1. KWnXw
Ìm³tUÀUv VIII
`mKw þ 1
tIc-f-kÀ¡mÀ
hnZym-`ym-k-h-Ip¸v
2009

2. tZio-b-Kmw
P-K-W-a-- -A-[n-m-bI Pbtl
`mc-X -`m-Ky-hn-[mXm,
]©m- -knÔp KpP-dm¯ admTm
{ZmhnU DXv¡-e- _w-Km,
hnÔy-ln-am-N-e -b-apmKwKm,
DÑePe-[n-X-cwKm,
Xh-ip-`-mta PmtK,
Xh-ip` Bin-j -am-tK,
Kmtl Xh- P-b-KmYm
P-K-W-aw-K-e-Zm-bI Pbtl
`mc-X- `m-Ky-hn-[mXm.
Pb-tl, Pb-tl, Pb-tl,
Pb Pb Pb Pbtl!
Prepared by :
State Council of Educational
Research and Training (SCERT)
Poojappura, Thiruvananthapuram - 12, Kerala.
Website www.scertkerala.gov.in
e-mail scertkerala@asianetindia.com
© Government of Kerala • Department of Education • 2009

3. {]nb-s¸« Ip«n-I-sf,
½psS Npäp-]m-Sp-apÅ Pohn-X-N-cy-I-fn Adnªpw
Adn-bm-sXbpw {]tbm-Kn-¡p¶ KWn-Xm-i-b-§Ä
Bi-b-X-e-¯n NÀ¨ sN¿m-pÅ {ia-amWv F«mw-
Xcw apXÂ Bcw-`n-¡p-¶-Xv. Bi-b-§-fpsS A]-{K-Y-
-¯neqsS kmam-y-h-Xv¡-c-W-¯nsâ Xe-¯n-te¡v
Nn´sb Xncn-¨p-hn-Sp-¶-Xnv Cu ]pkvX-I-¯n-eqsS
{ian-¡p-¶p.
Imcy-Im-cW _Ô-§Ä Is­-¯m-pw, kz´-ambn
A`n-{]m-b-§Ä cq]o-I-cn-¡m-pw, Bi-b-§sf
Nn{XoI-cn-¡mpw Hs¡ Ign-bp¶ Xc-¯n Nn´sb
Dt¯-Pn-¸n-¡m³ ]c-am-h[n {ian-¨n-«p-­v.
A[ym-]-I-cpsS nÀt±-i-t¯m-Sp-Iq-Sn ]co-£n-¨pw,
nco-£n-¨pw, Imc-W-§Ä Is­¯nbpw, kmam-y-h-
XvI-cn-¨pw, BËm-Z-]-c-amb KWn-X-]-Tw km[y-am-
Is« F¶v Biw-kn-¡p-¶p.
kvtl-]qÀhw
F.]n.-Fw. apl-½Zv lojv sF.F.-Fkv
Ub-d-IvSÀ C³ NmÀPv
Fkv.-kn.-C.-BÀ.-Sn.

4. ]mT-]p-kvXI cN
inev]-im-e-bn ]s¦-Sp-¯-hÀ
{io AnÂIp-amÀ Fw.-sI.
{ioaXn ^ukn -sI.-F.-Fw.
{io tKmhn-µ-cm-P³ Fw.-]n.
{ioaXn Pb-e-£van ]n.
tUm. -C.-Ir-jvW³
{io IrjvW-Ip-amÀ ]n.
{io -ap-c-fo-[-c³]nÅ sI.-BÀ
{io mcm-b-W-p®n Fw.-]n.
{io -{]-Im-i³ Sn.-]n
{io -cm-am-p-Pw BÀ.
tUm. km_pPn hÀKokv
{io kpnÂIpamÀ sI.-sI.
{io D-®n-Ir-jvW³ Fw.hn
{io hmkp sI.-Pn.
Nn{X-Imc³
{io- [t-i³ -Fw.-hn.
hnZ-Kv[À
tUm. cmP³ -F.-BÀ.
tUm. cma-N-{µ³ ]n.-Sn.
tZiob hnZ-Kv[À
tUm. -kp-{_-a-Wyw sI.
tUm. cmam-p-Pw BÀ.
A¡m-Z-anIv tImÀ-Un-t-äÀamÀ
{io tKmIp-e-Zm-k³ ]nÅ. kn
{io enUvk¬ cmPv. sP
kwØm hnZym-`ymk Kth-jW ]cn-io-e kanXn (SCERT)
hnZym-`h³, ]qP-¸p-c, Xncp-h--´-]pcw 695 012

5. DÅS¡w
1. kÀh-k-a-{Xn-tIm-W-§Ä ................ 7
2. Awi--Ôhpw Ap-]m-Xhpw ......... 27
3. yq-kw-Jy-IÄ ..........................47
4. ]W-hn-n-abw ............................ 61
5. _oP-K-WnXw ............................. 71

6. Blank

7. Awi-Ôhpw
Ap-]m-Xhpw
`n¶-§-fn-ÃmsX
Hcp nÝnX GIIw D]-tbm-Kn¨v
ofhpw aäpw Af-¡p-t¼mÄ
Ft¸mgpw F®Âkw-Jy-IÄ In«nÃ,
F¶ hkvXp-X-bn n¶mWv `n¶-
kwJy F¶ Bibw D­m-b-Xv. c­v
Af-hp-IÄ Xmc-Xayw sN¿p-t¼mÄ
th­{X sNdnb GIIw D]-tbm-Kn-
¨m c­n-tbpw F®Â kwJy-bm-
¡mtam F¶ Nn´-bmWv Awi-
_Ôw F¶ Bi-b-¯nv B[m-cw.
DZm-l-c-W-am-bn, Hcp Nc-Sp-sIm­v Af-
¡p-t¼mÄ Hcp hkvXp-hnsâ ofw
2
3
5 Dw asäm-¶nsâ ofw
5 Dw
BsW-¶ncn-¡-s«. Nc-Snsâ
1
5 `mKw
GI-I-am-sb-Sp-¯mÂ, BZy-t¯-Xnsâ
ofw 2 Dw c­m-a-t¯-Xnsâ ofw
3 Dw BIp-a-tÃm. of-§-fpsS Awi-
_Ôw 2 : 3 F¶p ]d-bmw.
c­p hkvXp-¡-fn H¶nsâ ofw
Hcp Nc-Snsâ
1
3 `mKhpw asäm-¶nsâ
ofw
1
5 `mKhpw BsW-¦ntem?
Nc-Snsâ
1
15 `mKw GI-I-am-sb-Sp-
¯m BZy-t¯-Xnsâ ofw 5 Dw
c­m-a-t¯-Xnsâ ofw 3 Dw BIpw.
Awi-Ôw 5 : 3 F¶p ]d-bmw.
CXpsIm­p-Xs¶ Awi--Ô-§Ä
s]mXpth F®Â kwJy-IÄ D]-
tbm-Kn-¨mWv ]d-bp-¶-Xv.
2
Hcp Imcyw, ]e coXn
Cu Nn{Xw tm¡q.
A B
AB F¶ hcsb A©p ka-`m-K-§-fm-¡n-bn-cn-¡p-¶p.
BZys¯ aq¶p-`mKw tNÀ¶-Xns AP F¶p hnfn-¨m AB,
AP, BP F¶o of§Ä X½n-epÅ _Ôw F§-s-sbÃmw
]dbmw?
A P B
$ AB bpsS
3
5 `mK-amWv AP..
$ AB bpsS
2
5 `mK-amWv BP.
$ AP bpsS
2
3 `mK-amWv BP.
$ BP bpsS 3
2 aS§mWv AP..
$ AP, BP Ch-bpsS of-§Ä X½n-epÅ Awi-Ôw 3 : 2
BWv.
asämcp coXn-bnepw ]dbmw:
$ AB bpsS
1
5 `mKw sIm­v Af-¶m AP bpsS ofw 3
Dw BP bpsS ofw 2 Dw BWv.
(Ggmw ¢mknse `mK-§Ä _Ô-§Ä F¶ A[ymbw
tm¡pI).
thsdmcp IW¡v: 50 cq] A½phpw A¸phpw ]¦n-s«-Sp-¯p.
A½p 30 cq]bpw, A¸p 20 cq]-bp-amWv FSp-¯-Xv.
A½p-hnv 50 Â 30 In«n.
30
50
=
3
5
BW-tÃm.
At¸mÄ A½p-hnv In«n-bXv BsI XpI-bpsS F{X `mK-amWv?
A¸p-hntm?

8. KWnXw þ ¢mkv 8
28
CXv GsXÃmw coXn-bn ]-dbmw?
$ BsI XpI-bpsS ........... `mKw A½p-hnpw ...........
`mKw A¸p-hnpw In«n
$ A¸p-hnv In«n-b-Xnsâ ........... aS§v A½p-hnv In«n.
$ A½p-hnv In«n-b-Xnsâ ........... `mKw A¸p-hnv
In«n.
$ A½p-hnpw A¸p-hnpw In«nb XpI-I-fpsS Awi-
_Ôw ........... BWv.
ChÀ hoXn¨ XpI ]¯v cq] tm«p-I-fm-sW-¦ntem?
$ A½p-hnv ........... tm«pw A¸p-hnv ........... tm«pw
In«n.
Cn Cu IW¡v tm¡q.
Nmt¡m-¨pw Znt-ipw IqSn Hcp ]¦p I¨-hSw
XpS§n. Nmt¡m-¨³ 5000 cq]bpw Znt-i³ 7000
cq]bpw BWv apXÂ apS-¡n-b-Xv.
BsI apS-¡p-ap-X-ensâ F{X `mK-amWv Nmt¡m-¨³
apS-¡n-bXv? Znt-itm?
Nmt¡m¨³ apS-¡n-b-Xnsâ F{X aS-§mWv Znt-i³
apS-¡n-bXv?
Znt-i³ apS-¡n-b-Xnsâ F{X -`m-K-amWv Nmt¡m¨³
apS-¡n-bXv?
F{X Bbncw cq]-bmWv Nmt¡m-¨³ apS-¡n-bXv? Znt-
itm?
Ch-cpsS apS-¡p-ap-X-ep-IÄ X½n-epÅ Awi- Ôw
F´mWv?
Hcp DZm-l-cWw IqSn-bm-Imw. Cu Nn{X-§Ä tm¡q:
henb s]³kn-ensâ ofw 7.5 skân-ao-äÀ; sNdnb
s]³kn-ensâ ofw 4.5 skân-ao-äÀ.
sNdnb s]³kn-ensâ F{X aS§v ofap­v henb
s]³kn-en-v?
Awi--Ôhpw `n¶-§fpw
Hcp hkvXp-hnsâ `mK-§Ä Xmc-Xayw
sN¿mpw Awi-Ôw D]-tbm-Kn-¡mw. DZm-l-
c-W-ambn, Cu Nn{X-¯n Idp¯- n-d-ap
Å `mKw hr¯-¯nsâ
3
8
`mK-amWv;
shfp¯ nd-apÅ `mKw hr¯-¯nsâ
5
8
`mK-hpw.
Ch c­pw tNÀ¶m apgp-h³ hr¯-am-bn.
Cu c­v `m-K-§-fp-tSbpw hen¸w X½n-epÅ
Awi-Ôw 3 : 5.
C§s tm¡p-t¼mÄ 3 : 5 F¶ Awi-
_Ôw
3
8
,
5
8
F¶ c­p `n¶-kw-Jy-I-sf-
bmWv kqNn-¸n-¡p-¶-Xv.

9. Awi-Ôhpw Ap-]m-Xhpw
29
7.5
4.5 =
75
45 =
5
3
At¸mÄ sNdnb s]³kn-ensâ of-¯nsâ
5
3 aS-§mWv
henb s]³kn-ensâ ofw. henb s]³kn-ensâ of-
¯nsâ
3
5 `mK-amWv sNdnb s]³kn-ensâ ofw F¶pw
]d-bmw.
1.5 skân-ao-äÀ of-apÅ Hcp NcSp sIm­v A-f-¶mtem?
henb s]³kn-ensâ ofw 5 NcSpw sNdnb s]³kn-ens
â ofw 3 NcSpw Bbn-cn-¡pw.
henb s]³kn-ensâbpw sNdnb s]³kn-en-sâbpw
of-§Ä X½n-epÅ Awi- Ôw F´mWv?
s]³kn-ep-IÄ c­pw tNÀ¯p-h-¨m BsI ofw
12 skân-ao-äÀ BW-tÃm. henb s]³kn-ensâ ofw
CXnsâ
7.5
12
=
5
8
`mK-hpw, sNdnb s]³kn-ensâ ofw
4.5
12
=
3
8
`mKhpw BsW¶pw thW-sa-¦n ]d-bmw.
]t£ s]³kn-ep-IÄ tNÀ¯p hbvt¡­ Bhiyw
hcp-¶n-Ã-tÃm.
CXp-t]mse NphsS -]-d-ªn-cn-¡p¶ Imcy-§Ä `n¶-
§Ä D]-tbm-Kn-¨pw, Awi- Ôw D]-tbm-Kn¨pw ]e
coXn-bn ]d-bmtam F¶p tm¡q:
$ tZmi-bp-­m-¡m³ 6 I¸v Acnbpw 2 I¸v Dgp¶pw
FSp-¯p.
$ ¢mkn 26 s]¬Ip-«n-Ifpw 24 B¬Ip-«n-I-fp-ap-­v.
$ kvIqfn 500 hnZ-ymÀ°n-Ifpw 15 A[-ym-]-Icpw D­v.
$ Hcp Nc-Sp-sIm­v Af-¶-t¸mÄ Hcp s]³kn-ensâ
ofw Nc-Snsâ 1
2 `mK-am-Wv. asämcp s]³kn-ensâ
ofw Nc-Snsâ
1
3 `mKhpam-Wv.
eLqI-cWw
Nn{X-¯nse hr¯-¯nsâ c­p `mK-§fpw
X½n-epÅ Awi-Ôw F´mWv?
hr¯-¯nsâ
2
8 `mK-amWv sNdnb IjWw.
henb IjWw
6
8 `mKamWv. AXn-mÂ
Awi-Ôw 2 : 6 F¶p ]dbmw.
]t£
2
8 =
1
4 Dw
6
8 =
3
4 BWtÃm
At¸mÄ Awi-Ôw 1 : 3 F¶pw ]dbmw.
hr¯s¯ 4 ka-`m-K-§-fm¡n 1 `mK-hpw,
an¨-apÅ 3 `mK-§fpw FSp-¯mepw CtX
IjW-§Ä Xs¶-btà In«p-¶Xv?
At¸mÄ 2 : 6 F¶ Awi_Ôhpw 1:3
F¶ Awi- -Ôhpw Xpe-y-am-Wv. 4 : 12 F¶
Awi--Ôhpw Chbv¡v Xpe-y-amWv.
s]mXpth ]d-ªmÂ
a : b = ma: mb
km[m-c-W-bmbn Awi- -Ô-§Ä ]d-bp-t
¼mÄ s]mXp LS-I-§Ä Hgn-hm¡n Gähpw
sNdnb kwJ-y-IÄ D]-tbm-Kn-¨m-Wv ]d-bp-
¶-Xv.

10. KWnXw þ ¢mkv 8
Awi--Ôhpw nc¡pw
CÍen D­m-¡p-t¼mÄ 2 I¸v Acn¡v
1 I¸v Dgp¶v F¶mWv s]mXpth FSp-¡p-
¶-Xv. Acnbpw Dgp¶pw FSp-¡p-¶Xv 2 : 1
F¶ Awi--Ô-¯n-emWv F¶p ]d-bmw.
Hmtcm 2 I¸v Acn¡pw 1 I¸v Dgp¶v FSp-
¡Ww F¶m-WtÃm CXnsâ AÀ°w.
ChnsS 2 v 1 F¶ nc-¡n-mWv {]m[m-
yw.
CXp-t]mse cmk-h-kvXp-¡Ä tNÀ¯v
kwbp-à-§Ä D­m-¡p-t¼mÄ Af-hp-IÄ
X½n-epÅ Awi-Ôw (nc-¡v) amdn-bmÂ
kwbp-à-¯nsâ kz`mhw Xs¶ amdn-tb
¡mw.
30
KWn-Xhpw {]tbmKhpw
Awi--Ôw- F¶ Bi-b-¯nv ]e hnh-c-W-§Ä I­-
tÃm. kµÀ`-¯npk-cn¨v Ch {]tbm-Kn-¡Ww. Nne
DZm-l-c-W-§Ä tm¡q.
hoSnv Nmb-a-Sn-¡m³ oebpw shÅbpw nd-§Ä
3 : 2 F¶ Awi- -Ô-¯n tNÀ¡-Ww. 35- en-äÀ
Nm-b-¡q-«p-­m¡m³ F{X enäÀ o-ebpw F{X
enäÀ sh-Åbpw tNÀ¡Ww?
ChnsS X¶n-«p-ÅXv BsI-bpÅ Nmb-¡q-«nsâ Af-hm-
Wv. oebpw shÅbpw X½n-epÅ Awi- Ôw
3 : 2 F¶-Xn n¶v, Nmb-¡q-«nsâ F{X `mK-amWv
oe, F{X- `m-K-amWv shÅ F¶v a-Ên-em-¡m-a-tÃm.
3
5
`mKw oe.
2
5
`mKw shÅ.
Nmb-¡q«v 35 enäÀ thW-sa-¦ntem?
oe-bpsS Afhv = 35´
3
5
= 21 enäÀ
shÅ-bpsS Afhv = 35 ´
2
5
= 14 enäÀ
asämcp coXn-bnepw Btem-Nn-¡mw. oebpw shÅbpw
tNÀ¡p-¶-Xnsâ Awi- Ôw 3 : 2 Bb-Xn-m Ch-bpsS
an{in-X-¯n oe-bp-sSbpw shÅ-bp-sSbpw
Af-hv, Hcp nÝnX Af-hnsâ 3 aS§pw 2 aS§pw
Bbn-cn-¡-Ww.
Cu nÝnX Afhv 1 enädmsW¦n BsI 5 enäÀ
an{inXw In«pw. th­Xv 35 enä-dm-W-tÃm. At¸mÄ
oe, 3 enä-dnsâ 7 aS§pw shÅ, 2 enä-dnsâ 7 aS§pw
Bbn-cn-¡pw.
Cn asämcp IW¡v.
Hcp kvIqfn B¬Ip-«n-I-fp-tSbpw s]¬Ip-«n-I-fp-tSbpw
F®w X½n-epÅ Awi-Ôw 12 : 13 BWv.
kvIqfn 360 B¬Ip-«n-I-fp-­v. s]¬Ip-«n-I-fpsS
F®w F{X-bmWv?
ChnsS X¶n-«pÅ Awi- -Ô-¯n n¶v hmbn-s¨-Sp-t
¡-­-sX-´mWv?
s]¬Ip-«n-I-fpsS F®-am-WtÃm I­p-]n-Sn-t¡-­-Xv.
AXv B¬Ip-«n-I-fpsS F®-¯nsâ F{X aS-§mWv?

11. Awi-Ôhpw Ap-]m-Xhpw
31
X¶n-«p-Å-Aw-i- -Ô-¯nÂn¶v s]¬Ip-«n-I-fpsS
F®w B¬Ip-«n-I-fpsS F®-¯nsâ 13
12 aS-§m-Wv.
B¬Ip-«n-I-fpsS F®w 360 BsW¶pw X¶n-«p-­v.
At¸mÄ s]¬Ip-«n-I-fpsS F®w = 360 ´ 13
12 = 390.
asämcp coXn-bnepw Btem-Nn-¡mw. B¬Ip-«n-I-fpsS
bYmÀ° F®w 360; Awi- -Ô-¯n CXns kqNn-
¸n-¡p¶ kwJy 12.
12 sâ F{X aS-§mWv 360?
At¸mÄ s]¬Ip-«n-I-fpsS F®w 13 sâ F{X aS-
§mWv?
Cu IW-¡p-IÄ kzbw sNbvXp tm¡q.
$ Hcp NXp-c-¯nsâ ofhpw hoXnbpw 5 : 3 F¶
Awi--Ô-¯n-em-Wv. ofw 2.5 aoädm-Wv. hoXn F{X
aoä-dmWv?
$ koÀ 4000 cq]bpw mcm-b-W³ 6000 cq]bpw
apS¡n H¶n-s¨mcp I¨-hSw XpS-§n. Hcp hÀjw
Ign-ª-t¸mÄ 3000 cq] em`w In«n. CXv apS-¡nb
XpI-bpsS Awi- -Ô-¯n hoXn-¨m Hmtcm-cp-
¯À¡pw F{X cq] In«pw?
$ ca-bpsS ssIbn 18 Nph-¶ -ap-¯p-Ifpw 12 ]¨-
ap-¯p-Ifpw D­v. Da-bpsS ssIbn Np-h¸pw ]¨-
bp-ambn 20 ap¯p-I-fm-Wp-ÅXv. c­p nd-¯n-epapÅ
ap¯p-I-fpsS Awi-Ôw ca-bptSXp Xs¶. Da-bpsS
ssIbn F{X Nph¶ ap¯p-I-fp­v? ]¨
ap¯pI-tfm?
aäp Nne tNmZy-§Ä
tPmbn-bpw Pbpw Hcp XpI 3 : 5 F¶- Awi_Ô-
¯n hoXn¨p. Pbv 2000 cq] A[nIw In«n. F{X
cq]-bmWv hoXn-¨Xv? Hmtcm-cp-¯À¡pw In«nbsX{X?
ChnsS In«nb XpI Adnbn-Ã. ]t£ Awi- Ôw
3 : 5 F¶p X¶n-«p-Å-Xn n¶v, tPmbn¡v Cu XpI-bpsS
3
8 `mKhpw Pbv
5
8 `mK-hp-amWv In«n-b-sX¶v
a-Ên-em-¡mw.
sSen-hn-j³ KWnXw
sSen-hn-j³ skäp-I-fpsS hen¸w s]mXpsh
14 C©v, 17 C©v , 21 C©v F¶n-§-s-
bmWv ]d-bp-¶-Xv. F´mWv CXnsâ AÀYw?
sSen-hn-j³ kv{Io³ Hcp NXp-c-am-W-tÃm.
AXnsâ hnIÀW-¯nsâ Af-hp-I-fmWv
Ch-sb-Ãmw. (F´p-sImt­m sSen-hn-j³
hyh-km-b-¯n C¶pw skân-ao-ä-dnp
]Icw C©v Xs¶-bmWv D]-tbm-Kn-¡p-¶Xv.)
CXp-sIm­p am{Xw sSen-hn-jsâ hen¸w
nÝ-bn-¡mtam? hnIÀWw Xpey-am-bmepw
hoXnbpw ofhpw hyXym-k-am-Im-a-tÃm.
kv{Ionsâ hen¸w F{X-X-s¶-bm-bmepw
AXnsâ ofhpw Db-chpw X½n-epÅ
Awi-Ôw km[m-cW sSen-hn-j³ skäp
I-fn 4 : 3 BWv. Cu Awi-Ôw 16 : 9
Bb (wide screen) skäp-Ifpw Ct¸mÄ
D­m-¡n-¯p-S-§n-bn-«p-­v. hnIÀW-¯nsâ
hen¸w Xpe-y-amb c­v sSen-hn-jsâ
kv{Iop-I-fn Cu hyX-ymkw tm¡q.
4 : 3 16 : 9
Cu Awi_-Ô-¯nv NXp-c-¯nsâ apJmw-i-
Ôw (aspect - ratio) F¶mWv t]cv.
17 C©v hen-¸-apÅ 4 : 3 sSen-hn-j³
kv{Ionsâ ofhpw Db-chpw I­p-]n-Sn-
¡mtam? ss]Y-tKm-d-kns HmÀ¡p-I.

12. KWnXw þ ¢mkv 8
32
Ch-bpsS hyXym-k-amWv Pbv IqSp-XÂ In«nb 2000
cq].
5
8 tâbpw
3
tâbpw hyXymkw F{X-bmWv?
8 5
3
2
þ
1
=
= 8 8 8 4
At¸mÄ hoXn¨ XpI-bpsS 1
4 `mK-amWv 2000.
Cn hoXn¨ XpI-bpw Hmtcm-cp-¯-cp-tSbpw hoXhpw
I­p-]n-Sn-¡m-a-tÃm?
asämcp coXn-bnepw Btem-Nn-¡mw. tPmbn¡pw Pbpw
In«n-bXv Hcp nÝnX XpI-bpsS 3 aS§pw 5 aS§pw
BW-tÃm. AXnm Pbv IqSp-X In«n-b-Xv, Cu
XpI-bpsS 2 aS-§m-Wv. AXv 2000 cq]-bm-Wv. At¸mÄ,
Cu nÝnX XpI 1
2 ´ 2000 = 1000 cq]-bm-I-W-a-tÃm.
Cn tPmbn¡v In«n-bXv 3 ´ 1000 = 3000 cq]-bpw,
Pbp In«n-bXv 5 ´ 1000 = 5000 cq]bpw F¶p ImWm³
hnj-a-an-Ã.
thsdmcp tNmZyw tm¡q:
Hcp ¢mknse B¬Ip-«n-I-fp-tSbpw s]¬Ip-«n-I-fp-tSbpw
F®w X½n-epÅ Awi- Ôw 2 : 3 BWv.
B¬Ip-«n-I-fnse 1
4 `mKw t]À ¢mknÂn¶p
t]mbm Cu Awi- Ôw F{X-bmIpw?
Cu ¢mkn B¬Ip-«n-I-fpsS F®-¯nsâ 1
1
2 aS-§m-
WtÃm s]¬Ip-«n-I-fpsS F®w. B¬Ip-«n-I-fnse 1
4
`mKw t]À ¢mknÂn-¶p-t]m-bm 3
4 `mKw t]À Ah-ti-
jn-¡pw. 3
4 sâ 2 aS-§mWv 1
1
2 . A-t¸mÄ Awi-
_Ôw 1 : 2 BWv.
atä-sXÃmw coXn-bn CXp sN¿mw?
Hcp tNmZyw IqSn:
ABC F¶ {XntIm-W-¯n AB : BC = 1 : 2 Dw
BC : AC = 3 : 5 BWv. AB : AC F{X-bmWv?
ChnsS AB F¶ hi-¯nsâ ofw, AC F¶ hi-
¯nsâ F{X aS-§mWv (A-sÃ-¦n `mK-am-Wv) F¶p
I­p-]n-Sn-¡-Ww.
At§m«pw Ct§m«pw
Hcp NXp-c-¯nsâ ofw 33 skân-aoädpw
hoXn 1 skân-ao-ädpw BWv. c­m-as¯
NXp-c-¯nsâ ofw 11 skân-ao-ädpw hoXn
6 skân-ao-ädpw BWv. Cu NXp-c-§-fpsS
Npä-f-hp-IÄ X½n-epÅ Awi-_Ôw
F´mWv? Ah-bpsS ]c-¸-f-hp-IÄ X½n-ep
Å Awi-Ôw F´mWv? CtX khn-ti-
j-X-bpÅ thtd NXp-c-§-fpt­m?

13. Awi-Ôhpw Ap-]m-Xhpw
33
AB bpsS ofw BC bpsS F{X- `m-KamWv?
AB : BC = 1 : 2 F¶-Xn n¶v
AB =
1
2
BC
Cn BC bpsS ofw AC bpsS F{X `mK-amWv?
BC : AC = 3 : 5 F¶-XnÂn¶v
BC =
3
5 AC
At¸mÄ AB bpsS ofhpw AC bpsS ofhpw X½n-ep
Å _Ôw F´mWv?
AB =
1
2
´
3
5 AC =
3
10 AC
AXm-b-Xv, AB =
3
10 AC
CXnÂn¶v
AB : AC = 3 : 10
F¶p ImWm-atÃm. CXpw atä-sX-¦nepw coXn-bnÂ
sN¿mtam?
Cn Cu IW-¡p-IÄ sNbvXp tm¡q.
$ Hcp {]tZ-is¯ nc-£-c-cpsS F®hpw km£-c-cpsS
F®hpw X½n-epÅ Awi-Ôw 1 : 19 BWv.
BsI P-kwJy 64000 BsW-¦n nc-£-c-cpsS
F®-sa{X? km£-c-cpsS F®-sa{X?
$ Hcp ]ip-h-fÀ¯Â tI{µ-¯nse ]ip-¡-fn Id-h-
bp-Å-hbp-tSbpw Idh-bn-Ãm-¯-h-bptSbpw F®w
X½n-epÅ Awi- Ôw 8 : 3 BWv. Id-h-bn-Ãm-¯-
h-bpsS F®w 144 BsW-¦n Id-h-bp-Å-h-bpsS
F®-sa{X? BsI ]ip-¡-fpsS F®-sa{X?
$ Hcp kvIqfnse B¬Ip-«n-I-fpsS F®hpw s]¬Ip-
«n-I-fpsS F®hpw 14 : 15 F¶ Awi- -Ô-¯n-emWv.
s]¬Ip-«n-I-fpsS F®w B¬Ip-«n-I-fpsS
F®-t¯-¡mÄ 27 IqSp-X-em-Wv. F¦n B¬Ip-
«n-I-fpsS F®-sa{X? s]¬Ip-«n-I-fpsS F®-sa{X?
$ Hcp NXp-c-¯nsâ ofhpw hoXnbpw 8 : 5 F¶
Awi- -Ô-¯n-emWv. ofw hoXn-tb-¡mÄ 10.5
aoäÀ IqSp-X-emWv. NXp-c-¯nsâ ofhpw hoXnbpw
F{X-bmWv?
$ Hcp tbmK-¯n ]s¦-Sp¯ ]pcp-j³am-cp-tSbpw
kv{XoI-fp-tSbpw Awi-Ôw 3 : 5 BWv. Iptd¡gn-
ª-t¸mÄ ]pcp-j-·m-cn ]IpXnt]cpw kv{XoI-fn
 aq¶n-sem¶p t]cpw Xncn-sI t]mbn. Ct¸mÄ
Zimwi-kw-Jy-IÄ
Hcp NXp-c-¯nsâ ofw 2.5 aoädpw hoXn
1.5 aoä-dpamWv. ofhpw hoXnbpw GXv
Awi--Ô-¯n-emWv?
0.5 aoä-dnsâ 5 aS-§mWv ofw; hoXn
3 aS§pw. At¸mÄ ofhpw hoXnbpw 5 : 3
F¶ Awi--Ô-¯n-em-Wv.
asämcp coXn-bn-epw -B-tem-Nn¡mw Cu NXp-c-
¯n ofw, hoXn-bpsS
2.5
1.5
aS§m-sW¶v
]d-bmw.
2.5
1.5
=
25
15
=
5
3
BW-tÃm. At¸mÄ Awi-Ôw 5:3
ofw 3.5 aoä-dpw, hoXn 2.25 aoä-dp-amb NXp-c-
¯nsâ ofhpw hoXnbp w X½n-epÅ
Awi-Ôw F´mWv?

14. KWnXw þ ¢mkv 8
34
Cu Nn{Xw tm¡q:
CuÀ¡n sIms­mcp {XntIm-Ww. CXnsâ hi-§Ä
8 skân-ao-äÀ, 4 skân-ao-äÀ, 6 skân-ao-äÀ F¶n-§-s-
bm-Wv.
At¸mÄ {XntImWw D­m-¡m³ F{X of-apÅ
CuÀ¡n-emWv D]-tbm-Kn-¨Xv?
8 + 4 + 6 = 18, AtÃ?
Xmgs¯ hiw CXnsâ F{X `mK-amWv?
8
18 =
4
9
aäp- c-­p-h-i-§tfm?
4
18 =
2
9
6
18 =
3
9
At¸mÄ hi-§Ä Ah-bpsS XpI-bpsS
4
9 `mKw,
2
9
`mKw,
3
9 `mKw F¶n-§-s-bm-Wv.
Awi--Ôhpw hnkvXoÀWhpw
Htc Npä-f-hpÅ c­p NXp-c-§fn H¶nsâ
ofhpw hoXnbpw X½n-epÅ Awi- Ôw
2 : 1. c­m-a-t¯-Xnsâ ofhpw hoXnbpw
X½n-epÅ Awi- Ôw 3 : 2. GXn-mWv
]c-¸-fhv IqSp-XÂ?
Npä-fhv Xpe-y-am-b-Xn-m hoXn-bp-sSbpw
of-¯nsâbpw XpI Xpe-y-am-Wv. CXv s
Fs¶-Sp-¯m BZ-ys¯ NXp-c-¯nsâ hi-
§Ä
1
3 s,
2
3 s. AXn-m ]c-¸-fhv
2
9 s2.
c­m-as¯ NXp-c-¯nsâ ]c-¸-fthm?
2 2 3 6
5 5 25
s´ s = s .
2
9 ,
6
25 Ch-bn hep-tX-XmWv?
2 ´ 25 < 6 ´ 9 Bb-Xn-mÂ
2
9 <
6
25 .
At¸mÄ c­m-as¯ NXp-c-¯n-mWv IqSp-
XÂ ]c-¸p-Å-Xv.
Cn CtX Npä-fhpw hi-§-fpsS Awi-
_Ôw 1 : 3 Dw Bb NXp-c-sa-Sp-¯mtem?
GXn-mWv ]c-¸-fhv IqSp-XÂ?
Cu NXp-c-§-fp-sS-sbÃmw ofhpw hoXnbpw
X½n-epÅ hy-X-ymkw I­p-]n-Sn¨p tm¡q.
Htc Npä-f-hpÅ Iqtd¡qSn NXp-c-§-sf-Sp¯p
]cn-tim-[n-¡q.
]pcp-j-·m-cp-sSbpw kv{XoI-fp-sSbpw F®w X½n-ep
Å Awi- Ôw F{X-bmWv?
$ Hcp kvIqfn temhÀ ss{]a-dn-bnepw A¸À ss{]-
a-dn-bnepw ]Tn-¡p¶ Ip«n-I-fpsS F®w X½n-epÅ
A wi- Ô w 2 : 3 BWv.- A-¸À ss{]a-dn-bnepw
sk¡âdnbnepw ]Tn-¡p¶ Ip«n-I-fpsS F®w X½n-ep
Å Awi- Ôw 4 : 5 BWv. temhÀ ss{]a-dn-bnepw
sk¡âdnbnepw ]Tn-¡p¶ Ip«n-I-fpsS
F®w X½n-epÅ Awi- Ôw F´mWv?
$ ^m¯n-a, KwK, loc F¶n-hÀ tNÀ¶v c­p- ]m¡äv
anTm-bn-hm-§n. 140 anTmbnIfp-­m-bn-cp-¶p. AhÀ
3 t]cpw IqSn AXv hoXn-¨p. ^m¯na FSp¯ anTm-bn-
I-fpsS F®hpw KwK FSp¯ anTm-bn-I-fpsS
F®hpw X½n-epÅ Awi- Ôw 3 : 4 BWv. KwK
FSp¯ anTm-bn-I-fpsS F®hpw loc FSp-¯-hbpw
6 : 7 F¶ Awi- -Ô-¯n-em-Wv. F¶m Hmtcm-cp-
¯À¡pw In«nb anTm-bn-I-fpsS F®w F{X?
Af-hp-IÄ aq¶m-bmÂ

15. Awi-Ôhpw Ap-]m-Xhpw
35
CXp asämcp coXn-bn¸-d-bmw.
hi-§-fpsS Awi- Ôw 4 : 2 : 3 BWv.
Iptd-¡qSn IrXy-am-bn-¸-d-bm³ {XntIm-W¯np
t]cnSmw:
C
6 sk.ao.
4 sk.ao.
A B
8 sk.ao.
Cu {XntIm-W-¯n AB, BC, CA Ch-bpsS ofw X½n-ep
Å Awi- Ôw 4 : 2 : 3 BWv.
asämcp coXn-bnepw CXp ImWmw. 2 skân-ao-äÀ of-ap
Å Hcp Nc-Sp-sIm­v Af-¶mÂ, AB, BC, CA Ch-bpsS
ofw 4, 2, 3 F¶p In«pw.
Cn Cu {XntImWw tm¡q:
R
3.5 sk.ao.
2.5 sk.ao.
P Q
4 sk.ao.
Awi-Ôw amdn-bmÂ
Cu t^mt«m tm¡q:
CXnsâ hoXn 1.5 skân-ao-ä-dpw, Dbcw 2.25
skân-ao-ä-dp-am-Wv. AXm-bXv hoXnbpw Db-chpw
X½n-epÅ Awi- Ôw 2 : 3.
ofhpw hoXnbpw Cc-«n-bm¡n-bmtem?
hoXn 3 skân-ao-ä-dpw, Dbcw 4.5 skân-ao-ä-
dpw. Awi-Ôw ]g-bXp Xs¶.
Cn hoXn 3 skân-ao-äÀ B¡p-t¼mÄ Db-chpw
Cc-«n-bm¡p-¶-Xnp ]I-cw, Db-chpw
1.5 skân-ao-äÀ Iq«n 3.75 skân-ao-äÀ
B¡n-bmtem?
Awi-Ôw 4 : 5 Bbn.
Nn{X-¯n-s´p kw`-hn¨p?
hi-§Ä X½n-epÅ Awi- Ôw F´mWv?
hi-§-fpsS XpI (A-Xm-b-Xv, Npä-f-hv) F{X-bmWv?
PQ + QR + RP = 4 + 3.5 + 2.5 = 10 sk.ao.
Hmtcm hihpw CXnsâ F{X `mK-amWv?
4
10 =
2
5
3.5
10 =
35
100 =
7
20
2.5
10 =
25
100 = 1
4
Cu `n¶-§-fp-sS-sbÃmw tOZw Xpey-a-Ã-tÃm. ]ns¶-
§s Awi- Ôw I­p-]n-Sn¡pw?

16. KWnXw þ ¢mkv 8
CXnse ABD, ACDF¶o {XntIm-W-§-fpsS
]c-¸-f-hp-IÄ X½n-epÅ Awi- -Ôw F-
´mWv?
A bn n¶v BC bntebv¡v ew_w hc-bv
¡pI
36
tOZ-§Ä Xpey-am-¡-Ww. 20 B¡n-bmtem?
2
5 =
8
20
1
4 =
5
20
At¸mÄ, PQ, QR, RP Ch-bpsS ofw Npä-f-hnsâ
8
20 ,
7
20 ,
5
20 F¶n-§-s-bm-Wv. AXm-b-Xv,
PQ : QR : RP = 8 : 7 : 5
asämcp coXn-bnepw CXv I­p-]n-Sn¡mw:
1
2
(A-sÃ-¦nÂ
0.5) skân-ao-äÀ of-apÅ Nc-Sp-sIm­v Af-¶m PQ,
QR, RP Ch-bpsS ofw 8, 7, 5 F¶m-bn-cn-¡pw.
asämcp DZm-lcWw tm¡mw: Ben 40000 cq]bpw
tPmkv 20000 cq]bpw tPm¬ 50000 cq]bpw apX-en-d
¡n Hcp GP³kn Bcw-`n-¨p. ChÀ apXÂ apS-¡n-bXv
GXv Awi- -Ô-¯n-emWv?
BsI apS-¡n-bXv 40000 + 20000 + 50000 = 110000 cq]-
bm-W-tÃm. AXn Ben-bp-tSXv F{X `mKw?
40000
110000 =
4
11
tPmk-n-tâtXm?
20000
110000 =
2
11
tPmWn-tâXv?
50000
110000 =
5
11
At¸mÄ Ben, tPmkv, tPm¬ Ch-cpsS apS-¡p-ap-XÂ
4 : 2 : 5 F¶ Awi_-Ô-¯n-em-Wv.
asämcp coXn-bnepw CXp ImWmw. ]Xn-m-bncw cq]
h¨p IW-¡m-¡n-bmÂ, Ben-bpsS apS-¡p-ap-XÂ 4 ]Xn-
m-bn-cw, tPmk-^n-tâXv 2 ]Xn-m-bn-cw, tPmWn-tâXv
5 ]Xn-m-bncw F¶n-§-s-bm-W-tÃm.
Cn Cu IW-¡p-IÄ kzbw sNbvXp-tm¡q:
$ Hcp aÕ-c-¯nse H¶m-av 1000 cq]-bpw, c­m-av
600 cq]-bpw, aq¶m-av 400 cq]-bp-amWv k½m-w.
k½m--¯p-I-IÄ X½n-epÅ Awi-_Ôw
F´mWv?
hnkvXoÀW_Ôw
Nn{Xw tm¡q.
A
B D C
A
B D C
A
B D P C
Cu ew_-¯nsâ ofw h Fs¶-Sp-¯mÂ
DABD bpsS ]c-¸-fhv
=
1
2 h ´ BD
DACD bpsS ]c-¸-fhv
=
1
2
h ´ CD
At¸mÄ
DABD bpsS ]c-¸-fhv
DACD bpsS ]c-¸-fhv
=
BD
CD
AXm-bX,v Cu ]c-¸-f-hp-I-fpsS Awi-Ôw
BD, CD F¶o of-§-fpsS Awi- Ôw
Xs¶-bm-Wv.
At¸mÄ, Hcp {XntIm-Ws¯ Htc ]c-¸-f-hp
Å c­p {XntIm-W-§-fmbn `mKn-¡p-¶-
sX-§s?
Hcp `mK-¯nsâ ]c-¸v, c­m-as¯ `mK-
¯nsâ ]c-¸nsâ Cc-«n-bm-I-W-sa-¦ntem?

17. Awi-Ôhpw Ap-]m-Xhpw
37
$ 10 aoäÀ Npä-f-hpÅ Hcp {XntIm-W-¯nsâ c­p
hi-§Ä 1
2
2 aoädpw 1
3
2 aoä-dp-am-Wv. Cu {XntIm-
W-¯nsâ aq¶p hi-§-fp-tSbpw ofw X½n-epÅ
Awi- Ôw F´mWv?
$ 1 Intem{Kmw Acnbpw 250 {Kmw ]ghpw 750 {Kmw
iÀ¡-cbpw tNÀ¯v D®n-b¸w D­m-¡n. tNcp-h-
IÄ X½n-epÅ Awi- Ôw F´mWv?
aäp Nne DZm-l-c-W-§Ä tm¡mw:
Hcp {XntIm-W-¯nsâ Npä-fhv 60 skân-ao-ä-dm-Wv. hi-
§Ä 3 : 4 : 5 F¶ Awi- -Ô-¯n-em-Wv. hi-§-fpsS
of-sa{X?
hi-§-fpsS Awi- Ôw 3 : 4 : 5 Bb-Xn-mÂ, hi-§-
fpsS ofw Npä-f-hnsâ 3
12 `mKw, 4
12 `mKw, 5
12 `mKw
F¶n-§sbmW-tÃm. Npä-fhv 60 skân-ao-ä-dm-Wv.
At¸mÄ hi-§-fpsS ofw
60 ´ 3
12
= 15
60 ´ 4
12
= 20
60 ´ 5
12
= 25
skân-ao-äÀ hoX-am-Wv.
atä-sX-¦nepw coXn-bn CXp sN¿mtam?
hi-§Ä Hcp nÝnX Af-hnsâ 3, 4, 5 aS-§m-WtÃm.
At¸mÄ Npä-fthm?
Cn Cu IW¡p tm¡q:
hnP-bpw tKm]pw apIp-µpw Hcp Icm-dp-]Wn Gsä-
Sp-¯p. hnP-b³ 3 Znh-k-hpw, tKm]³ 5 Znh-k-hpw, apIp-
µ³ 6 Znh-k-hp-amWv tPmen sNbvX-Xv. Icm-dp-XpI hoXn-
¨-t¸mÄ tKm]p hnP-b-t-¡mÄ 500 cq] IqSp-XÂ
In«n. Icm-dp-XpI F{X-bmWv? Hmtcm-cp-¯À¡pw F{X
cq] In«n?
{XntIm-W-a[yw
Hcp {XntImWw hc-¨v, AXnsâ hi-§-fp-sS-
sbÃmw a[y- n-µp-¡Ä AS-bm-f-s¸-Sp-
¯pI:
A
F E
B C
D
Cn Cu a[y- n-µp-¡Ä Hmtcm-¶n-tbpw
FXnÀioÀj-hp-ambn tbmPn-¸n-¡pI:
A
F E
B C
D
Cu hc-Isf {XntIm-W-¯nsâ a[y-a-tc-J-I
Ä (medians) F¶mWv ]d-bp-¶-Xv. Cu
aq¶p a[y-a-tcJ-Ifpw {XntIm-W-¯n--Is¯
Hcp _nµp-hn¡qSn IS-¶p-t]m-Ip-¶ntÃ?
A
F E
B C
D
G
Cu _nµp-hnv {XntIm-W-¯nsâ a[y-nµp
(centroid) F¶mWv t]cv.
Cu _nµp a[y-a-tc-J-I-sf-sbÃmw 2 : 1 F¶
Awi--Ô-¯n-emWv `mKn-¡p-¶-Xv. AXm-b-
Xv, ½psS Nn{X-¯nÂ
AG
GD
=
BG
GE
=
CG
GF
Cu _nµp-hnv asämcp {]tXy-I-X- Iq-Sn-bp-
­v. CXp-t]m-semcp Nn{Xw ImÀUvt_mÀ
Un hc¨v sh«n-sb-Sp-¡p. Cu _nµp-hnÂ
s]³knÂap- h-¨v {XntIm-Ws¯
NmbmsX, Ncn-bmsX nÀ¯mw.
AXm-bXv {XntIm-W-¯nsâ a[y- n-µp,
AXnsâ Kpcp-Xzm-IÀj-W-tI{µw (centre of
gravity) BWv.

18. KWnXw þ ¢mkv 8
38
tPmen sNbvX Znh-k-§-fpsS Awi- Ôw 3 : 5 : 6
BW-tÃm. Cu Awi- -Ô-¯n¯s¶-bmWv XpI
hoXn-¡p-¶-Xpw. At¸mÄ Hcp nÝn-X -Xp-I-bpsS
3 aS§v hnP-bpw 5 aS§v tKm]pw 7 aS§v apIp-µpw
In«n.
At¸mÄ Cu XpI-bpsS 2 aS-§mWv tKm]v hnP-b-t-
¡mÄ IqSp-XÂ In«n-b-Xv.
CXv 500 cq]-bm-sW¶v ]d-ªn-«p-­-tÃm.
B nÝn-X-XpI
1
2
´ 500 = 250 cq]-bm-Wv. Cn Hmtcm-cp-
¯À¡pw In«n-bXv F{X-bm-sW¶v I­p-]n-Sn-¡m-atÃm:
hnP-bp In«n-bXv = 3 ´ 250 = 750 cq
tKm]p In«n-bXv = 5 ´ 250 = 1250 cq
apIp-µp In«n-bXv = 6 ´ 250 = 1500 cq
CXp-t]mse Cu IW-¡p-Ifpw sN¿m-a-tÃm.
$ Hcp {XntIm-W-¯nsâ tIm-Wp-IÄ 1 : 3 : 5 F¶
Awi--Ô-¯n-em-Wv. Hmtcm tImWpw F{X-bmWv?
$ shSn-a-cp-¶p-­m-¡m³ ImÀ_¬, kÄ-^À,
s]m«m-kyw sst{Säv Ch 3 : 2 : 1 F¶ Awi- -
Ô-¯n-emWv tNÀt¡-­-Xv. 1.2 Intem{Kmw shSn-a-cp-
¶p-­m-¡m³ Ch Hmtcm¶pw F{X hoXw
thWw?
$ Hcp {Xn-tIm-W-¯nsâ hi-§-Ä 2 : 3 : 4 F¶
Awi- -Ô-¯n-em-Wv. Gähpw henb hi-¯nv
Gähpw sNdnb hi-t¯-¡mÄ 20 skân-ao-äÀ ofw
IqSp-X-em-Wv. Hmtcm hi-¯n-tâbpw ofw I­p-]n-
Sn-¡p-I.
$ GsX-¦nepw {XntIm-W-¯nsâ hi-§-fpsS ofw
1 : 2 : 3 F¶ Awi--Ô-¯n BIptam? F´p-sIm
­v?
$ ABC F¶ {XntIm-W-¯n AB : BC = 2 : 3 Dw
BC : CA = 4 : 5Dw BWv. AB : BC : CA F´mWv?
A[ym-]-I-hn-ZymÀYn Awi-Ôw
A[-ym-]I hnZ-ymÀYn Awi-Ôw F¶Xv
Hcp cmP-y-¯nse hnZ-ym-`-ym-k-s¯-¡p-dn-¨pÅ
Hcp {][m kqNn-I-bm-Wv. 2005 se IW-
¡-p-k-cn¨v, tIc-f-¯n-se -kvIq-fp-I-fn CXv
1 : 27 BWv. AXm-b-Xv,- i-cm-icn
27 hnZymÀYn-IÄ¡v 1 A[-ym-]-I³ hoX-ap-
­v. C´y sam¯-am-sb-Sp-¯m Cu Awi-
_Ôw GXm­v 1 : 40 BWv.

19. Awi-Ôhpw Ap-]m-Xhpw
39
$ tamfn, ^o-k, Hma F¶n-hÀ tNÀ¶v Hcp
X¿Â¡S XpS-§n. AhÀ apS-¡nb XpI 5 : 7 : 8
F¶ Awi- -Ô-¯n-em-Wv. Hcp hÀjs¯ em`w
AhÀ CtX Awi--Ô-¯n hoXn-s¨-Sp-¯p. Hma-
¡v tamfn-tb-¡mÄ 10,800 cq] IqSp-XÂ In«n.
F¶m Hmtcm-cp-¯À¡pw In«nb XpIsb{X?
amdp-¶Xpw amdm-¯Xpw
Nn{X-§Ä tm¡q:
Htc Nn{Xw Xs¶ c­p hen-¸-¯n AtÃ?
BZys¯ Nn{X-¯nsâ hoXn 1 skân-ao-ädpw Dbcw 1.5
skâo-ao-ä-dp-am-Wv.
c­m-as¯ Nn{X-¯ntem?
hoXn 2 skân-ao-ädpw Dbcw 3 skânao-ä-dpw.
AXmbXv hoXnbpw Db-chpw Cc-«n-¨p.
ChnsS Nn{X-§-fpsS hoXnbpw Db-chpw amdn.
amdm-Xn-cn-¡p-¶Xv F´mWv?
BZys¯ Nn{X-¯nÂ, hoXn-bpsS 3
2 aS-§m-WtÃm
Dbcw.
c­m-as¯ Nn{X-¯ntem?
CXv `n¶-§Ä D]-tbm-Kn-¡msX ]d-bmtam?
c­p Nn{X-¯nepw hoXnbpw Db-chpw X½n-epÅ Awi-
_Ôw 2 : 3 Xs¶-btÃ?
Cu Nn{X-¯ntem?
Awi--Ôhpw ck-X-{´hpw
ck-X-{´-¯n ]ZmÀY-§sf aqe-I-§Ä
F¶pw kwbp-à-§Ä F¶pw Xcw Xncn-¨n-
«p-­-tÃm.
GXp kwbp-à-¯nepw AXn-e-S-§p¶ aqe-
I-§-fpsS {Zh-y-amw (mass) Hcp nÝnX
Awi--Ô-¯n-em-bn-cn-¡p-sa¶v ]Xn-s«mw
qäm-­n tPmk^v {]qÌv F¶ imkv{X-
Ú³ Is­¯n.
DZm-l-c-W-ambn tIm¸À ImÀ_-tW-änÂ
Ft¸mgpw ImÀ_-tW-änsâ {Zh-y-am--¯nsâ
5.3 aS§v tIm¸-dpw, 4 aS§v HmIvkn-Ppw
Bbn-cn¡pw F¶v ]co-£-W-§-fn-eqsS
At±lw Is­¯n.
aqe-I-§-fpsS Xosc sNdnb IWn-I-IÄ
k¦-ev]n-¨m C¯cw Xmc-X-ayw F®Â
kwJ-y-I-fn-eqsS Bhmw F¶ Nn´-bm-IWw,
]c-amWp F¶ Bi-b-¯n-te¡v bn-¨-Xv.
]s¯m-¼Xmw qäm-­n tPm¬ Um«³
F¶ imkv{X-Ú-mWv C¯cw Hcp
kn²m´w Ah-X-cn-¸n-¨-Xv.
Um«sâ kn²m-´-a-p-k-
cn¨v aqe-I-§-fpsS
Xosc sNdnb IWn-I-I-fmb
]c-am-Wp¡Ä
(atoms ) tN˦m-Wv
kwbp-à-§Ä D­m-Ip-
¶-Xv. GXp kwbp-à-
¯nepw AXnse hnhn[
aqe-I-§-fpsS ]c-am-Wp-
¡-fpsS F®w Hcp
nÝnX Awi- -Ô-¯n-em-
Wv.

20. KWnXw þ ¢mkv 8
40
asämcp IW¡p tm¡q:
A¸p-hnsâ A½ CÍen D­m-¡p-¶-Xnv km[m-c-W-bmbn
2 I¸v Acnbpw 1 I¸v Dgp-¶p-amWv FSp-¡p-¶-Xv.
hncp-¶p-ImÀ D­m-bn-cp¶ Hcp Znhkw 3 I¸v Acnbpw
1
1
2 I¸v Dgp-¶p-amWv FSp-¯-Xv.
ChnsS Acn-bpsS Afhv 1
1
2 aS-§m-¡n. Dg-p¶nsâ
Afthm?
c­-h-k-c-¯n-epw Acn-bp-sSbpw Dg-p-¶n-sâbpw Af-hp-
IÄ X½n-epÅ Awi- Ôw F´mWv?
C§s Awi- Ôw amdmsX Af-hp-IÄ amdp-¶-Xnv
KWn-X-¯n-semcp t]cp­v: Ap-]mXw (proportion).
At¸mÄ, Nn{X-¡-W-¡nÂ, BZys¯ c­p NXp-c-§-
fpsS hoXnbpw Db-chpw Bp-]m-Xn-I-amWv (proportional).
CÍ-en-¡-W-¡ntem?
c­-h-k-c-§-fn-ep-apÅ Acn-bp-sSbpw Dgp-¶n-sâbpw
Afhv Bp-]m-Xn-I-am-Wv.
aq¶-f-hp-IÄ Bp-]m-Xn-I-am-hp¶ kµÀ`-§-fpap­v.
Cu Nn{X§Ä tm¡q:
6 sk.ao.
8 sk.ao.
3 sk.ao
4 sk.ao
4 sk.ao.
2 sk.ao
Hmtcm {XntIm-W-¯nepw hi-§-fpsS of-¯n-sâ Awi-
_Ôw F´mWv?
Awi- Ô]-cn-tim-[
Nn{X-¯nse c­p NXp-c§-fp-tSbpw hi-
§Ä Xa-an-epÅ Awi-Ôw Xpey-amtWm?
1.2 sk.ao.
1.5 sk.ao.
1.6 sk.ao.
2 sk.ao.
BZys¯ NXp-c-¯nÂ, sNdnb hi-¯nsâ
1.5
aS-§mWv henb hiw. c­m-as¯
1.2
NXp-c-¯ntem? sNdnb hi-¯nsâ
2
1.6
aS-
§mWv henb hiw.
Awi-_Ôw Xpey-am-I-W-sa-¦n Cu
kwJy-IÄ Xpey-am-I-W-a-tÃm.
Xpey-amtWm? ]cn-tim-[n¨p tm¡q.
asämcp coXn-bnepw CXv Btem-Nn-¡mw.
c­m-as¯ NXp-c-¯nsâ henb hiw
BZys¯ NXp-c-¯nsâ henb hi-¯nsâ
2
1.5
aS-§m-Wv. sNdnb hitam? c­m-as¯
NXp-c-¯nsâ sNdnb hiw BZys¯ NXp-c-
¯nsâ sNdnb hi-¯nsâ
1.6
1.2
aS-§mWv.
Awi-Ôw Xpey-am-sW-¦n Chbpw Xpey-am-
It­?
`n¶-§Ä eLp-I-cn¨v tm¡q.
BZyw In«nb `n¶-kwJy F´mWv ImWn-
¡p-¶Xv?
c­p NXp-c-§-fnepw sNdnb hi-¯nsâ
5
4
aS-§mWv henb hiw.
c­m-aXp In«nb `n¶-kw-Jytbm?
BZys¯ NXp-c-¯nsâ hi§-fpsS
4
3
aS-
§mWv c­m-as¯ NXp-c-¯nsâ hi-§Ä.

21. Awi-Ôhpw Ap-]m-Xhpw
41
Nph-sS-bpÅ Hmtcm tPmSn NXp-c-¯nepw hi-§-fpsS
ofw Bp-]m-Xn-I-am-bh GsXm-s¡-bm-sW¶v Duln-
¡p-I. Af¶p tm¡n Dulw icn-bmtWm F¶v ]cn-tim-[
n-¡p-I.
Cn Iptd {XntIm-W-tPm-Sn-I-fm-Is«.
_oP-K-Wn-X-k-lmbw
c­p NXp-c-§-fpsS hi-§Ä Bp-]m-Xn-I-amtWm
F¶v F-§s ]cn-tim-[n¡pw?
BZy-t¯-Xnsâ hi-§-fpsS ofw a, b F¶n-cn-
¡s«. c­m-a-t¯-Xn-tâXv p, q F¶pw.
a
b
p
q
Hmtcm NXp-c-¯nepw Hcp hiw atä hi-
¯nsâ F{X aS§v (AsÃ-¦n `mKw)
BsW¶p tm¡-Ww. AXm-bXv
a
b
,
p
q
Ch Xpey-amtWm F¶p tm¡-Ww.
AsÃ-¦n c­p NXp-c-§-fn-tebpw sNdnb
hi-§fpw henb hi-§fpw Htc aS§v
(`mKw) BtWm F¶v tm¡Ww.
AXm-b-Xv,
a
b
=
p
q BtWm F¶v tm¡-
Ww.
a
b
=
p
q BI-W-sa-¦nepw
a
p =
b
q BI-W-
sa-¦nepw aq = bp BI-W-a-sÃm. adn¨v,
Cu KpW--^-e-§Ä Xpey-am-bm `n¶-
§fpw Xpey-am-Ipw. At¸mÄ Bp-]m-Xn-I-amtWm
F¶p tm¡m³ Cu KpW--^-e-
§Ä ]cn-tim-[n-¨m aXn.
hi-§Ä 2.5 skân-ao-ädpw 1.5 skân-ao-ädpw
Bb Hcp NXpcw; hi-§Ä 2 skân-ao-ädpw
1.2 skân-ao-ädpw Bb asämcp NXpcw. hi-
§-fpsS ofw Bp-]m-Xn-I-amtWm?

22. KWnXw þ ¢mkv 8
42
ap¡v Is­¯mw
$ amXyp 30000 cq]bpw Ìo^³ 50000 cq]bpw apS¡n
Iq«p-I-¨-hSw Bcw-`n-¨p. Hcp amkw-sIm­v 2400 cq]
BZmbw In«n. amXyp 900 cq]bpw Ìo^³ 1500
cq]bpw BZmbw hoXn-s¨-Sp-¯p. apS-¡p-ap-X-ep-IÄ
X½n-epÅ Awi- Ôw F{X? hoX-§-fpsS Awi-
_-Ôtam? apS-¡p-ap-Xepw hoXhpw Bp-]m-Xn-I-amtWm?
$ cmap 8 aWn-¡qÀ tPmen sNbvX-t¸mÄ 400 cq]
In«n. s_¶n 6 aWn-¡qÀ tPmen sNbvXp 300 cq]
In«n. tPmen ka-b-¯nv Bp-]m-Xn-I-ambmtWm
Iqen- In-«n-bXv?
$ 10 enäÀ oebpw 15 enäÀ shÅbpw IeÀ¯nb
Nmb-¡q-«npw 12 enäÀ oebpw 17 enäÀ shÅbpw
IeÀ¯nb Nmb-¡q-«npw Htc nd-am-bn-cn-¡ptam?
F´p-sIm­v?
amä-¯nse ØncX
3 skân-ao-äÀ hi-apÅ Hcp ka-N-Xp-c-¯nsâ Npä-fhv
F{X-bmWv?
5 skân-ao-äÀ hi-apÅ Hcp ka-N-Xp-c-¯nsâ Npä-fthm?
hi-¯nsâ ofw F´m-bmepw AXnsâ mep-a-S-§m-
WtÃm ka-N-Xp-c-¯nsâ Npä-f-hv. CXp thsdmcp coXn-bnepw
]dbmw:
ka-N-Xp-c-¯nsâ hi-¯nsâ ofhpw Npä-fhpw X½n-ep
Å Awi- Ôw 1 : 4 BWv.
hi-¯nsâ ofw amdp-¶-Xn--p-k-cn¨v ka-N-Xp-c-¯nsâ
Npä-fhpw amdpw; ]t£ Ch X½n-epÅ Awi- Ôw
amdp-¶n-Ã. At¸mÄ ap¼p ]dª Imcyw asämcp coXn-bnepw
]d-bm-atÃm:
ka-N-Xp-c-§-fnÂ, hi-¯nsâ ofhpw Npä-fhpw Bp-
]m-Xn-I-am-Wv.
ka-N-Xp-c-¯np ]Icw ka-`pP-{Xn-tImWw FSp-¯mepw
CXp icn-btÃ?
asämcp IW¡p tm¡mw: Hcp s]³kn-env 1
1
2
cq]bmWv hne. C¯-c-¯nepÅ 10 s]³kn-epI-fpsS
Bp]m-Xn-I-a-Ãm¯ hfÀ¨
c­v Af-hp-IÄ Ap-]m-Xn-I-am-sW-¦nÂ,
AXn Hc-fhv IqSp-¶-Xn--p-k-cn¨v atä
Afhpw IqSpw. F¶m H¶p IqSp-¶-Xn--
p-k-cn¨v asäm¶v IqSp¶p F¶Xp sIm­p
am{Xw c­-f-hp-IÄ Bp-]m-Xn-I-amIW-sa-
¶n-Ã.
DZm-l-c-W-ambn ka-N-Xp-c-¯n hi-¯nsâ
ofw IqSp-¶-Xn--p-k-cn¨v ]c-¸-fhpw IqSpw.
]t£, 1 skân-ao-äÀ hi-apÅ ka-N-Xp-c-
¯nsâ ]c-¸-fhv 1 NXp-c{i skân-ao-ädpw
2 skân-ao-äÀ hi-apÅ ka-N-Xp-c-¯nsâ
]c-¸-fhv 4 NXp-c-{i-skân-ao-ä-dp-am-WtÃm;
BZys¯ NXp-c-¯n hi-¯nsâ ofhpw
]c-¸-fhpw X½n-epÅ Awi- Ôw 1 : 1 Dw
c­m-a-t¯-Xn CXv 2 : 4 = 1 : 2 Dw BWv.
c­p kµÀ`-§-fn-sebpw Awi--Ô-§Ä
Xpey-a-Ãm-¯-Xn-m Cu amäw Bp-]m-Xn-
I-a-Ã.
ka-N-Xp-c-¯nsâ hihpw Npä-fhpw Bp-
]m-Xn-I-amtWm?

23. Awi-Ôhpw Ap-]m-Xhpw
43
hnesb´v? 20 s]³kn-ep-I-fp-tStbm? s]³kn-ensâ
F®hpw BsI hnebpw Bp-]m-Xn-I-amtWm?
Cu DZm-l-c-W-§-sfÃmw _oP-K-Wn-X-cq-]-¯n FgpXn
tm¡mw:
$ ka-N-Xp-c-¯nsâ hi-¯nsâ ofw x skânao-ädpw
Npä-fhv y skân-ao-ädpw BsW-¦nÂ
y = 4x
$ ka-`p-P-{Xn-tIm-W-¯nsâ hi-¯nsâ ofw
x skânaoädpw Npä-fhv y skân-ao-ädpw BsW-¦nÂ
y = 3x
$ s]³kn-ensâ F®w x Dw BsI hne y cq]bpw
BsW-¦nÂ
y =
3
2
x
s]mXpth ]d-ªm x, y F¶ c­-f-hp-IÄ Bp-]m-
Xn-I-am-bmWv amdp-¶-sX-¦nÂ, hyXykvX kµÀ`-§-fn-ep
Å Ah-bpsS hne-IÄ X½n-epÅ _Ôw
y = kx
F¶m-bn-cn¡pw. ChnsS k F¶Xv x sâbpw y bptSbpw
hne-IÄ amdp-¶-Xn--p-k-cn¨v amdp-¶n-Ã. CXns Cu
amä-¯nse Bp-]m-Xn-I-Øncw (constant of
proportionality) F¶mWv ]d-bp-¶Xv.
½psS BZys¯ DZm-l-c-W-¯nse Bp-]m-Xn-I-Øn-c-
amb 4 F¶ kwJy hi-§-fpsS F®-amWv. c­m-as
¯ DZm-l-c-W-¯nse 3 Dw hi-§-fpsS F®w Xs¶.
aq¶m-as¯ DZm-l-c-W-¯n Bp-]m-XnIØncw
3
1
= 1
BWv. CXv Hcp s]³kn-ensâ hne-bm-Wv.
2
2 $ cmP³ 10000 cq], 6% km[m-cW ]eni In«p¶
_m¦n nt£-]n-¡p-¶p. XpI _m¦n- nt£-
]n¨ hÀj-§-fpsS F®-t¯bpw BsI In«p¶
]en-i-tbbpw _Ôn-¸n-¡p¶ ka-hmIyw Fgp-XpI.
BsI In«p¶ ]en-i, hÀj-§-fpsS F®-¯nv
Bp-]m-Xn-I-amtWm? km[m-cW ]en-ibv¡p
]Icw Iq«p-]-en-i-bm-bmtem?
`mc-a-f-¡m³ Ap-]mXw
`mc-a-f-¡m³ D]-tbm-Kn-¡p¶
kv{]n§v{Xmkv I­n«ntÃ?
sImfp-¯n `mcw Xq¡p-t¼mÄ
{Xmkn-p-Ånse kv{]n§v Xmtgm«v
hen-bp-¶p. kv{]n§n LSn-¸n-¨n-
«pÅ Hcp kqNnI, {Xmknsâ Hcp
hi-¯pÅ kvsIbn-en `mcw
ImWn-¡p-¶p.
F´mWn-Xnsâ XXzw? `mcw Xq¡p-t¼mÄ
kv{]n§n-p-­m-Ip¶ hen-hnsâ ofw `mc-
¯nv Bp-]m-Xn-I-am-sW¶v, ]Xn-tgmw
qäm-­n Cw¥-­n Pohn-¨n-cp¶ tdm_À«v
lqIv F¶ imkv{X-Ú³ Is­-¯n.
CXp-]-tbm-Kn-¨v, Cu {Xmkn `mc-§Ä AS-bm-
f-s¸-Sp-¯m³ Ffp-¸-am-Wv. sImfp-¯nÂ
`mc-sam¶pw CÃm-¯-t¸mÄ kqNn-I-bpsS
Ømw AS-bm-f-s¸-Sp-¯p-¶p. ]n¶oSv Hcp
nÝn-X-`mcw Xq¡p-t¼mgpÅ Ømhpw
AS-bm-f-s¸-Sp-¯p-¶p. DZm-l-c-W-ambn
1 Intem{Kmw Xq¡nb-t¸mÄ 2 skânaoäÀ
hen-hp-­mbn F¶p Icp-Xp-I. At¸mÄ
BZ-ys¯ AS-bm-f-¯n n¶v 2, 4, 6, 8
skânaoäÀ AI-e-¯n 1, 2, 3, 4 F¶n-
§s Intem-{Km-an `mcw AS-bm-f-s¸-Sp-¯m-a-
tÃm. Cu AI-e-§sfsbÃmw ]¯p Xpey-
`m-K-§-fm-¡n-bm 1.1 Intem-{Kmw, 2.4
Intem{Kmw F¶n-§-s-bpÅ `mc-§fpw AS-bm-
f-s¸-Sp-¯mw.

24. KWnXw þ ¢mkv 8
{]]-©m-p-]mXw
{]]-©-¯nse FÃm hkvXp-¡fpw ]c-kv]
cw BIÀjn-¡p-¶p-sh-¶Xv `uXnI
imkv{X-¯nse ASn-Øm--X-Xz-am-Wv.
Cu BIÀj-W-i-àn, hkvXp-¡-fpsS {Zhy-am--
§-fpsS (mass) KpW--^-e-¯nv Bp-
]m-Xn-I-hpw, Ah X½n-epÅ Zqc-¯nsâ
hÀK-¯nv hn]-co-Xm-p-]m-X-¯nepw BsW-
¶mWv Cu XXzw ]d-bp-¶-Xv.
]Xn-tgmw qäm-­n Pohn-¨n-cp¶ {]kn²
imkv{X-Ú-mb sFk¡v yq«-mWv Cu
XXzw Ah-X-cn-¸n-¨-Xv.
44
$ tacnbv¡v Hmtcm hÀjhpw 200 cq] IqSp-XÂ
i¼fw In«pw. tPmen sN¿p¶ hÀj-§-fpsS
F®hpw i¼-f-¯n BsI-bp-­m-Ip¶ hÀ[-
hpw X½n-epÅ _Ôw Hcp ka-hm-Iy-ambn Fgp-
XpI. BsI i¼-f-hÀ[-hv tPmen sNbvX hÀj-
§-fpsS F®-¯nv Bp-]m-Xn-I-amtWm?
$ Db-c-¯nÂn¶v Xmtgm«p hogp¶ Hcp hkvXp-hnsâ
t sk¡âv Ign-bp-t¼m-gpÅ thKw v aoäÀ/sk¡âv
BsW¦nÂ
v = 9.8t
thKw ka-b-¯nv Bp-]m-Xn-I-amtWm?
$ Db-c-¯nÂn¶v Xmtgm«p hogp¶ Hcp hkvXp
t sk¡âv sIm­v k©-cn-¡p¶ Zqcw s aoä-dm-sW-
¦nÂ
s = 4.9t2
k©-cn-¡p¶ Zqcw ka-b-¯nv Bp-]m-Xn-I-amtWm?
hn]-co-X-amäw
30 NXp-c{i skân-ao-äÀ ]c-¸-f-hpÅ F{X NXp-c-§Ä
hcbv¡mw?
hi-§-fpsS ofw 10 skân-ao-ädpw hoXn 3 skân-ao-
ädpw Bbn hcbv¡mw; AsÃ-¦n 6 skân-ao-ädpw
5 skân-ao-ä-dpw. 12.5 skân-ao-äÀ, 2.4 skân-ao-äÀ F¶n-
§s `n¶-kw-Jy-Ifpw D]-tbm-Kn-¡mw.
At¸mÄ C§-s-bpÅ F{X NXp-c-§Ä thW-sa-
¦nepw hc-bv¡m-a-tÃm. Cu NXp-c-§-fn Hcp hi-¯nsâ
ofw IqSp-t¼mÄ atä hi-¯nsâ ofw Ipd-bp-I-bmWv
sN¿p-¶-Xv. AXn-m hi-§-fpsS ofw Bp-]m-Xn-I-a
à F¶p XoÀ¨-bm-Wv. F´mWv Af-hp-IÄ X½n-epÅ
_Ôw?
C¯-c-samcp NXp-c-¯nsâ hi-§-fpsS of-§Ä x, y
Fs¶-Sp-¯mÂ
xy = 30

25. Awi-Ôhpw Ap-]m-Xhpw
45
Cu ka-hmIyw
y =
30
x
= 30 ´
1
x
F¶pw Fgp-Xm-a-tÃm. AXm-b-Xv, y F¶Xv
1
x v Bp-
]m-Xn-I-am-Wv.
x, y F¶ GsX-¦nepw c­-f-hp-IÄ amdp-¶-Xv
y =
k
x
F¶ _Ôw (C-Xn k F¶ kwJy amdp-¶n-Ã) Ap-k-
cn-¨m-sW-¦nÂ, Ch hn]-co-Xm-p-]m-X-¯n-emWv
(inversely proportional) F¶mWv ]d-bp-¶-Xv.
ap¼v I­ Bp-]m-Xn-I-amb Af-hp-IÄ, tc-p-]m-X-
¯n-emWv (directly proportional) F¶pw ]d-bm-dp-­v.
Cn Nne tNmZy-§fmImw.
$ Htc Npä-f-hpÅ hnhn[ NXp-c-§-fpsS ofhpw
hoXnbpw hn]-co-Xm-p-]m-X-¯n-emtWm?
$ 200 Intem-ao-äÀ k©-cn-¡p¶ Hcp hml--¯nsâ
icm-icn thK-hpw, bm{X sN¿m-s-Sp-¡p¶ ka-bhpw
_Ôn-¸n¨v Hcp ka-hmIyw Fgp-Xp-I. icm-icn
thKhpw ka-bhpw hn]-coX Ap-]m-X-¯n-emtWm?
$ 1000 cq] Iptd-t¸À¡v Xpey-ambn hoXn-¡-Ww.
hoXn-¡p-¶-h-cpsS F®hpw Hmtcm-cp-¯À¡pw
In«p¶ ]Whpw tc-p-]m-X-¯n-tem, hn]-co-Xm-
p-]m-X-¯ntem BtWm?
$ Hcp I¼-n-bpsS 50 {Kmw, 100 {Kmw, 150 {Kmw,
200 {Kmw s]bvÌp-I-fpsS hne-hn-hcw tiJ-cn-¡pI.
s]bvÌp-I-fpsS Xq¡hpw hnebpw tc-p-]-mX-¯n-emtWm?
hn]-co-X-ew
Nn{X-¯n CS-Xp-h-i¯v `mcw IqSp-X-epÅ
hkvXp-hm-Wv. F¶mepw Xpew km[n-¨-
sX-§s?
W w
D d
C§s Xpew sN¿W-sa-¦n Ccp-h-i-
¯p-apÅ `mc-§fpw Xpe-- n-µp-hnÂn¶v
Ah-bn-te-¡pÅ AI-e-§fpw hn]-coX
Ap-]m-X-¯n-em-bn-cn-¡-Ww.
AXm-b-Xv, Nn{X-¯nse tKmf-§-fpsS `mcw
W, w F¶pw, Xpe-- n-µp-hn n¶pÅ
Zqcw D, d F¶pw FSp-¯mÂ
W ´ D = w ´ d
_n.kn c­mw qäm-­n Pohn-¨n-cp¶
BÀ¡n-an-Uokv F¶ {]kn² imkv{X-Ú-
mWv Cu XXzw I­p-]n-Sn-¨Xv. Dt¯m-e-
I-XXzw (lever principle) F¶mWv Cu XXzw
Adn-b-s¸-Sp-¶-Xv.
Cu XXz¯nsâ Hcp {]tbm-K-am-bn, of-ap
Å tIm D]-tbm-Kn-¨m sNdnb _ew
D]-tbm-Kn-¨pXs¶ `mcw IqSnb hkvXp-¡Ä
DbÀ¯mw F¶pw At±lw Is­-¯n. (C-cp
¼p hSn D]-tbmKn¨v ]md-Ifpw aäpw
s]m¡p-¶Xv I­n-«pt­m?)
“nev¡m-sm-cn-S-hpw, th­{X of-apÅ
tImepw Ds­-¦n `qan-bpsS Ømw Xs¶
amämw.” F¶Xv BÀ¡n-an-Uo-knsâ Hcp
{]kn-²-amb {]kvXm-h--bm-Wv.

26. KWnXw þ ¢mkv 8
BÀ¡n-an-Unkv
{]mNo--Im-es¯ Gähpw al-mmb
imkv{X-Úpw temI-¯nse F¡m-e-s
¯bpw anI¨ KWn-X-im-kv{X-Ú-cn Hcm-fp-
amWv BÀ¡n-an-Un-kv.
At±-l-¯nsâ Imes¯ aäv {Ko¡p imkv{X-
Ú-cn n¶v hyXy-kvX-am-bn, ssk²m-
´nI Nn´-IÄ¡v ]pdsa kmt¦-XnI nÀam-
W-§fpw BÀ¡n-an-Uokv S-¯n-bn-cp-¶p.
Xsâ P·-tZ-i-amb kndm-¡p-kns tdmam-
¡mÀ B{I-an-¨-t¸mÄ Dt¯m-e-I-XXzw D]-
tbm-Kn-¨p-­m-¡nb b{´-§Ä sIm­v I¸-
ep-I-sf-t¸mepw adn-¨n-«-Xmbn A¶s¯ Ncn-
{X-Im-c-·mÀ Fgp-Xn-bn-«p-­v.
Xmgv¶ nc-¸n-epÅ shÅw DbÀ¯n apI-fnse
hb-ep-I-fn Pe-tk-Nw S-¯m³
BÀ¡n-an-Uokv I­p-]n-Sn¨ b{´-amWv
BÀ¡n-an-Uokv kv{Iq.
C¯cw b{´-§Ä C¶pw hyh-km-bim-e-
I-fnse h³b-{´-§-fnepw lrZ-b-i-kv{X-{In-bbn
 cà-Nw{I-aWw km[y-am-¡p¶ sNdp
b{´-§-fnepw D]-tbm-Kn-¡p-¶p-­v.
46
$ ]pcp-j-·m-cpsS 100 aoäÀ, 200 aoäÀ, 400 aoäÀ,
800 aoäÀ Hm« aÕ-c-§-fpsS temI dn¡mÀUv ]cn-tim-[
n-¡p-I. Cu Zqc-§fpw ka-b-§fpw tc-p-
]m-X-¯n-emtWm?
$ Hcp {XntIm-W-¯nsâ aq¶p hi-§-fpsS ofhpw
Ah-bn-te-¡pÅ Db-c-§-fpsS ofhpw hn]-co-Xm-
p-]m-X-¯n-emtWm?