Marathi

20 {XZX{e©H$m

bjnyd©H$ dmMm d g‘OyZ ¿¶m :

1 {Xdg = 24 Vmg
1 AmR>dS>m = 7 {Xdg
1 df© = 12 ‘{hZo
12 ‘{hZo = 365 {Xdg (52 AmR>dS>o 1 {Xdg)

J«oJar¶Z df}

31 {XdgmMo ‘{hZo 30 {XdgmMo ‘{hZo 28/29 {XdgmMm ‘{hZm

OmZodmar, ‘mM©, ‘o, Owb¡, Am°JñQ>, E{àb, OyZ, gßQ>|~a, Zmoìh|~a = ’o$~«wdmar 1 ‘{hZm
Am°³Q>mo~a, {S>g|~a = EHy$U 7 ‘{hZo EHy$U 4 ‘{hZo. ""EOygZmo''

^maVr¶ gm¡a dfm©Vrb ‘{hZo (‘amR>r ‘{hZo)

31 {Xdg 30 {Xdg 30/31 {Xdg

d¡fmI, Ooð>, AmfmT>, lmdU, ^mÐnX AmpídZ, H$m{V©H$, ‘mJ©erf© M¡Ì
(AJ«hm¶U), nm¡f, ‘mK, ’$mëJwZ

 nwT>rb V³Ë¶mVrb VmaIm§Zm EH$M dma AgVmo.

CXm. gmo_ 1 8 15 22 29 `m gd© VmaIm§Zm EH$M dma `oB©b.
_§Ji 2 9 16 23 30 (H$maU dma hm gmV {XdgmZ§Va VmoM `oVmo.)

~wY 3 10 17 24 -

 nmM doim ¶oUmao dma

‘{hݶmVrb EHy$U {Xdg nmM doim ¶oUmè¶m VmaIm d dma

28 nmM doim EH$hr dma ¶oV Zmhr.

29 1 VmaIoMm

30 1 d 2 VmaIoMm
31 1, 2 d 3 VmaIoMm

J{UV nŠH$m nm`m 415

ñH$m°ba JmB©S>

brn df} : ho 366 {Xdg åhUOo 52 AmR>dS>o d 2 {XdgmMo AgVo.
 ¶m dfu ’o$~«wdmar ‘{hZm 29 {XdgmMm AgVmo.
H$go AmoiImb?
1. Á¶m dfm©À¶m g§»¶obm 4 Zo {Z:eof ^mJ OmVmo, Vo brn df© AgVo.

CXm. 1992, 1996, 2000, 2004, 2008, 2012, 2016, 2020
(darb dfmªMo {ZarjU nmhVm brn df© ho Xa Mma dfmªZr ¶oVo.)
2. Á¶m eVH$ dfm©bm 400 Zo ^mJ OmVmo Vo gwÕm brn df©M AgVo.
CXm. 1600, 2000 hr brn df© AmhoV. ‘mÌ 1700, 1800, 1900 ¶m dfm©bm 400 Zo {Z:eof ^mJ OmV
Zgë¶m‘wio hr brn df© ZmhrV.
 gd©gmYmaU df© : 365 {XdgmMo AgVo (52 AmR>dS>o 1 {Xdg)
 brn df© AgVmZm : àOmgÎmmH$ {XZr Omo dma AgVmo ˶mÀ¶m nmR>r‘mJrb dma ñdmV§Í¶{XZr AgVmo.
 brn df© ZgVmZm : 26 OmZodmar amoOr Omo dma AgVmo ˶mÀ¶m‘mJrb Xþgam dma 15 Am°JñQ>Mm AgVmo.
CXm. 26 OmZo. 2018, ewH«$dma
15 Am°JñQ> 2018, ~wYdma
EH$mM dfm©Vrb H$mbmdYrgmR>r : ¶m àg§Jr dma EH$M AgVmo.
1. ñdmV§Í¶{XZ (15 Am°JñQ>), {Q>iH$ nwʶ{VWr (1 Am°JñQ>), ~mb{XZ (14 Zmoìh|~a), {ejH$ {XZ (5
gßQ>|~a) ¶m {Xder EH$M dma AgVmo.
 bjmV R>odm : ({Q>iH$ {ejH$ Pmbo ~mbH$m§Zm ñdmV§Í¶ {Xbo.)
2. ‘hmamï´>{XZ (1 ‘o), Jm§Yr O¶§Vr (2 Am°³Q>mo.), emhÿ O¶§Vr (26 OyZ), ZmVmi (25 {S>g|~a) ¶m {Xder
EH$M dma AgVmo.
 bjmV R>odm : (‘hmamï´>mV Jm§Yr ZmVmimV emhÿ)
3. ‘hÎdmMo {XZ bjmV R>odm :

3 OmZodmar ~m{bH$m{XZ 1 Am°JñQ> bmo. {Q>iH$ nwʶ{VWr

26 OmZodmar àOmgÎmmH$/JUV§Ì{XZ 9 Am°JñQ> H«$m§Vr {XZ

30 OmZodmar hþVmË‘m {XZ 15 Am°JñQ> ñdmV§Í¶{XZ

27 ’o$~«wdmar ‘amR>r ^mfm {XZ 20 Am°JñQ> gmjaVm {XZ

28 ’o$~«wdmar amï´>r¶ {dkmZ {XZ 29 Am°JñQ> amï´>r¶ H«$sS>m {XZ

8 ‘mM© OmJ{VH$ ‘{hbm {XZ 16 gßQ>|~a AmoPmoZ g§ajU {XZ

7 E{àb OmJ{VH$ Amamo½¶ {XZ 2 Am°³Q>mo~a ‘. Jm§Yr O¶§Vr

416 nŠH$m nm`m J{UV

ñH$m°ba JmB©S>

21 H$mJXH$m‘

bjnyd©H$ dmMm d g‘OyZ ¿¶m :

12 H$mJX = 1 S>PZ
1 S>PZ = 12 ZJ
1 XñVm = 24 H$mJX (2 S>PZ)
1 J«mog = 144 H$mJX (12 S>PZ)
1 ar‘ = 480 H$mJX (20 XñVo)

H$mJX 12 24 144 480

1. S>PZ XñVm J«mog ar‘

bjmV R>odm :

S>PZ - XñVm - J«mog - ar‘

("H$mS>XJmoar' - H$m = H$mJX, S> = S>PZ, X = XñVm, Jmo = J«mog, ar = ar‘)

2. CXm. 1 S>PZ = 12 ZJ

nmd S>PZ = 3 ZJ 3 3 = 1 S>PZ
AYm© S>PZ = 6 ZJ 33
nmD$U S>PZ = 9 ZJ

gìdm S>PZ = 12 + 3 = 15 ZJ

XrS> S>PZ = 12 + 6 = 18 ZJ

nmdUo XmoZ S>PZ = 12 + 9 = 21 ZJ

CXm. 1 XñVm = 24 ZJ 6 6 = 24 H$mJX
66
nmd XñVm = 6 H$mJX
AYm© XñVm = 12 H$mJX
nmD$U XñVm = 18 H$mJX
1 XñVm = 24 H$mJX
gìdm XñVm = 24 + 6 = 30 H$mJX
XrS> XñVm = 24 + 12 = 36 H$mJX
nmdUo XmoZ XñVo = 24 + 18 = 42 H$mJX

430 nŠH$m nm`m J{UV

ñH$m°ba JmB©S>

1 J«mog = 144 H$mJX 36 36 = 144 H$mJX
36 36
nmd J«mog = 36 H$mJX
AYm© J«mog = 36 + 36 = 72 H$mJX
nmD$U J«mog = 108 H$mJX
gìdm J«mog = 180 H$mJX
XrS> J«mog = 216 H$mJX
nmdUo XmoZ J«mog = 252 H$mJX

1 ar‘ = 480 H$mJX

nmd ar‘ = 120 H$mJX 120 120 = 480 H$mJX
AYm© ar‘ = 240 H$mJX 120 120
nmD$U ar‘ = 360 H$mJX

gìdm ar‘ = 480 + 120 = 600 H$mJX

{XS> ar_ = 480 + 240 = 720 H$mJX

nmdUo XmoZ ar‘ = 480 + 120 + 120 + 120 = 840 H$mJX

‘hÎdmMo :

H$mJXmMr g§»¶m XoD$Z S>PZ/XñVm/J«mog/ar‘ H$mT>md¶mg gm§{JVbr Va ÷ {dMmaboë¶m Jmoï>rMo
ZJ H$am.
CXm. 480 H$mJX = {H$Vr S>PZ
480  12 = 40 S>PZ
CXm. {H$Vr?

H$mJX S>PZ XñVm J«mog ar‘
480
480 480 480 480
=
12 24 144 480

40 20 3.33 1

¶mCbQ> S>PZ/XñVm/J«mog/ar‘ ¶m§Mr g§»¶m X|D$Z H$mJX {H$Vr Agm àíZ {dMmaë¶mg nwT>rb
H¥$Vr H$am.
CXm. 1) 2 XñVo = {H$Vr H$mJX?

= 2  24 = 48 H$mJX
2) 3 S>PZ = {H$Vr H$mJX?

= 3  12 = 36 H$mJX

J{UV nŠH$m nm`m 431

ñH$m°ba JmB©S>

22 ZmUr d ZmoQ>m

bjnyd©H$ dmMm d g‘OyZ ¿¶m :
 ZmUr : 1 é., 2 é., 5 é., 10 é. hr ZmUr AmnU ì¶dhmam‘ܶo dmnaVmo.
 ZmoQ>m : 1 é., 2 é., 5 é., 10 é., 20 é., 50 é., 100 é., 200 é., 500 é., 2000 é. EdT>çm

ZmoQ>m X¡Z§{XZ ì¶dhmam‘ܶo AmnU dmnaVmo.
 gwQ>o H$aUo/‘moS> KoUo :

ZmoQ> XoD$Z ZmUr KoUo
ZmUr XoD$Z ZmoQ>m KoUo.
CXm. 50 é. Mr ZmoQ> XoD$Z 5 é. Mr Xhm ZmUr ¶oVrb d 10 é., 20 ZmUr XoD$Z 200 é. Mr ZmoQ> KoVm ¶oB©b.

H$Yr H$Yr én¶mMr nQ> H$mTy>Z CXmhaU 25 25 = 100 n¡go
gmoS>{dVmZm Imbrb Jmoï>r bjmV ¿¶m. n¡go n¡go
CXm. 100 é. 25 25
1 é. = 100 n¡go n¡go n¡go

a¸$‘ nQ> AnyUmªH$mV Xem§e A. én¶o, n¡go n¡emV
nmd én¶m 0.25 é. 0 é. 25 n¡. 25 n¡.
1
4

AYm© én¶m 1 0.50 é. 0 é. 50 n¡. 50 n¡.
nmD$U én¶m 2

3

4 0.75 é. 0 é. 75 n¡. 75 n¡.

gìdm én¶m 1 1.25 é. 1 é. 25 n¡. 125 n¡.
1 1.50 é. 1 é. 50 n¡. 150 n¡.
1.75 é. 1 é. 75 n¡. 175 n¡.
4 2.5 é. 2 é. 50 n¡. 250 n¡.

XrS> én¶m 1
12

nmdUo XmoZ én¶o 3
1

4

AS>rM én¶o 1
22

436 nŠH$m nm`m J{UV

ñH$m°ba JmB©S>

5 ì¶mdhm[aH$ J{UV

23 Z’$m-VmoQ>m

bjnyd©H$ dmMm d g‘OyZ ¿¶m :
Z’$m-VmoQ>m ¶m KQ>H$m§VJ©V CXmhaUo ObX gmoS>{dVm ¶oʶmgmR>r Imbrb g§~moY d gyÌo bjmV KoUo JaOoMo
Amho.

1. IaoXr qH$‘V = dñVy Á¶m {H§$‘Vrbm AmnU XþH$mZmVyZ {dH$V KoVmo Vr qH$‘V.
 dñVy IaoXr doir dmhVyH$ IM©, OH$mV, h‘mbr qH$dm Q>nmb, Hw$arAa IM©, XþéñVr IM© hm gd©

IM© IaoXr {H$§‘VrV Yabm OmVmo.
CXm. {ZH$sVmZo 5000 é. bm OwZm b°nQ>m°n IaoXr Ho$bm d XþéñVrgmR>r 3000 é. IM© Ho$bo. åhUOo
b°nQ>m°nMr IaoXr qH$‘V = 5000 + 3000 = 8000 é. IaoXr {H§$‘Vrbm "‘yi qH$‘V' Agohr åhUVmV.
2. {dH«$s qH$‘V : Á¶m {H§$‘Vrbm dñVy {dH$br OmVo Vr qH$‘V.
CXm. {ZH$sVmZo b°nQ>m°n 10,000 é. bm {dH$bm. hr a¸$‘ åhUOo {dH«$s qH$‘V hmo¶.
3. Z’$m : åhUOo ’$m¶Xm. IaoXr {H§$‘Vrnojm {dH«$s qH$‘V OmñV Agob, Va Z’$m hmoVmo.
CXm. {ZH$sVmZo 8000 é. Mm b°nQ>m°n 10,000 é. bm {dH$bm åhUOo {Vbm 2000 é. Z’$m/’$m¶Xm Pmbm.
4. VmoQ>m : IaoXr {H§$‘Vrnojm {dH«$s qH$‘V H$‘r Agob Va VmoQ>m hmoVmo.
CXm. {ZH$sVmZo 8000 é. Mm b°nQ>m°n Oa 5000 é. bm {dH$bm, Va {VMo 3000 é. Mo ZwH$gmZ hmoB©b
åhUOoM VmoQ>m hmoB©b.
bjmV R>odm : Z’$m/VmoQ>m hm Zoh‘r IaoXr {H§$‘Vrda R>a{dbm OmVmo. {dH«$s {H§$‘Vrda R>a{dbm OmV
Zmhr.

 gyÌo :

Z’$m hmoV Agob Va,
1. IaoXr qH$‘V = {dH«$s - Z’$m
2. {dH«$s qH$‘V = IaoXr + Z’$m

J{UV nŠH$m nm`m 445

ñH$m°ba JmB©S>

à. 2 ñdamZo X. gm. X. eo. 7 XamZo 20,000 é. 5 dfm©gmR>r ~±Ho$V R>odbo, Va {Vbm {H$Vr ì¶mO

{‘iob? Feb. 2017

1) 5000 é. 2) 700 é. 3) 7000 é. 4) 500 é.

‘wÔb = 20,000 Xa = 7 H$mbmdYr = 5 df} gaiì¶mO = ?

g. = ‘  X H$
100

g. = 20000  7 5
100

g. = 200  35

gaiì¶mO = 7000 é. C. n¶m©¶ H«$. 3 

à. 3 amhþbZo 15,000 é. 3 dfm©gmR>r H$mhr XamZo gw^mfbm dmnam¶bm {Xbo. gw^mfZo ˶mbm 4500

é. ì¶mO {Xbo, Va ì¶mOmMm Xa X. gm. X. eo. {H$Vr Agob? Feb. 2018

1) 10% 2) 15% 3) 9% 4) 8%

‘wÔb = 15,000 H$mbmdYr = 3 gaiì¶mO = 4500

CXm. 1 Zwgma C. n¶m©¶ H«$. 1 

à. 4 é. 500 ‘wÔbmMo X. gm. X. eo. 8 XamZo 3 dfm©Mo gaiì¶mO {H$Vr? Feb. 2018

1) é. 240 2) é. 120 3) é. 500 4) é. 400 

‘wÔb = 500 Xa = 8 H$mbmdYr = 3 df} gaiì¶mO = ?

g = ‘  X H$ g = 500  8 3
100 100

g. = 120 é. C. n¶m©¶ H«$. 2

à. 5 amo{hVbm X. gm. X. eo. 7 XamZo EH$m ‘wX²>XbmMo 3 dfm©Mo gaiì¶mO 2520 é. {‘imbo, Va

‘wÔb {H$Vr? Feb. 2018

1) 15,000 2) 21,600 3) 1,500 4) 12,000

Xa = 7 ‘wXV = 3 df} gaiì¶mO = 2520 ‘wÔb = ?

g. = ‘  X H$ 2520 = ‘  7 3
100 100

120
‘  21 2520  100
100 = 2520  ‘wÔb =
21

= 120  100 = 12,000 é.

J{UV nŠH$m nm`m 465

ñH$m°ba JmB©S>

28 {ÌH$moU, Mm¡ag d Am¶V

bjnyd©H$ dmMm d g‘OyZ ¿¶m :

AmH¥$VrMo Zmd {eamoq~Xÿ ~mOy/^yOm H$moZ JwUY‘©

{ÌH$moU ABC 3 3 3 1) VrZ ~mOy§Zr ~§XrñV AmH¥$Vr.
~mOy AB, ABC, 2) {VÝhr H$moZm§Mr ~oarO 2
A A, B, C ~mOy BC, BCA,
BC ~mOy AC H$mQ>H$moZ/1800 AgVo.
BAC 3) à˶oH$ {eamoq~XÿOdi 2

~møH$moZ hmoVmV.

Mm¡ag PQRS 4 4 4 1) Mm¡agmÀ¶m Mmahr ~mOy g‘mZ

PS P, Q, R, S ~mOy PQ,  Q P S , ‘mnmÀ¶m AgVmV.

QR ~mOy QR, PQR, 2) Mm¡agmMo Mmahr H$moZ EH$ê$n
AgVmV.
Am¶V LMNO ~mOy PS, QRS,
~mOy SR PSR 3) Mm¡agmMm à˶oH$ H$moZ
LO H$mQ>H$moZ/900 ‘mnmMm AgVmo.

MN 4) Mm¡agmbm 2 H$U© AgVmV.

H$U© PR d H$U© QS

4 4 4 1) Am¶VmÀ¶m g‘moamg‘moarb

L, M, N, O ~mOy LM, LMN, ~mOy g‘mZ ‘mnmÀ¶m AgVmV.
~mOy MN, MNO, 2) Am¶VmMo Mmahr H$moZ 900
~mOy NO, NOL, ‘mnmMo (H$mQ>H$moZ) AgVmV.
~mOy LO MLO

Mm¡H$moZ ABCD 4 4 4 1) Mm¡H$moZmÀ¶m AmVrb gd©

A D A, B, C, D ~mOy AB, ABC, H$moZm§Mr ~oarO 3600 AgVo.

~mOy BC, BCD, 2) Mm¡H$moZmMm à˶oH$ H$moZ

~mOy DC, CDA, g‘mZ ‘mnmMm ZgVmo.

B C ~mOy AD BAD

J{UV nŠH$m nm`m 479

ñH$m°ba JmB©S>

Z‘wZm àíZ

à. 1 gmo~VÀ¶m {ÌH$moUmÀ¶m AmH¥$Vr‘Yrb

1 2 3 45 {ÌH$moUm§Mr EHy$U g§»¶m {H$Vr?
1) 10 2) 12
3) 15 4) 14 
gd© A§H$m§Mr ~oarO H$am.
1 + 2 + 3 + 4 + 5 = 15
C. n¶m©¶ H«$. 3

à. 2 P gmo~VÀ¶m AmH¥$VrV EHy$U {H$Vr {ÌH$moU AmhoV?

Q 1 1) 10 2) 15
2
3) 13 4) 16
23
4 àW‘ H$moU˶m {eamoq~XþVyZ {H$aUo OmVmV, Vo emoYm.
3
¶oWo P ‘YyZ 5 {H$aUo OmVmV.

R 5 - 1 = 4 A§H$ {bhmdo.

H«$‘e: Q ‘YyZ 4 {H$aUo

4 - 1 = 3 A§H$ {bhm.

1 A§H$ H$m°‘Z ¶oB©b d gdmªMr ~oarO

1 + 2 + 3 + 2 + 3 + 4 = 15

C. n¶m©¶ H«$. 2 

à. 3 1 2 3 AmH¥$VrV EHy$U Mm¡agm§Mr g§»¶m {H$Vr?

2 1) 10 2) 13
3
3) 14 4) 16

dJ©g§»¶m§Mr ~oarO
12 = 1
22 = 4
33 = 9

14 C. n¶m©¶ H«$. 3 

à. 4 1 2 3 4 AmH¥$VrV EHy$U Mm¡agm§Mr g§»¶m {H$Vr?
1) 20 2) 18
2 3) 22 4) 28
3 Aer H¥$Vr H$am. (3  4) + (2  3) + (1  2)
12 + 6 + 2 = 20 C. n¶m©¶ H«$. 1 

480 nŠH$m nm`m J{UV

ñH$m°ba JmB©S>

àW‘ Mma nXar Hw§$nUmgmR>r 1200 ‘r. Vma Va EH$ nXar Hw§$nUmgmR>r = 300 ‘r. Vma

100

50 50

100 é§Xr 50 ‘r. C. n¶m©¶ H«$. 2 

à. 17 EH$m Am¶VmMr bm§~r hr é§XrÀ¶m XþßnQ> Amho d ˶mMr n[a{‘Vr 120 g|‘r. Amho, Va ˶m

Am¶VmMr bm§~r {H$Vr?

1) 20 g|‘r. 2) 60 g|‘r. 3) 80 g|‘r. 4) 40 g|‘r.

Am`VmMr n[a{_Vr = 120 g|_r. 40
åhUyZ 2 ~mOy§Mr bm§~r = 60 g|_r.

é§Xr =  _mZy, bm§~r = 2x 20 20

x + 2x = 60 g|_r. 40
3x = 60 g|_r.

x = 60 = 20 g|_r.
3
44
åhUyZ x = (é§Xr) = 20 g|_r.

bm§~r XþßnQ> åhUyZ bm§~r = 2 20 = 40 g|_r. C. n¶m©¶ H«$. 4 

gamd Vº$m

~mJ/‘¡XmZ bm§~r é§Xr n[a{‘Vr {H$Vr nXar à{V ‘r. IM© EHy$U IM©
AmH$ma ‘r. ‘r. Hw§$nU
Am¶VmH¥$Vr 100
50 2  bm. + 2  é§Xr  4 nXar é. 50 60,000 é.
Mm¡ag 50
2  100 + 2  50 300 4 1200  50
{ÌH$moU 44
n[aK = 300 = 1200 ‘r.

- 4  50 = 200 5 nXar é. 25 25,000 é.

200  5 = 1000 1000  25

- 44 ‘r. 3 nXar é. 50 6600 é.

44  3 132 50

= 132 ‘r.

J{UV nŠH$m nm`m 495

ñH$m°ba JmB©S>

34 B{ï>H$m{MVr, KZ, n¥ð>o, H$S>m, {eamoq~Xÿ

bjnyd©H$ dmMm d g‘OyZ ¿¶m :

{MVr (Solid) :

""H$moU˶mhr AmH$mamda ˶mM AmH$mamMo Wamda Wa {Xë¶mg EH$ AmH¥$Vr V¶ma hmoVo, ˶m AmH¥$Vrg {MVr
Ago åhUVmV.

B{ï>H$m{MVr :
B{ï>H$m åhUOo drQ>.
{dQ>, H$mS>onoQ>r, nwñVH$, H$nmQ> ¶m§gmaIm AmH$ma AgUmè¶m AmH¥$VtZm B{ï>H$m{MVr åhUVmV.

SR

PQ

BC

AD

{eamoq~Xÿ = A, B, C, D, P, Q, R, S = EHy$U 8
H$S>m = PS, SR, RQ, QP, AB, BC, CD, DA, PA, SB, RC, QD = EHy$U 12
n¥ð>o = PQRS, ABCD, QDCR, PSBA, PQDA, SBCR = EHy$U 6
 B{ï>H$m{MVrMr g‘moamg‘moarb n¥ð>o d H$S>m g‘mZ bm§~rÀ¶m AgVmV.
 B{ï>H$m{MVrÀ¶m n¥ð>m§Mm AmH$ma Am¶VmH¥$Vr/Mm¡agmH¥$Vr AgVmo.

KZ :

bm§~r, é§Xr d C§Mr g‘mZ AgUmè¶m AmH¥$Vrg KZ åhUVmV.

gmo~VMr AmH¥$Vr KZmMr Amho. O R
Q V
KZmbm B{ï>H$m{MVrà‘mUo 6 n¥ð>o, P
12 H$S>m d 8 {eamoq~Xÿ AgVmV. S
U
KZmMr gd© n¥ð>o AmH$mamZo Mm¡agmH$ma

(Mmahr ~mOy gma»¶m ‘mnmÀ¶m) AgVmV.

T

518 nŠH$m nm`m J{UV

ñH$m°ba JmB©S>