcmd,5, c]t [vdI6[muj,uglAo-nj

8A’lkddao D

I6[ dD I6[ 0D I6[ 7D I6[ CD
@D 9qj’d;fg[yj’;jkI6[.f,uglAo-nj0t|kodao D

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49

[qfmu 18 8e]k9eo;o

I. dyf9tdeD
!D ok’ 7exko wxIhko0kpv5xtdvodkoIPo ]k;wfhwxvjko]k7klyo7hkc8]j t

-toyf16jdtfkofa’j ]5j,oUD 9q’j -;j p ok’ 7exko 7yfw]j.lj8k8t]k’]5,j oUD

9eo;ol=fe 1 2 3 4 5 6 7 8 9 10
]k7kl=fe 1000

9eo;o[yd 1 2 3 4 5 6 7 8 9 10
]k7k[yd 1500

9eo;oxB,0Po 1 2 3 4 5 6 7 8 9 10
]k7kxB,0Po 2000
ສະຫງສ.ວວນ.ລິສຂະສິດ
@D 76xt9eshv’wfhgvqkxB,vkj o1kp.shoadIPo7qo]t & sq; grnvj wxvjkoD ]k;7yfw]j
c]t ltcf’gxao dik2ydfaj’]5,j oU G
9q’j la’gdfc]h;8v[7e4k,D
9eo;ooadIPo

20 { oadIPo & 7qo 761kp
15 xB,.sh9adsq; L
10 { xB, !WW sq; 1kp.shoad
IPo9ad7qo9bj’r=fu\qf L
5 { 4hkoadIPo @& 7qo 76
1 9tgvqkxB,,k1kp.sh9adsq; L
0
9eo;oxB,
5 25 50 75 100

50

II. .97;k,D

8e]k9eo;oc,jo dkorq;rao]ts;jk’9eo;o 3fpltcf’dkorq;raovvd

fh;p8k8t]k’F dik2ydF D D D

8q;1jk’G oadIPoshv’ xD$ 3po\kd[ko.l[j h;’z6h]t @ 7A’ 4hk,u & 7qo

9t3po\kd[ko];,ma’\qf9ad7A’ L

9eo;o7qo

ສະຫງສ.ວວນ.ລິສຂະສິດ
{ .olt4kotdkooUc,jodkorq;rao]ts;jk’7qoda[9eo;o7A’

dik2yd 8k8t]k’9eo;o

5 9eo;o7qo 9eo;o7A’

4 1 2
2 4
3 3 6
4 8
2 5 10

1 9eo;o7A’

0
2 4 6 8 10

III. [qfg/udsafD
!D 9q’j 0Po8nj,.l8j k8t]k’]5j,oU c]h;ltcf’fh;pdik2ydD

oE|adwdjgxao kg 1 2 3 4 5 6 7 8 9 10

]k7kwdjgxao du[ 15000 du[

@D ok’9aol5fk wx8ts^kf]k;wfh-NxkF -nh-UoF -nhzad c]t -N\kdw,h fa’j 8k8t]k’
]5j,oU G 9qj’0Po8n,j .l8j k8t]k’.sh7q[ D

oE|ad kg 1 2 3 456
]k7k0v’
]kpdkogxaodu[ 25 000 20 000 9000 25 000

-Uo

xk

zad

\kdw,h

51

[qfmu 19 I6[c[[8kj ’M.odkoltg|u0=h,6o

I. dyf9tdeD

!D 9qj’la’gdfdik2ydc]h;8v[7e4k,]5j,oU G

dD dik2yd]5j,oUltcf’g4y’ 0D dik2ydoUltcf’g4y’];’pk;

v5oj str5,0v’xtgmf]k; 0v’c,joE16j[k’xtgmf.o3]d D

X;afcmd16j;P’9ao? rkp.o|bj’xu ];’pk; Xkm?
v5ostr6, X c= ?

36

34 7000 6650 6400
32
30 6000
28 5410
26 5000 4880
4444
24 3766
22 4000 vk2iydk
vkg,]ydkg|nv
20 3000
vkg,]ydk.8h
18
16 2000 ສະຫງສ.ວວນ.ລິສຂະສິ-u-u]ggດk[[;]u]u
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12 1000 c,oj E

10 oyo ,u-yf vk,k vq[ oE vk
-u[[u 3-o 0v’ ,q;
8

6

4
2 gfnvo
0

,a’dvo ,uok rbfltrk d=]tdqf daopk rt9yd

d5,rk g,lk ,u45ok ly’sk 85]k mao;k

{ c8]j tgfnvo,uv5ostr6,9advq’lk L { c,oj E.flAod;jk\6j c]t ,u9addy3]c,af L
{ v5ostr6,8ej l5fgmqkj .f c]t16j.ogfnvo.f L { c,oj E.fpk;d;kj \6j c]t,u9addy3]c,afL
{ v5ostr6,l6’l5fgmqjk.f c]t 16j.ogfnvo.f L

@D 9qj’la’gdf8k8t]k’c]h;8v[7e4k,D

oaddy]kmjuc]oj g[u # g[u & g[u @ g[u ! g[u $ { oaddy]kc8j]t7qoc]jo
g;]k.-hc]oj .ow]pt wxwfh9addy3]c,af.o
* dy3]c,af Xgxaookmu? 60 30 120 20 40 !@W okmuD
w]ptmuwj xwfh.o !@W okmu 12 24 6 36 18 { .ow]ptmk’ * dy3]
c,afc8]j t7qo.-hg;]k
c]jogmqjk.f L

52

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xEoE,ao
gxuf[=]ydko @$ -qj;3,’

{ c8j]txhkpltcf’g4y’spa’L
II. .97;k,D

dkoltg|u0=h,6olk,kfltcf’wfhs^kpI6[c[[g-ajoG
dik2ydS 8k8t]k’F xhkp37ltokF D D D
III. [qfg/udsafD
!D I6[rk[;af8t9addko0tspkp8q;0v’dq[

dD w0jmujskd=wfhIa[dkoxtlq,g-nhv
0D w0mj u[j ao95|+]6d c]t de]a’9tgxao]6dI;dD
7D I;dmujskd+vvd9kdw0j D
CD I;dmuj,u2ao2n,c]h; D
9D I;dmugj duf,u0ks^a’ D
lD I;dmugj duf,u0k|hk D
-D sk’0v’I;d7jvpMl6oskpwxD
pD dq[lq,[6oc[[ D

{ ;af8t9addko0tspkp8q;0v’dq[,u9adw]pt L
{ w]pt 7 c]t l ,u7;k,\kp;jkco;.f L
@D 9’jq l’a gdf8k8t]k’lt48y 0y v’oda IPoxt4,q l,q [o6 m’jq dk’ldq IPo @WW*{@WW(D
shv’ x! x@ x# x$ x& { oadIPoc8j]tshv’

,ugmqkj .fL
9eo;o ] p ] p ] p ] p ] p { oadIPopy’ -AoxD#

&( @) &@ @* *W @& && #@ &$ @$ ,u9ad7qo L

53

[qfmu 20 I6[c[[8kj ’M.odkoltg|u0=h,6oX8+?

I. dyf9tdeD xtgrflaf]hP’

!D 9qj’la’gdf dik2yd]5j,oUc]h;8v[7e4k, D

9eo;o38

60
50
40
30
20
10
0

wdj gxaf \6 Cq; 7;kp

!@ ,{ laf]hP’c8j]txtgrf,u9ad38L
shv’Ia[c0d
shv’dyog0Qk

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@D 9q’j la’gdfg[y’j czoza’0v’gInvos^a’oUc]h;8v[7e4k, D

!# ,

gInvo7q;

shv’oE

shv’ovo

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mk’g0Qk

{ ];’dh;k’F ];’pk;0v’gInvo,ugmqjk.fc,afL
{ ,ushv’9adshv’F 7nshv’.fcfDj c8j]tshv’d;hk’F pk;gmqjk.fc,af L
{ ,u9adxt86 L
{ .shvtmy[kplaopt]addPj ;da[xt86 D

54

II. .97;k,D

0=h,6omu.j sh,koAo,us^kpI6[c[[F lyj’le7aogIqk8hv’I6hvjkom5dI6[c[[0v’

dkoltg|u0=h,6ooAoD

III. [qfg/udsafD

!D 9q’j la’gdfI6[]t[q[l5]ypt9addt;ko c]t 0=h,6o.o8k8t]k’ c]h;8v[7e4k,G

8kg;ao fk;rtl5d

fk;glqk

fk;pq,

fk;rtsaf

fk;gdf fk;,t]bf8tp6
fk;r5f
|j;p3]d fk;va’7ko ສະຫງສ.ວວນ.ລິສຂະສິດ

-njfk; w]ptsjk’9kdf;’vkmyf Xrao]hko km ? glAozjko.9dk’(km)

fk;r5f &) $ )(W
fk;rtl5d !W) !@ !@W
|;j p3]d !&W !@ (&*
fk;va’7ko @&) * )WW
fk;rtsaf ((_ !$@ *WW
fk;glqk ! $@) !!_ #WW
fk;,t]bf8tp6 @ )(@ $_ WWW
fk;gdf $ &WW $) WWW
fk;pq, & _!& * WWW

dD |j;p3]d,uw]ptskj ’9kdf;’vkmyfgmqjk.f c]t ,uglAozkj o.9dk’gmqkj .f L
0D fk;c8]j tf;’,wu ]ptskj ’9kdf;’vkmfy gmkjq .f c]t,gu loA zkj odk’gmkjq .f L

@D .olqdIPo @WW*{@WW( 16j3I’IPoxt4q,csj’|bj’,uoadIPofaj’oUG x! &W 7qoS
xD@ $& 7qoS x# $W 7qoS x$ #& 7qoS c]t x& #W 7qoS 9qj’ltcf’9eo;o
oadIPo0v’3I’IPofaj’dkj ;fh;pdik2yd D

55

[qfmu 21 dkooe.-h;q’;Po

I. dyf9tdeD
!D9q’j c8h,I6[.ljxN,0Po3fpoe.-h;q’;Po

09 ສະຫງສ.ວວນ.ລິສຂະສິດ

14444442344c44m443

d

C7

56

ສະຫງສ.ວວນ.ລິສຂະສິດ@D 9q’j oe.-h;q’;Pogrnjvc8h,I6[c,’dtg[Nv]5,j oU G

{ gryoj c8h,I6[c,’dtg[Nvfh;p;q’,qoma’\qfXsq;F38Fxud?
{ .shoadIPoc8h,I6[c,’dtg[Nv7no3fp,u0t|kfgfu, c]t ,umu8j A’8k,.9D

57

II. .97;k,D ສະຫງສ.ວວນ.ລິສຂະສິດ
gryoj .-h;q’;Pogrnvj 0uf;q’,qo3fpxt8y[affa’j oU G
{ deoqf95f.9dk’0v’;q’,qo c]h;;k’
g0a,;q’;Po.lDj

{ pt0k;q’;Povvd8k,];’pk;0v’
]aflt\u;q’,qomugj Iqk9tc8h,D

{ 0uf;q’,qofh;pdko.-hoY;,nXoY;3xh
c]t oY;-Uxuoj 8k,I6[ c]t dqfg0a,
|hvp|b’j ?D

III. [qfg/udsafD
!D 9q’j c8h,;q’,qoshkvaomujI;j ,g,af.9dk’ O vaofP;dao

{ ;q’,qo d ,u]aflt\u ! cm
{ ;q’,qo 0 ,u]aflt\u @ cm
{ ;q’,qo 7 ,u]aflt\u # cm
{ ;q’,qo C ,u]aflt\u $ cm
{ ;q’,qo 9 ,u]aflt\u & cm

@D 9qj’c8h,;q’,qomuj,u]aflt\upk; # -a’8ugg,afc]h;0ufglahozkj o.9dk’lv’glaho8ah’
lkddaog-yj’8af;q’,qo16lj ugj ,af 9kdoAo.sh0uf $ g,af8+.ldj aoc]h;9twfhI6[spa’ L

58

ສະຫງສ.ວວນ.ລິສຂະສິດ#D 9qj’c8h,I6[.ljxB,0Po 3fpg]u,j 9kd|bj’;q’,qo c]t c[’j ;q’,qooAovvdgxao *
lj;o gmqjkdao 3fp.shc8j]tl;j ogmqkj da[]aflt\u0v’,ao c]h;\kp * g,afF 9kd
c8j]tg,af.sh0uf;q’,qo|bj’;q’D 4k,;jk.odkoc8h,oAo,u9ad;q’,qo L

$D 9q’j c8h,I6[]5j,oU.lxj nh,0Po D

59

[qfmu 22 dkoc8h,I6[7no

I. dyf9tdeD
!D 9q’j c8h,I6[]5j,oU.lxj nh,0Pomu,j u8kdk3iD

ABC

GH D
I

F

E

I6[ d I6[ 0 ສະຫງສ.ວວນ.ລິສຂະສິດ
@D 9qj’]vdc[[I6[]5j,ouh.lxj nh,0PoD

60

II. .97;k,D
dkoc8h,I6[7noc,jo dko]vdc[[F dkogdF c]t c8h,I6[gdqjk7no3fp.sh7n

da[I6[gfu, lj;o0t\kf0v’I6[vkf9tohvp]q’ s^n .spj0BoF0Boda[7;k,8hv’dkoD
III. [qfg/udsafD
!D 9q’j ]vdc[[I6[oU.ljg9hp8kdk3i c]h;mklu8k,.9F 9kdoAo .sh-vdsk0Ao8vo

0v’dkoc8h,D

ສະຫງສ.ວວນ.ລິສຂະສິດ
@D grnjvc8h,I6[8k,c[[]5,j oU 8hv’c8h,I6[9t85]af ABCD dvj o 8+wx.sh0ufmvj o
dqj’muj,ug,af.9dk’ D , A , B , C muj,u]aflt\u DE = DC ; AF = AE ; BF = BG ;
CG = CH 8k,]efa[ D

G

F AB

D C H
E
61

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I. dyf9tdeD
!D gsfdko8jk’M mujgduf0Bo.ow]ptzjko,k D

xu !&** lhk’mkfs^;’ !_*_ !)_#
7qomevyf0Bof;’9ao /ia’j 1bf7v’]k;
!_W! 0t[;odko8+l6h
0v’r+dtf;f !_(&
lhk’8A’ lxx]k;

!$_@ 7^yflt38[3d]q,gfuomk’ !_#W ສະຫງສ.ວວນ.ລິສຂະສິດ
0hk,,tsklt,5fvaf]a’8yd g]uj,,u3m]trk[

{ 9qj’0Poxu0v’gsfdko8jk’MoU.l.j ocdo9eo;oD 2 000

1 400 1 500 1 600 1 700 1 800 1 900

1975

lhk’8A’ lxx]k;

@D 9q’j 8nj,g7njv’\kp > ; < .l[j jvo9Eg,af.sh4nd8hv’D

45 . . . 70 89 . . . 98 384 . . . 196 476 . . . 602 45 . . . 649

649 . . . 642 386 . . . 368 713 . . . 731 805 . . . 850 89 . . . 368

62

#D 9q’j .-h8q;g]d.o8k8t]k’grnvj 0Pogxao9eo;omujs^kpl5f c]t |hvpl5f D

8q;g]dmuj.sh.-h 9eo;omusj ^kpl5f 9eo;omuj|hvpl5f
&F(F*F@ (*&@ @&*(
$F)F(F*F&
&F$F#FWF@
*F&F!F@F#F$
_F)F(F*F&

$D 9qj’0Po9eo;o8nj,.lj[vj o9Eg,af D 351 < . . . < 353 ສະຫງສ.ວວນ.ລິສຂະສິດ
9012 < . . . < 9014
79 < . . . < 81 567 < . . . < 569
8476 < . . . < 8478 9821 < . . . < 9823
43721 < . . . < 43723 4001 < . . . < 4003
307 < . . . < 309
5760 < . . . < 5762

&D 9q’j 0Po9eo;o16|j hk4af c]t s^a’4af9eo;o]5j,oU D

2680 < 2681 < 2682 . . . < 345 < . . .
. . . < 3642 < . . . . . . < 7011 < . . .
. . . < 8042 < . . . . . . < 90765 < . . .
. . . < 90012 < . . . . . . < 84561 < . . .
. . . < 84775 < . . . . . . < 654430 < . . .

63

II. .97;k,D

dkoxP[mP[ @ 9eo;o D
!D lv’9eo;o,u9eo;o8q;g]d[+gmqkj daoF 9eo;omujs^kpd;kj c,jo9eo;omu,j u9e

o;o8q;g]ds^kpd;jkD
8q;1jk’ G
!*& ,u lk,8q;g]d
*& ,u lv’8q;g]d
c,oj !*& > *&

@D lv’9eo;omuj,u8q;g]dgmqjkdaoD gIqkla’gdfg[yj’8q;g]dmu|bj’9kdg[nhv’-hkpF

4hk;jk 8q;g]dmu|bj’gmqjkdao8hv’g[yj’8q;g]dmulv’S 4hk8q;g]dmulv’pa’gmqjk

daovud8hv’g[yj’8q;g]dmulk, c]t 8+Mwx D

8q;1kj ’G # _ ( da[ #_W ສະຫງສ.ວວນ.ລິສຂະສິດ

3 = 3 8q;mu|bj’

9 = 9 8q;mulv’

c,oj G 3 9 7 > 3 9 0 7 > 0 8q;mulk,

dko0v[0AoD
gIqklk,kf0v[0Ao9eo;o.f|bj’fh;p9eo;o|hk4af c]t s^a’4af
8q;1kj ’ G 1 2 4 < 1 2 5 < 1 2 6

III. [qfg/udsafD 2689
!D 9qj’8nj,9eo;o.lgj lAo9eo;o]5,j oU G

2680

2681

64

@D 9qj’oe.-h8q;g]d.o8k8t]k’]5j,oU0Po.shwfh * 9eo;oD

8q;g]dmuj.sh.-h * 9eo;omu]j Po8k,lefa[c8|j hvpsks^kp

8,9,4 489 ; 498 ; 849 ; 894 ; 948 ; 984
3,7,8
5,2,6

#D 9q’j la’j gdfc]h;ln[8+0Po.shwfhc4;]t & 9eo;oD

2680 < 2685 < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1290 < 1300 < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3476 < 3576 < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1234 < 2234 < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ສະຫງສ.ວວນ.ລິສຂະສິດ
$D 9qj’la’gdfc]h;ln[8+0Po.shwfhc4;]t & 9eo;oD

108 > 106 > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1276 > 1271 > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8765 > 8755 > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1470 > 1370 > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

&D 9q’j 8n,j g7njv’\kp > ; < c]t = .l[j vj o9Eg,af.sh4nd8hv’D

4782 _ 1234 . . . 4978 _ 732 120 X 12 . . . 4 X 500
5372 _ 1489 . . . 9567 _ 2734 947 + 176 . . . 37+795
47 X 10 . . . 36 X 12 ;
^4320 ' 2h # 13 . . . . 12#^4320 ' 2h 765 + 4372 . . . 567 + 5786
(7564 + 432) _ 3476 . . . (9476 _ 376) + 594

65

[qfmu 24 dko-vdI6h0v[g0f c]t mu,j k0v’0=h,6o

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crfxyoj xq;

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]k[xk ! 9ko !W WWW du[ S -5[zad ! 9ko & WWW du[D
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66

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dkolezaf8q;9y’ grnjv.shI6h;jk 0=h,6owfh,k9kd.lS c,jo.zgxaoz6h.sh0=h,6oS 0=h,6o
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!D 8ts^kfg-Qk { mjkgfnvj ( 3,’ #W okmu _ 3,’ #W okmu ສະຫງສ.ວວນ.ລິສຂະສິດ
@D 8ts^kfg-Qk { [hkogdyo ) 3,’ WW okmu !@ 3,’ WW okmu
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!D xhkpwfh.sh0=h,6ovao.fcfj L
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67

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10 000 k 18 000 k

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68

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!D

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kg hg dag g dg cg mg

1 0 00 00

10 01

0, 0 01

0, 0

0, 1

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III. [qfg/udsafD

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15 hg . . . . . . . . 15g ; 850 dg . . . . . . . . 85 cg ; 830 kg . . . . . . . . 560 hg ;
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10 g . . . . . . . . 10 hg ; 54 hg . . . . . . . . 540 g ; 370 mg . . . . . . . . 37 dg .

71

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72

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oeh|ad Xkg ?

60

50 45

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xu @WW@ ( xu @WW& !_
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73

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74

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1979 6 9 .....
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2000 ..... ..... .....
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120 km, 180 km.

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