Patigonit (inception plus) [www.chakritips.com]

52935wU †evW© eB‡qi cÖvmw½K Uwc‡Ki GK Af~Zc~e© msKjb ‡evW© eB †_‡K wewmGm Gi MwYZ cÖkœ I mgvavb
Avcbviv nq‡Zv 40Zg wewmGm mn weMZ wewmGm¸‡jv‡Z
jÿ¨ K‡i‡Qb, wewmGm wcÖwjwgbvwi cixÿvi cÖkœ¸‡jv‡Z
wKQz A¼ ûeû beg-`kg †kÖwYi eB †_‡K w`‡q †`qv nq|
ZvB wm‡jev‡mi AšÍf©y³ UwcKmg~‡ni mv‡_ wgj †i‡L
lô/mßg/Aóg/beg-`kg †kÖwYi eB‡qi wb‡¤œv³ Aa¨vqmg~n
†_‡K 41Zg wewmGm wcÖwjwgbvwi cixÿvi Rb¨ cÖkœ DËimn
†`qv n‡jv|
GKUv welq g‡b ivL‡eb, GB cÖkœ¸‡jvi evB‡iI †evW©
eB‡q Av‡iv A‡bK cÖkœ Av‡Q| ‡h‡nZz GB eBwU †evW©
eBwfwËK ïay wewmGm wcÖwjwgbvwi cixÿvi wm‡jevm
wbf©i, †m‡nZz Avgiv ‡Póv K‡iwQ wm‡jev‡mi mv‡_
m¤úwK©Z †evW© eB‡qi Aa¨vqmg~‡ni mKj AsK
GKmv‡_ Dc¯’vcb Ki‡Z hv‡Z wcÖwjwgbvwi cixÿvq †evW©
eB †_‡K cÖkœ Avm‡j Aek¨B Bb‡mckb cøvm †_‡K
Kgb c‡o|
wewmGm wcÖwjwgbvwi cixÿvi wm‡jevm
I †evW© eB‡qi ‡gjeÜb
µg Uwc‡Ki bvg ‡gvU
b¤î
01 ev¯Íe msL¨v, j.mv.¸-M.mv.¸, kZKiv,
mij I †h․wMK gybvdv, AbycvZ I
mgvbycvZ Ges jvf-ÿwZ
03
wm‡jev‡mi UwcK ‡kÖwYi bvg Aa¨v‡qi bvg
ev¯Íe msL¨v
mßg †kÖwY 1.1, 1.2
beg-`kg †kÖwY 1
1. wb‡Pi †Kvb msL¨v `ywU mn †g․wjK msL¨v? [6ô †kÖYx-
(Aby:-1.2)]
K. 27,54 L.63,91
M. 189,210 N. 52, 97 DËi N
☼ mgvavb:
Ackb¸‡jvi g‡a¨ N Gi 52 I 97 msL¨vwUi g‡a¨ 97
msL¨vwU †g․wjK nIqvq 52 I 97 Gi M.mv.¸ 1|
KviY G‡ì g‡a¨ †Kvb mvaviY ¸YbxqK †bB| wKš‘
Aci wZbwU Ack‡bi msL¨vØ‡qi mvaviY ¸YbxqK 3|
ZvB DËi: N |
2. 30 †_‡K 70 Gi g‡a¨ †g․wjK msL¨v¸‡jv KZwU?
K. 10wU L. 9wU
M. 11wU N. 15wU DËi L
3. `yBwU µwgK msL¨vi e‡M©i AšÍi 37 n‡j, msL¨v `yBwU
wbb©q Ki| [7g †kÖYx Aby : 1.2]
K. 17 I 18 L. 18 I 19
M. 19 I 20 N. 20 I 21 DËi L
☼ mgvavb:
awi, µwgK msL¨v `yBwU x, (x+1)
cÖkœg‡Z,
(x+1)2
-x2
= 37 x2
+2x+1-x2
= 37
 2x=37-1 2x=36 x=
2
36
 x = 18
 msL¨v `yBwU x = 18 Ges x+1 = 18+1=19
myZivs msL¨v `yBwU h_vµ‡g, 18 I 19|
kU©KvU: 37 Gi mv‡_ 1 †hvM K‡i 2 w`‡q fvM Ki‡j eo
msL¨vwU 19 Ges 37 †_‡K 1 we‡qvM K‡i 2 w`‡q fvM
Ki‡j †QvU msL¨vwU 18 †ei n‡e| A_ev, `ywU µwgK
msL¨vi †hvMdjB µwgK msL¨v `yBwUi e‡M©i AšÍi|
A_©vr (18+19) = 37|
4. Ggb `yBwU ÿz`ªZg µwgK msL¨v wbY©q Ki hv‡ì e‡M©i
AšÍi GKwU c~Y© eM©msL¨v| [7g †kÖYx Aby : 1.2]
K. 5 I 6 L. 6 I 7
M. 4 I 5 N. 4 I 6 DËi M
☼ mgvavb:
awi, µwgK msL¨v `yBwU x, (x+1)
Zv‡ì e‡M©i AšÍi GKwU = (x+1)2
-x2
= x2
+2x+1-x2
= 2x+1
x = 1 n‡j 2.1+1 = 2+1=3; hv c~Y©eM© bq|
x = 2 n‡j 2.2+1 = 4+1=5; hv c~Y©eM© bq|
x = 3 n‡j 3.2+1 = 6+1= 7; hv c~Y©eM© bq|
x = 4 n‡j 4.2+1 = 8+1=9; hv c~Y©eM© |
 msL¨v `yBwU x = 4
x+1 = 4+1 = 5
 msL¨v `yBwU h_vµ‡g 4 I 5|
kU©KvU : Ack‡bi msL¨v `ywUi eM© K‡i eowU †_‡K †QvUwU
we‡qvM Kiæb| †hgb 52
– 42
= 9 = 32
5. wb‡Pi †KvbwU Ag~j` msL¨v ?
(K) 30
.
. (L)
9
16
‡evW© eB †eBR&W MwYZ cvwUMwYZ wewmGm wcÖwjwgbvwi cixÿvi wm‡jev‡mi mv‡_
m¤úwK©Z †evW© eB‡qi mKj AsK GKmv‡_...
ev¯Íe msL¨vUwcK †eBRW&
(‡evW© eB †_‡K)

(M) 3
27
8
(N)
3
5
DËi N
☼ mgvavb:
(K) 30
.
. =
9
3
=
3
1
(L)
9
16
=
3
4
(M) 3
27
8
=
3
2
(N)
3
5
=
3
5
GLv‡b, ïaygvÎ N Ackb ev‡` Ab¨ me¸‡jv‡K fMœvsk
AvKv‡i cÖKvk Kiv hvq ZvB N Ack‡bi
3
5
B Agyj`|
6. a, b, c, d PviwU µwgK ¯^vfvweK msL¨v n‡j wb‡Pi
†KvbwU c~Y©eM© msL¨v ?
(K)abcd (L) ab+cd
(M) abcd+1 (N) abcd-1 DËi M
☼ mgvavb:
awi, a=1, b =2, c = 3 Ges d=4
Zvn‡j,
(K)abcd = 1.2.3.4 = 24 c~Y©eM© bq|
(L) ab+cd= 1.2+3.4 = 2+12 = 14 c~Y©eM© bq
(M) abcd+1 = 1.2.3.4+1=24+1=25 c~Y©eM©
(N) abcd-1= 1.2.3.4-1=24-1=23c~Y©eM bq
GLv‡b, 1,2,3 4 Gi RvqMvq 2,3,4,5 A_ev
3,4,5,6 †h †Kvb msL¨v wb‡jI M AckbwU cÖwZevi c~Y©eM©
n‡e|
7. 1 ‡_‡K 10 ch©šÍ †g․wjK msL¨v KqwU ?
(K) 3 (L) 4
(M) 5 (N) 6 DËi L
☼ mgvavb:
1 †_‡K 10 ch©šÍ 4wU †g․wjK msL¨v n‡jv 2,3,5,7
8. ev¯Íe msL¨vi †ÿ‡Î
(i) we‡Rvo msL¨vi eM© GKwU we‡Rvo msL¨v|
(ii) `yBwU †Rvo msL¨vi ¸Ydj Gi ¸wYZK †Rvo
msL¨v|
(iii) c~Y©eM© bq Ggb msL¨vi eM©g~j g~j` msL¨v|
wb‡Pi ‡KvbwU mwVK
(K)iI ii (L) i I iii
(M)iiI iii (N) i, ii, I iii DËi N
9. wZbwU µwgK ¯^vfvweK msL¨vi ¸Ydj me©`vB wb‡Pi
†Kvb msL¨v Øviv wefvR¨ n‡e ?
(K) 5 (L) 6
(M) 7 (N) 11 DËi L
☼ mgvavb:
awi, msL¨v 3wU h_vµ‡g, 1,2,3 A_ev, 3,4,5
GLb G‡ì ¸Ydj 1.2.3 = 6 ev 3.4.5 = 60
‡`Lv hv‡”Q Dfq msL¨v‡K 6 w`‡q fvM Kiv hvq|
Zvn‡j ejv hvq GiKg †h †Kvb 3wU ¯^vfvweK µwgK
msL¨vi ¸Ydj‡K me©`vB 6 w`‡q fvM Kiv hv‡e|
10. aI b `yBwU µwgK †Rvo msL¨v n‡j wb‡Pi †KvbwU
we‡Rvo msL¨v ?
(K)a2
(L) b2
(M)a2
+1 (N) b2
+2 DËi M
☼ mgvavb:
a=2 Ges b = 4 a‡i Ackb¸‡jv cixÿv Ki‡j †`Lv
hvq ïaygvÎ a2
+1 = 22
+1 = 4+1 = 5 B we‡Rvo|
11. aI b `yBwU c~Y©msL¨v n‡j a2
+b2
Gi mv‡_ wb‡Pi
‡KvbwU †hvM Ki‡j †hvMdj GKwU c~Y©eM© msL¨v n‡e ?
(K) -ab (L) ab
(M) 2ab (N) ab DËi M
☼ mgvavb:
c~Y©eM© msL¨v n‡Z n‡j (a+b)2
Gi m~Îvbymv‡i ivwkwUi
gvb a2
+2ab+b2
n‡Z n‡e| cÖ‡kœ †h‡nZz a2
+b2
‡`qv
Av‡Q ZvB †hvM Ki‡Z n‡e2ab
12. 56728 Rb •mb¨ †_‡K Kgc‡ÿ KZRb •mb¨ mwi‡q
ivL‡j ev Zv‡ì mv‡_ KZc‡ÿ Avi KZRb •mb¨ †hvM
w`‡j •mb¨`j‡K eM©vKv‡i mvRv‡bv hv‡e? [mßg †kÖwY
(Abykxjbx 1.2), cÖkœ bs- 15]
K. 68 Rb L. 84 Rb
M. 56 Rb N. 62 Rb DËi L
13. ‡Kv‡bv we`¨vj‡qi 2704 Rb wkÿv_©x‡K cÖZ¨wnK
mgv‡ek Kivi Rb¨ eM©vKv‡i mvRv‡bv n‡jv| cÖ‡Z¨K
mvwi‡Z wkÿv_©xi msL¨v wbY©q Ki| [mßg †kÖwY
(Abykxjbx 1.2), cÖkœ bs- 16]
K. 50 Rb L. 52 Rb
M. 48 Rb N. 46 Rb DËi L
14. GKwU mgevq mwgwZi hZRb m`m¨ wQj cÖ‡Z¨‡K ZZ
20 UvKv K‡i Pvu`v †Ìqvq †gvU 20480 UvKv n‡jv| H
mwgwZi m`m¨msL¨v wbY©q Ki| [mßg †kÖwY (Abykxjbx
1.2), cÖkœ bs- 17]
K. 32 Rb L. 40 Rb
M. 46 Rb N. 52 Rb DËi K
15. ‡Kv‡bv evMv‡b 1800wU PvivMvQ jvMv‡Z wM‡q 36wU MvQ
†ewk n‡jv| cÖ‡Z¨K mvwi‡Z PvivMv‡Qi msL¨v wbY©q
Ki|[mßg †kÖwY (Abykxjbx 1.2), cÖkœ bs- 18]
K. 42wU L. 38wU
M. 36wU N. 40wU DËi K

2 12, 18, 20, 105
2 6, 9, 10, 105
3 3, 9, 5, 105
5 1, 3, 5, 35
1, 3, 1, 7
2 16, 24 , 32
2 8, 12, 16
2 4, 6 , 8
2 2, 3, 4
1, 3, 2
2 20, 25, 30, 36, 48
2 10, 25, 15, 18, 24
3 5, 25, 15, 9, 12
5 5, 25, 5, 3, 4
1, 5, 1, 3, 4
j. mv. ¸ I
M. mv. ¸
lô †kÖwY 1.3
cÂg †kÖwY 5
mßg †kÖwY 1.2
16. 12,18,20 I 105 Gi j.mv.¸ KZ? [6ô †kÖwY-(Aby:1.3
D`v:4]
K.1400 L.1350
M.1260 N.1630 DËi M
☼ mgvavb:
wb‡Y©q j.mv.¸ = 223537 = 1260
17. ‡g․wjK ¸Ybxq‡Ki mvnv‡h¨ 30,36,40 Gi j.mv.¸
wbY©©q Ki |
K. 360 L. 420
M. 370 N. 380 DËi K
☼ mgvavb:
GLv‡b, 30 = 235
 30 Gi †g․wjK ¸YbxqK¸‡jv 2,3,5
36 = 2233
 36 Gi †g․wjK ¸YbxqK¸‡jv 2,2,3,3
Ges 40 = 2225
 40 Gi †g․wjK ¸YbxqK¸‡jv 2,2,2,5
 30,36,40 Gi j.mv.¸ = 222335 = 360
A_©vr wb‡Y©q j.mv.¸ 360
18. †Kvb ÿz`ªZg msL¨vi mv‡_ 5 †hvM Ki‡j †hvMdj
16,24 I 32 w`‡q wb:‡k‡l wefvR¨ n‡e?
K. 71 L. 91
M. 101 N. 103 DËi L
☼ mgvavb:
wb‡Y©q ÿz`ªZg msL¨vwU n‡e 16, 24 I 32 Gi j.mv.¸
†_‡K 5 Kg |
 16, 24 I 32 Gi j.mv.¸
= 2  2 2 2  3  2 = 96
wb‡Y©q ÿz`ªZg msL¨vwU ( 96 - 5) ev 91 |
19. ‡Kvb ÿz`ªZg msL¨v‡K 8,12,18 Ges 24 Øviv fvM
Ki‡j cÖ‡Z¨Kevi fvM‡kl 5 n‡e?
K. 55 L. 62
M. 68 N. 77 DËi N
☼ mgvavb:
wb‡Y©q ÿz`ªZg msL¨vwU n‡e 8, 12 , 18 I 24 Gi
j.mv.¸ A‡cÿv 5 †ewk
8, 12, 18 I 24 Gi j.mv.¸ = 22233 =72
wb‡Y©q ÿz`ªZg msL¨vwU = 72 + 5 = 77 |
20. ‡Kvb ÿz`ªZg msL¨v‡K 20,25,30, 36 Ges 48 w`‡q
fvM Ki‡j h_vµ‡g 15, 20, 25, 31,I 43 fvM‡kl
_vK‡e? (6ô †kÖwYAby:1.3 Gi 8)
K. 3236 L. 3448
M. 3515 N. 3595 DËi N
☼ mgvavb:
GLv‡b, fvRK I fvM‡kl Gi ga¨Kvi cv_©K¨
20- 15 = 5
25 - 20 = 5
30 - 25 = 5
36 - 31 = 5
48 - 43 = 5 hv cÖwZ †ÿ‡ÎB GKwU wbw`©ó msL¨v
wb‡Y©q ÿz`ªZg msL¨vwU n‡e 20, 25, 30, 36 I 48
Gi j.mv.¸ †_‡K 5 Kg
20 , 25, 30, 36, 48 Gi j.mv.¸ = 2235
534 = 3600
wb‡Y©q ÿz`ªZg msL¨vwU = 3600 - 5 = 3595
DËi: 3595|
21. Pvi A‡¼i †Kvb ÿz`ªZg msL¨v 12, 15, 20 I 35 Øviv
wb:‡k‡l wefvR¨ [6ô †kÖwY- ( Aby: 1.3 )]
2 8, 12 , 18 , 24
2 4, 6 , 9 , 12
2 2, 3 , 9 , 6
3 1, 3, 9 ,3
1, 1, 3, 1
j.mv.¸ j.mv.MyUwcK †eBRW&

2 12, 15, 20, 35
2 6, 15, 10,35
3 3, 15, 5, 35
5 1, 5, 5, 35
1, 1, 1, 7
420) 1000 (2
840
160
2 10, 20, 24, 32
2 5, 10, 12,16
2 5, 5, 6, 8
5 5, 5, 3, 4
1, 1, 3, 4
K. 1260 L. 1280
M. 1160 N. 1180 DËi K
☼ mgvavb:
12, 15, 20 I 35 Gi j.mv.¸ wbY©q Ki
 j.mv.¸ = 22357 = 420
Pvi A‡¼i ÿz`ªZg msL¨v 1000
 12, 15, 20, 35 Gi j.m.¸ 420 Øviv wbt‡k‡l
wefvR¨ msL¨vwU n‡e 1000 - 160 = 840
A_ev, 1000 + ( 420 - 160) = 1000 + 260
= 1260
wKš‘ 840 msL¨vwU wZb A‡¼i e‡j MÖnb †hvM¨ bq|
wb‡Y©q ÿz`ªZg msL¨v 1260 |
22. 5 A‡¼i †Kvb e„nËg msL¨v‡K 16, 24, 30 I 36
w`‡q fvM Ki‡j cÖ‡Z¨Kevi fvM‡kl 10 n‡e?[6ô †kÖwY-
(Aby: 1.3 )]
K. 99390 L. 99370
M. 99360 N. 99350 DËi L
☼ mgvavb:
5 A‡¼i e„nËg msL¨v = 99999
16,24,30 Ges 36 Gi j.mv.¸ = 720
Gici 720 w`‡q 99999 †K fvM Ki‡j fvM‡kl 639
myZivs 5 A‡¼i e„nËg msL¨v hv 16,24,30 I 36 w`‡q
wefvR¨ n‡e99999-639 = 99360
wKš‘ cÖ‡kœ ejv n‡q‡Q †h cÖwZevi 10 Aewkó _vK‡e ZvB
wb‡Y©q msL¨vwU n‡e 99360+10 = 99370
DËi: 99370
[GB wbqgUv Lye gb‡hvM w`‡q co‡j GiKg †h †Kvb AsK
cviv hv‡e|]
23. ‡Kv‡bv evm ÷ªvÛ †_‡K 4wU evm GKwU wbw`©ó mgq
ci h_vµ‡g 10wK.wg., 20wK.wg., 24 wK.wg., I
32 wK.wg., c_ AwZµg K‡i| Kgc‡ÿ KZ `~i
c_ AwZµg Kivi ci evm 4wU Avevi GK‡Î
wgwjZ n‡e?[6ô †kÖwY(Aby:1.3)]
K. 480 L. 460
M. 490 N. 450 DËi K
☼ mgvavb:
wb‡Y©q `~iZ¡ n‡e PviwU evm KZ…©K AwZµvšÍ `~i‡Z¡i j.mv.¸
 10, 20, 24 I 32 Gi j.mv.¸ = 222534
= 480
480 wK.wg. `~iZ¡ AwZµg Kivi ci evm PviwU GK‡Î
wgwjZ n‡e| DËi: 480 wK.wg.|
24. 28, 48 Ges 72 Gi M.mv.¸ KZ? [6ô ‡kÖYx-(Aby:1.3)
D`v:3)]
K. 4 L. 6
M. 8 N. 14 DËi K
☼ mgvavb:
28 = 227
48 = 22223
72 = 22233
wb‡Y©q M.mv.¸ = 22 = 4
25. †Kvb e„nËg msL¨v Øviv 365 I 463 †K fvM
Ki‡j fvM‡kl h_vµ‡g 5 I 7 _v‡K| e„„nËg
msL¨vwUi gvb KZ?
K. 24 L. 52
M. 45 N. 34 DËi K
☼ mgvavb:
†h‡nZz e„nËg msL¨v Øviv 365 I 463 †K fvM
Ki‡j fvM‡kl h_vµ‡g 5 I 7 _v‡K| Kv‡RB
wb‡Y©q msL¨vwU n‡e (365 - 5) ev 360 Ges
(463-7) ev 456 Gi M.mv.¸|
720) 99999 (138
720
2799
2160
6399
5760
639
M.mv.¸ M.mv.MyUwcK †eBRW&

24) 36 (1
24
12) 24 (2
24
0
360)456(1
360
96) 360 (3
288
72) 96 (1
72
24) 72 (3
72
0
96)180(1
96
84)96 (1
84
12) 84 (7
84
0
12) 60 (5
60
0
GLb,
 360 I 456 Gi M.mv.¸. 24 |
wb‡Y©q e„nËg msL¨vwU 24 |
26. †Kvb e„nËg msL¨v Øviv 57, 93 Ges 183 †K fvM
Ki‡j †Kv‡bv fvM‡kl _vK‡e bv?
K. 3 L. 5
M. 7 N. 8 DËi K
☼ mgvavb:
GLv‡b, wb‡Y©q e„nËg msL¨vwU n‡e 57, 93 I 183 Gi
M.mv.¸|
GLv‡b, 57 = 319, 93 = 331 Ges 183 = 361
57, 93 I 183 Gi M.mv.¸ 3|
wb‡Y©q e„nËg msL¨vwU 3 |
27. ‡Kvb e„nËg msL¨v Øviv 100 I 184 †K fvM Ki‡j
cÖ‡Z¨Kevi fvM‡kl 4 _vK‡e? [6ô †kÖwY- ( Aby: 1.3 )] +
(WvK Awa`: Dc: †cvógvóvi -2016)
K. 14 L. 10
M. 12 N. 16 DËi M
☼ mgvavb:
‡h‡nZz e„nËg msL¨v Øviv 100 I 184 †K fvM Ki‡j
cÖ‡Z¨Kevi fvM‡kl 4 _v‡K, †m‡nZz wb‡Y©q msL¨vwU n‡e
(100-4) ev 96 Ges (184-4) ev 180 Gi M.mv.¸
96 Ges 180 Gi M.mv.¸ 12
wb‡Y©q e„nËg msL¨vwU 12 (DËi)|
28. †Kvb e„nËg msL¨v Øviv 27, 40 I 65 †K fvM Ki‡j
h_vµ‡g 3,4,5 fvM‡kl _vK‡e? (Kviv ZË¡veavqK
¯^ivógš¿:-10) + [6ô †kÖwY- (Aby:1.3)]
K.10 L.12
M.14 N.16 DËi L
☼ mgvavb:
‡h‡nZz e„nËg msL¨v Øviv 27, 40 I 65 †K fvM Ki‡j
h_vµ‡g 3, 4, 5 fvM‡kl _vK‡e, †m‡nZz wb‡Y©q msL¨vwU
n‡e ( 27 - 3) ev, 24, ( 40 - 4) ev 36 Ges 65 - 5
ev 60 Gi M.mv.¸
‡h‡nZz †kl fvRK 12  wb‡Y©q e„nËg msL¨v 12 |
29. GKwU †jvnvi cvZ I GKwU Zvgvi cv‡Zi •`N©¨
h_vµ‡g 672 †m.wg. Ges 960 †m.wg. cvZ `ywU
†_‡K †K‡U †bIqv GKB gv‡ci me‡P‡q eo UzKivi
•`N©¨ KZ n‡e? cÖ‡Z¨K cv‡Zi UzKivi msL¨v wbY©q
Kiæb| [6ô ‡kÖYx-(Aby:1.3) )]
K. 96, 7, 12 L. 96, 7, 16
M. 96, 7, 8 N. 96, 7, 10 DËi N
☼ mgvavb:
GLv‡b, 672 I 960 Gi M.mv.¸ B n‡e me‡_‡K
eo UzKivi •`N©¨|
672 I 960 Gi M.mv.¸ = 96
Ges †jvnvi cvZ = 67296 = 7 UzKiv
Ges Zvgvi cvZ 96096 = 10 UzKiv|
30. `yÕwU msL¨vi ¸Ydj 3380 Ges M.mv.¸ 13| msL¨v
`yÕwUi j.mv.¸ KZ? [36Zg wewmGm] [6ô †kÖwY(Aby:1.3)
Gi 13 ]
K.260 L.265
M.270 N.250 DËi K
☼ mgvavb:
`ywU msL¨vi ¸Ydj I Zv‡`i j.mv.¸ Ges M.mv.¸ - Gi
¸Ydj ci¯úi mgvb nq|
ZvB GLv‡b M.mv.¸ 13 Ges ¸Ydj 3380 nIqvq
msL¨v`ywUi j.mv.¸ n‡e
13
3380
= 260|
31. †Kv‡bv evMv‡b 1296 wU AvgMvQ Av‡Q| evMv‡bi
‣`N©¨ I cÖ‡¯’i Dfq w`‡Ki cÖ‡Z¨K mvwi‡Z mgvb
msL¨K AvgMvQ _vK‡j cÖ‡Z¨K mvwi‡Z Mv‡Qi
msL¨v wbY©q Ki|
(K) 36 (L) 28
(M) 34 (N) 25 DËi K

☼ mgvavb:
evMv‡bi •`N©¨ I cÖ‡¯’i Dfq w`‡Ki cÖ‡Z¨K mvwi‡Z
mgvb msL¨K AvgMvQ Av‡Q|
cÖ‡Z¨K mvwi‡Z AvgMv‡Qi msL¨v n‡e 1296 Gi
eM©g~j|
GLb,
wb‡Y©q AvgMv‡Qi msL¨v 36 wU|
32. GKwU ¯‥vDU `j‡K 9, 10 Ges 12 mvwi‡Z mvRv‡bv
hvq| Avevi Zv‡ì eM©vKv‡iI mvRv‡bv hvq| H ¯‥vDU
`‡j Kgc‡ÿ KZRb ¯‥vDU i‡q‡Q? (7g †kÖYx Aby:1.2
Gi D`v: 10)
K. 1600 L. 2500
M. 1800 N. 900 DËi N
☼mgvavb: ¯‥vDU `j‡K 9, 10 Ges 12 mvwi‡Z mvRv‡bv
hvq| d‡j ¯‥vDU Gi msL¨v 9,10 Ges 12 Øviv
wefvR¨| Giƒc ÿz`ªZg msL¨v n‡e 9, 10 Ges 12 Gi
j.mv.¸.|
GLv‡b,
9, 10 Ges 12 Gi j.mv.¸. = 22335 =
(22)  (33)5
cÖvß j.mv.¸. = (22)  (33)5 †K eM©vKv‡i
mvRv‡bv hvq bv|
(22)  (33)5 †K eM©msL¨v Ki‡Z n‡j Kgc‡ÿ
5 Øviv ¸Y Ki‡Z n‡e|
 9, 10 Ges 12 mvwi‡Z Ges eM©vKv‡i mvRv‡bvi Rb¨
¯‥vDU Gi msL¨v cÖ‡qvRb (22) 
(33)(55)=900
wb‡Y©q ¯‥vDU Gi msL¨v 900|
☼ weKí mgvavb:
9,10 I 12 Gi j.mv.¸ = 180
GLb 180 †K fv½‡j †`Lv hvq,
(22)(33)5 nq|
GLv‡b 5 Gi †Rvov bv _vKvq 180 †K c~Y© eM© ivwk
evbv‡Z 5 w`‡q ¸Y Ki‡Z n‡e|
myZivs DËi n‡e 1805 = 900
33. †Kvb ÿz`ªZg c~Y© eM© msL¨v 9, 15 Ges 25 Øviv
wefvR¨? [7g †kÖYx Aby : 1.2]
K. 125 Rb L. 225 Rb
M.350 Rb N. 102 Rb DËi L
☼ mgvavb:
ÿz`ªZg c~Y©eM© msL¨v 9,15 Ges 25 Gi j.mv.¸
9, 15, 25 Gi j.mv.¸. = 35315 = 225
wb‡Y©q c~Y©eM© msL¨v 225|
kZKiv
lô †kÖwY 2.2
beg-`kg †kÖwY 11.2
34. GKwU ivwk Aci GKwU ivwki 50%| ivwk `yBwUi
AbycvZ wbY©q Ki|
K. 2 : 1 L. 1 : 2
M. 3 :4 N. 5 : 6 DËi L
☼ mgvavb:
50% =
100
50
= A_©vr, GKwU ivwk 50 n‡j, Aci ivwkwU
n‡e 100 |
50 Ges 100 Gi AbycvZ n‡jv 50 : 100 = 1 : 2
wb‡Y©q ivwk `yBwUi AbycvZ = 1 : 2 |
35. gwbiv evwl©K cixÿvq 80% b¤î †c‡q‡Q| cixÿvq
†gvU b¤î 800 n‡j, gwbiv cixÿvq †gvU KZ b¤î
†c‡q‡Q?
K. 620 L. 720
M. 540 N. 640 DËi N
☼ mgvavb:
gwbivi cÖvß b¤î = 800 Gi 80% = 800
100
80
= 640 b¤î|
12 96 36
9
66 3 96
3 96
0
2 9, 10, 12
3 9, 5, 6
3, 5, 2
kU©KvU:
180 w`‡q fvM Kiv hvq Ggb eM© msL¨v Ack‡bi g‡a¨
Av‡Q ïaygvÎ 900| ZvB DËi n‡e 900|
3 9, 15, 25
5 3, 5, 25
3, 1, 5
kZKiv kZKivUwcK †eBRW&

36. d‡ji †`vKvb †_‡K 180 wU dRwj Avg wK‡b Avbv
n‡jv| `yB w`b ci 9 wU Avg cu‡P †Mj| kZKiv KZwU
Avg fvj Av‡Q? (D`vniY-12)
K. 90% L. 95%
M. 80% N. 5% DËi L
☼ mgvavb:
fv‡jv Avi iB‡jv ( 180 - 9) wU ev 171 wU|
kZKiv fv‡jv Avg Av‡Q
180
100171 
wU ev 95%
37. GKwU ¯‥z‡j wkÿv_©xi msL¨v 500 Rb| Gi g‡a¨ QvÎxi
msL¨v 40% n‡j, H ¯‥z‡ji QvÎmsL¨v wbY©q Ki|
K. 280 Rb L. 360 Rb
M. 300 Rb N. 420 Rb DËi M
☼ mgvavb:
QvÎxi msL¨v 500 Gi 40% = 500
100
40
= 200 Rb
 Qv‡Îi msL¨v = (500-200) = 300 Rb |
DËi: 300 Rb|
38. ‡WwfW mvgwqK cixÿvq 900 b¤^‡ii g‡a¨ 600 b¤î
†c‡q‡Q| †m kZKiv KZ b¤î †c‡q‡Q? †gvU b¤î Ges
cÖvß b¤^‡ii AbycvZ wbY©q Ki|
K. 66.67%, 3 : 2 L. 80%, 6 : 3
M. 65%, 5 : 4 N. †KvbwUB bq DËi K
☼ mgvavb:
900 b¤^‡ii g‡a¨ †cj = 600
 1 Ó Ó Ó =
900
600
100 Ó Ó Ó =
900
100600
= 66.67
†WwfW mvgwqK cixÿvq kZKiv b¤î †cj = 66.67%
Ges †gvU b¤î I cÖvß b¤^‡ii AbycvZ = 900:600 =
600
900
=
2
3
= 3:2 DËi: 66.67%, 3:2|
39. gymvbœv eB‡qi †`vKvb †_‡K GKwU evsjv iPbv eB 84
UvKvq µq Kij| wKš‘ eBwUi Kfv‡i g~j¨ †jLv wQj
120 UvKv| †m kZKiv KZ UvKv Kwgkb †cj?
K. 25% L. 30%
M. 35% N. 40% DËi L
☼ mgvavb:
Kfvi g~j¨ 120 UvKvi eB‡qi µqg~j¨ = 84 UvKv
 Ó Ó 1 Ó Ó Ó =
120
84
UvKv
 Ó Ó 100 Ó Ó Ó =
120
10084
UvKv = 70 UvKv
gymvbœv kZKiv Kwgkb †cj = (100-70)UvKv
= 30 UvKv ev 30%| DËi: 30%|
40. GKRb PvKzwiRxwei gvwmK Avq 15000 UvKv| Zvi
gvwmK e¨q 9000 UvKv| Zvi e¨q, Av‡qi kZKiv KZ?
K. 55% L. 60%
M. 55% N. 40% DËi L
☼ mgvavb:
Avq 15000 UvKv n‡j, e¨q n‡e = 9000 UvKv
Ó 1 Ó Ó Ó Ó =
15000
9000
Ó
Ó 100 Ó Ó Ó Ó =
15000
1009000
Ó
= 60 UvKv
Zvi e¨q Av‡qi kZKiv 60 UvKv ev 60%
DËi : 60%|
kU©KvU:
15000
1009000
= 60%|
41. ‡kv‡q‡ei ¯‥z‡ji gvwmK †eZb 200 UvKv| Zvi gv Zv‡K
cÖwZw`‡bi wUwdb eve` 20 UvKv †`b| Zvi cÖwZw`‡bi
wUwdb eve` LiP, gvwmK †eZ‡bi kZKiv KZ?
K. 15% L. 10%
M. 25% N. 5% DËi L
☼ mgvavb:
†eZb 200 UvKv n‡j wUwdb eve` LiP = 20 UvKv
 Ó 1 Ó Ó Ó Ó Ó =
200
20
Ó
 Ó 100 Ó Ó Ó Ó =
200
10020
Ó
= 10 UvKv
†kv‡q‡ei wUwdb eve` LiP Zvi gvwmK ‡eZ‡bi kZKiv
10%| DËi: 10%|
kU©KvU:
20 UvKv 200 UvKvi 10 fv‡Mi 1 fvM ev 10%
42. GKwU ¯‥z‡j wkÿv_©xi msL¨v 800 Rb| eQ‡ii ïiæ‡Z
5% wkÿv_©x bZzb fwZ© Kiv n‡j, eZ©gv‡b H ¯‥z‡j
wkÿv_©xi msL¨v KZ?
K. 820 Rb L. 810 Rb
M. 840 Rb N. 870 Rb DËi M

☼ mgvavb:
5% wkÿv_x© bZzb fwZ© nIqvq,
c~‡e©i wkÿv_©x 100 n‡j eZ©gvb (100+5) = 105 Rb
Ó Ó 1 Ó Ó Ó =
100
105
Rb
 Ó Ó 800 Ó Ó Ó =
100
800105 
Ó
= 840 Rb
eZ©gv‡b H ¯‥z‡j wkÿv_©xi msL¨v = 840 Rb|
GK jvB‡b: 800 Gi 105% = 840 Rb|
43. GKwU †kÖwY‡Z 200 Rb wkÿv_©xi g‡a¨ 5% Abycw¯’Z
wQj| KZRb wkÿv_©x Dcw¯’Z wQj?
K. 190 Rb L. 170 Rb
☼ mgvavb:
5% Abycw¯’Z n‡j, Dcw¯’Z = 95 Rb|
100 Rb wkÿv_©xi g‡a¨ Dcw¯’Z wQj 95 Rb
1 Ó Ó Ó Ó Ó =
100
95
Rb
200 Ó Ó Ó Ó Ó =
100
20095
Rb
= 190 Rb|
kU©KvU: 200 Gi 95% = 190 Rb| DËi: 190 Rb|
44. hv‡n` 10% Kwgk‡b GKwU eB µq K‡i †`vKvbx‡K
180 UvKv w`j, eBwUi cÖK…Z g~j¨ KZ?
K. 200 UvKv L. 150 UvKv
M. 180 UvKv N. 190 DËi K
☼ mgvavb:
10% Kwgk‡b, eBwUi cÖK…Z g~j¨ 100 UvKv n‡j µqg~j¨
(100-10) = 90 UvKv
GLb,
eBwUi µqg~j¨ 90 UvKv n‡j cÖK…Z g~j¨ = 100 UvKv
 Ó Ó 1 Ó Ó Ó Ó =
90
100
Ó
 Ó Ó 180 Ó Ó Ó Ó =
90
180100 
Ó = 200 UvKv
wb‡Y©q eBwUi cÖK…Z g~j¨ = 200 UvKv (DËi)
45. Kjvi `vg 14 K‡g hvIqvq 420 UvKvq c~e©v‡cÿv
10wU Kjv †ewk cvIqv hvq| cÖwZ WRb Kjvi eZ©gvb
`vg KZ?
K. 60 L. 70
M. 72 N. 80 DËi M
☼ mgvavb:
Kjvi `vg 14 % ev % K‡g hvIqvq,
100 UvKvq K‡g UvKv
1 Ó Ó
 420 Ó Ó = 60 UvKv
GB 60UvKv w`‡qB AwZwi³ 10 wU Kjv cvIqv hvq|
myZivs 10wU Kjvi eZ©gvb `vg = 60UvKv |
Zvn‡j 1wU Kjvi `vg = 6010= 6UvKv myZivs 12wU
Kjvi `vg = 612 = 72 UvKv|
jvf ÿwZ
mßg †kÖwY 2.2
Aóg †kÖwY 2.1
beg-`kg †kÖwY 3.5
46. 75 UvKvq 15wU ej‡cb wK‡b 90 UvKvq weµq Ki‡j
kZKiv KZ jvf n‡e? (7g †kÖwY, Aby:2.2 Gi
D`v:11)
K. 10 % L. 15 %
M. 20 % N. 25 % DËi M
☼ mgvavb:
jvf = weµqg~j¨ - µqg~j¨ =(90-75)=15 UvKv
 75 UvKvq jvf nq 15 UvKv
1 Ó Ó Ó
75
15
Ó
 100 Ó Ó Ó
75
10015 
Ó ev 20 UvKv
wb‡Y©q jvf 20%|
kU©KvU :
15 UvKv 75 UvKvi 5 fv‡Mi 1 fvM = 20%
%
7
2
7
2
7
100
7
100
1007
100

1007
420100


jvf ÿwZUwcK †eBRW&

47. GKRb gvQ we‡µZv cÖwZ nvwj Bwjk gvQ 1600 UvKvq
wK‡b cÖwZwU gvQ 350 UvKv K‡i weµq Ki‡jb| Zuvi
kZKiv KZ jvf ev ÿwZ n‡jv? [7g †kÖwY-2.2-
D`vniY-12]
K. jvf 12% L. ÿwZ 12%
M. jvf 12 % N. ÿwZ 12 % DËi N
☼ mgvavb:
4wU = 1600 n‡j 1wU = 400
GLb 400 †Z ÿwZ = 400-350 = 50 UvKv|
ÿwZi nvi = = 12 %
48. GKev· Av½yi 2750 UvKvq weµq Kivq 450 UvKv
ÿwZ n‡jv| H Av½yi 3600 UvKvq weµq Ki‡j KZ
jvf ev ÿwZ n‡Zv? (7g †kÖwY, Aby:2.2 Gi D`v:13)
M. 400 UvKv N. 500 UvKv DËi M
☼ mgvavb:
Av½y‡ii weµqg~j¨ = 2750 UvKv
ÿwZ = 450 Ó
µqg~j¨ = 3200 UvKv (‡hvM K‡i)
Avevi, weµqg~j¨ = 3600 UvKv
µqg~j¨ = 3200 UvKv
jvf = 400 UvKv(we‡qvM K‡i)
jvf 400 UvKv|
49. GKRb Pv we‡µZv GKev· Pv cvZv †KwR cÖwZ 80 UvKv
wn‡m‡e µq K‡ib| me Pv cvZv †KwR cÖwZ 75 UvKv
`‡i weµq Kivq 500 UvKv ÿwZ nq| wZwb KZ †KwR
Pv cvZv µq K‡iwQ‡jb?(7g †kÖwY2.2 Gi D`v:14)
K. 75 ‡KwR L. 84 †KwR
M. 96 †KwR N. 100‡KwR DËi N
☼ mgvavb:
‡KwR cÖwZ Pv cvZvi µqg~j¨ 80 UvKv
Ó Ó Ó Ó weµqg~j¨ 75 UvKv
 1 †KwR Pv cvZvq ÿwZ = 5 UvKv
GLb, 5 UvKv ÿwZ nq 1 †KwR‡Z
 1 Ó Ó Ó
5
1
Ó
 500 Ó Ó Ó
5
5001
Ó = 100 †KwR‡Z
Pv cvZv µq K‡iwQ‡jb 100 †KwR|
DËi: 100 †KwR|
50. GKRb wWgwe‡µZv cÖwZ WRb wWg 101 UvKv `‡i 5
WRb Ges 90 UvKv `‡i 6 WRb wWg wK‡b KZ `‡i
weµq Ki‡j Zuvi WRb cÖwZ 3 UvKv jvf n‡e? (7g
†kÖwY, Aby:2.2 Gi D`v:15)
M. 97 UvKv N. 98 UvKv DËi N
☼ mgvavb:
1 WRb wW‡gi µqg~j¨ 101 UvKv
 5 Ó Ó Ó 101  5 UvKv ev 505 UvKv
Avevi, 1 WRb wW‡gi µqg~j¨ 90 UvKv
 6 Ó Ó Ó 90  6 UvKv ev 540 UvKv
 (5+6) WRbev 11 WRb wW‡gi µqg~j¨ =(505 +
540) UvKv ev 1045 UvKv
 1WR‡bi Mo µqg~j¨ =
11
1045
UvKv ev 95 UvKv
WRb cÖwZ 3 UvKv jv‡f 1 WRb wW‡gi weµqg~j¨
(95+3) UvKv ev 98 UvKv DËi- 98 UvKv|
51. GKwU QvMj 10% ÿwZ‡Z weµq Kiv n‡jv| weµqg~j¨
450 UvKv †ewk n‡j 5% jvf n‡Zv| QvMjwUi µqg~j¨
KZ? (7g †kÖwY, Aby:2.2 Gi D`v:16)
☼ mgvavb:
10% ÿwZ‡Z weµqg~j¨ = 100-10 = 90 UvKv|
Avevi 5% jv‡f weµqg~j¨ = 100+5 = 105 UvKv|
2 weµqg~‡jï cv_©K¨ = 105-90 = 15UvKv|
`yB weµqg~‡jï cv_©K¨ 15 UvKv n‡j µqg~j¨ = 100
`yB weµqg~‡jï cv_©K¨ 450 UvKv n‡j µqg~j¨ =
15
450100 
= 3000 UvKv|
kU©KvU :
10% ÿwZ + 5% jvf = 15% cv_©K¨
UvKvq cv_©K¨ = 450 UvKv| 15% = 450
myZivs µqg~j¨ 100% = 3000 UvKv|
52. bvwej wgwó †`vKvb †_‡K cÖwZ †KwR 250 UvKv wn‡m‡e
2 †KwR m‡›`k µq Ki‡jv| f¨v‡Ui nvi 4 UvKv n‡j,
m‡›`k µq eve` †m †`vKvwb‡K KZ UvKv †`‡e? (7g
†kÖwY, Aby: 2.2 D:17)
M. 550 UvKv N. 600 UvKv DËi L
☼ mgvavb:
f¨vU Qvov ïay wgwói `vg = 2502 = 500 UvKv|
f¨vU = 500 Gi 4% = 20 UvKv|
2
1
2
1
400
10050
2
1

myZivs f¨vU mn †gvU `vg = 500+20 =520 UvKv|
53. bvwm‡ii gvwmK g~j‡eZb 27,650 UvKv| evwl©K †gvU
Av‡qi cÖ_g `yB jÿ cÂvk nvRvi UvKvi AvqKi 0
(k~b¨) UvKv| cieZ©x UvKvi Dci AvqK‡ii nvi 10
UvKv n‡j, bvwmi mv‡ne KZ UvKv AvqKi †`b? (7g
†kÖwY, Aby: 2.2 D:18)
M. 7180 UvKv N. 6600 UvKv DËi K
☼ mgvavb:
1 gv‡mi g~j †eZb 27,650 UvKv
 12 Ó Ó Ó (2765012) UvKv
= 3,31,800 UvKv
 Ki‡hvM¨ UvKvi cwigvY (3,31,800 - 2,50,000)
UvKv ev 81,800 UvKv
100 UvKvq AvqKi = 10 UvKv
1 Ó Ó
100
10
Ó
81,800 Ó Ó
100
8180010 
Ó ev 8,180 UvKv
bvwmi mv‡ne 8,180 UvKv AvqKi †`b|
54. GKRb †`vKvb`vi cÖwZ wgUvi 200 UvKv `‡i 5 wgUvi
Kvco wK‡b cÖwZ wgUvi 225 UvKv `‡i weµq Ki‡j
KZ UvKv jvf n‡q‡Q? (7g †kÖwY, Aby: 2.2 Gi 1)
☼ mgvavb:
1 wgUvi Kvc‡oi µqg~j¨ 200 UvKv
5 Ó Ó Ó (2005) Ó = 1000 UvKv|
Avevi, 1 wgUvi Kvc‡oi weµqg~j¨ 225 UvKv
5 Ó Ó Ó (5225) Ó
= 1225 UvKv
jvf = (1215 - 1000) UvKv = 125 UvKv
AZGe, jvf = 125 UvKv|
kU©KvU :
1wgUv‡i jvf 225-200 = 25UvKv n‡j
5 wgUv‡i jvf n‡e 255 = 125 UvKv|
55. GKRb Kgjvwe‡µZv cÖwZ nvwj 60 UvKv `‡i 5 WRb
Kgjv wK‡b cÖwZ nvwj 50 UvKv `‡i weµq Ki‡j
kZKiv KZ jvf ev ÿwZ nq? (7g †kÖwY, Aby: 2.2 Gi 2)
K. jvf 150 UvKv
L. ÿwZ 150 UvKv
M. jvf 140 UvKv
N. ÿwZ 140 UvKv DËi L
☼ mgvavb:
4wUi µqg~j¨ = 60UvKv n‡j,
1wUi µqg~j¨ = 604=15UvKv|
5 WRb ev 60wUi µqg~j¨ = 1560 = 900 UvKv|
Avevi, 4wUi `vg 50UvKv n‡j
1wUi weµqg~j¨ = 504 = 12.5 UvKv
Ges 60wUi weµqg~j¨ = 12.560 = 750UvKv
myZivs †gvU ÿwZ = 900-750 = 150 UvKv|
56. iwe cÖwZ †KwR 40 UvKv `‡i 50 †KwR PvDj wK‡b 44
UvKv †KwR `‡i weµq Ki‡j KZ jvf ev ÿwZ n‡e?
(7g †kÖwY, Aby: 2.2 Gi 3)
☼ mgvavb:
1 †KwR‡Z jvf = 44-40 = 4 UvKv
myZivs 50 †KwR‡Z jvf = 450 = 200 UvKv|
57. cÖwZ wjUvi wgéwfUv `ya 52 UvKvq wK‡b 55 UvKv `‡i
weµq Ki‡j kZKiv KZ jvf nq? (7g †kÖwY, Aby:
2.2 Gi 4)
☼ mgvavb:
jvf = (55-52) UvKv = 3 UvKv
52 UvKvq jvf nq =3 UvKv
1 Ó Ó Ó
52
3
Ó
100 Ó Ó Ó
52
1003
Ó =
13
75
= 5
13
10
UvKv
jvf = 5
13
10
%| DËi: jvf 5
13
10
%|
58. cÖwZwU PK‡jU 8 UvKv wn‡m‡e µq K‡i 8.50 UvKv
`‡i weµq K‡i 25 UvKv jvf n‡jv, †gvU KqwU PK‡jU
µq Kiv n‡qwQj? (7g †kÖwY, Aby: 2.2 Gi 5)
K. 40wU L. 48wU
M. 50wU N. 60wU DËi M
☼ mgvavb:
jvf (8.5- 8) = 0.5 UvKv
0.50 UvKv jvf nq 1 wU PK‡j‡U
 1 Ó Ó Ó
50
1
.
UvKv
25 Ó Ó Ó
50
251
.

= 50wU|
DËi: 50wU P‡Kv‡jU|

59. cÖwZ wgUvi 125 UvKv `‡i Kvco µq K‡i 150 UvKv
`‡i weµq Ki‡j †`vKvb`v‡ii 2000 UvKv jvf nq|
†`vKvb`vi †gvU KZ wgUvi Kvco µq K‡iwQ‡jb? (7g
†kÖwY, Aby: 2.2 Gi 6)
K. 70 wgUvi L. 80 wgUvi
M. 90 wgUvi N. 110 wgUvi DËi L
☼ mgvavb:
1 wgUvi Kvc‡o jvf nq = 150 - 125 = 25 UvKv
25 UvKv jvf nq = 1 wgUv‡i
1 Ó Ó Ó =
25
1
Ó
 2000 Ó Ó =
25
20001
Ó = 80DËi: 80wg.
60. GKwU `ªe¨ 190 UvKvq µq K‡i 175 UvKvq weµq
Ki‡j kZKiv KZ jvf ev ÿwZ n‡e? (7g †kÖwY, Aby:
2.2 Gi 7)
☼ mgvavb:
ÿwZ = (190-175) UvKv = 15 UvKv
190 UvKvq ÿwZ nq = 15 UvKv
1 Ó Ó Ó
190
15
Ó
100 Ó Ó Ó
190
10015 
Ó =
19
150
Ó
= 7
19
17
UvKv
DËi: 7
19
17
% ÿwZ |
61. 25 wgUvi Kvco †h g~‡j¨ µq K‡i, †mB g~‡j¨ 20
wgUvi Kvco weµq Ki‡j kZKiv KZ jvf ev ÿwZ
n‡e? (7g †kÖwY, Aby: 2.2 Gi 8)
K. jvf 20% L. ÿwZ 20%
M. jvf 25% N. ÿwZ 25% DËi M
☼ mgvavb:
g‡b Kwi, Kvc‡oi g~j¨ x UvKv
25 wgUvi Kvc‡oi µqg~j¨ x UvKv
 1Ó Ó Ó
25
x
Ó
Avevi, 20 wgUvi Kvc‡oi weµqg~j¨ x UvKv
 1Ó Ó Ó Ó
20
x
Ó
GLv‡b,
20
x
>
25
x
A_©vr µqg~‡j¨i ‡P‡q weµqg~j¨ †ewk
nIqvq jvf n‡q‡Q|
jvf =
20
x
-
25
x
=
100
45 xx 
=
100
x
UvKv
25
x
UvKvq jvf nq =
100
x
UvKv
1Ó Ó Ó
100
x

x
25
Ó
100Ó Ó Ó
100
x

x
25
 100 = 25 UvKv
jvf 25%| DËi: jvf 25%|
kU©KvU :
jv‡fi nvi = %
100
x2
[x=25-20=5]
= %%
100
25
100
x2
 = 25% DËi: 25 %|
[Help:20wgUv‡i jvf = 5wgUvi ZvB 100 wgUv‡i jvf
n‡e 25 wgUvi DËi; 25% |]
62. 5 UvKvq 8 wU AvgjwK µq K‡i 5 UvKvq 6 wU `‡i
weµq Ki‡j kZKiv KZ jvf ev ÿwZ n‡e ?[KwgDwbwU
†nj&_ †Kqvi †cÖvfvBWvi -2018(wj ] + [cjøx mÂq
e¨vsK- (K¨vk)-2018] +[7g †kÖwY-(Aby:2.2)]
K. 35 jvf L. 33 jvf
M. 30% jvf N. 33 jvf DËi L
wUcm: [100UvKvq 10wU wK‡b 100UvKv 8wU wewµi
AsKwUi gZB A_ev Gi Av‡Mi AsKwUI wKš‘ GKB|
†hLv‡b UvKvi cwigvY mgvb Av‡Q | 1UvKv = 1UvKv,
100UvKv = 100UvKv, 5UvKv = 5UvKv me GKB wbq‡g
n‡e| Gfv‡e wbqg wgwj‡q AsK Ki‡j Kg K‡ó †ewk wKQy
†kLv hvq]
☼ mgvavb:
GLv‡b `y †ÿ‡ÎB µqg~j¨ 5UvKv nIqvq UvKvi K_v ev`
w`‡q ïay msL¨v a‡iB wn‡me Kiv hvq
jvf = 8-6 = 2wU| 6wU †Z jvf 2wU n‡j jv‡fi nvi
n‡e 3 fv‡Mi 1 fvM ev 33 |
Avevi j¤^v ÷vB‡j wPšÍv Ki‡Z PvB‡jGfv‡e Ki‡Z cv‡ib,
%
2
1
%
3
1
%
2
1
%
3
1

8wUi µqg~j¨ = 5UvKv 1wUi µqg~j¨ =
8
5
UvKv|
GKBfv‡e 1wUi weµqg~j¨ =
6
5
UvKv
Gici we‡qvM K‡i jvf Ges †mLvb †_‡K kZKiv jvf
‡ei Kiv hv‡e wKš‘ Zv‡Z mgq †ewk jvM‡e |
63. GKwU Mvwoi weµqg~j¨ MvwowUi µqg~‡j¨i
5
4
As‡ki
mgvb| kZKiv KZ jvf ev ÿwZ n‡e?(7g
†kÖwY,Aby:2.2Gi 10)
K.20% ÿwZ L.25% jvf
M.80%ÿwZ N.20%jvf DËi K
[Help :5UvKvq wK‡b 4 UvKvq wewµ K‡i 1 UvKv ÿwZ,
5UvKvq 1UvKv ÿwZ n‡j 100 UvKvq ÿwZ 20 UvKv ev
20%]
64. GKwU `ªe¨ 400 UvKvq weµq Ki‡j hZ ÿwZ nq 480
UvKvq weµq Ki‡j, Zvi wZb¸Y jvf nq| `ªe¨wUi
µqg~j¨ wbY©q Ki? (7g †kÖwY, Aby: 2.2 Gi 11)
☼ mgvavb:
awi, 400 UvKvq weµq Ki‡j x UvKv ÿwZ nq|
 `ªe¨wUi µq g~j¨ (400+x) UvKv
Avevi, 480 UvKvq weµq Ki‡j 3x UvKv jvf nq
 `ªe¨wUi weµg~j¨ (480-3x) UvKv
cÖkœg‡Z,
400 + x = 480-3x
x = 20  `ªe¨wUi µq g~j¨ = (400+x) UvKv
= (400+20) =420 UvKv|DËi: 420 UvKv
65. GKwU Nwo 625 UvKvq weµq Ki‡j 10% ÿwZ nq|
KZ UvKvq weµq Ki‡j 10% jvf n‡e?
☼ mgvavb:
awi, NwowUi µqg~j¨ 100 UvKv
10% ÿwZ‡Z weµqg~j¨ (100-10) = 90 UvKv|
weµqg~j¨ 90 UvKv n‡j µqg~j¨ = 100 UvKv
 Ó 1 Ó Ó Ó =
90
100
Ó
 Ó 625 Ó Ó Ó =
90
625100 
Ó
=
9
6250
UvKv
Avevi, 10% jv‡f, weµqg~j¨ (100+10) = 110 UvKv
µqg~j¨ 100 UvKv n‡j weµqg~j¨ 110 UvKv
 Ó 1 Ó Ó Ó
100
110
Ó

9
6250
Ó Ó Ó
9100
6250110


Ó
=
9
6875
UvKv ev 763
9
8
UvKv| DËi:
kU©KvU : 90%= 625; 1% =
90
625
110%=
90
110625
=
9
6875
= 763
9
8
UvKv|
66. gvBkv cÖwZ wgUvi 20 UvKv `‡i 15 wgUvi jvj wdZv
µq Ki‡jv| f¨v‡Ui nvi 4 UvKv| †m †`vKvwb‡K 500
UvKvi GKwU †bvU w`j| †`vKvwb Zv‡K KZ UvKv †diZ
†`‡eb? (7g ‡kÖYx, Aby- 2.2, cÖkœ bs- 13)
☼ mgvavb:
f¨vU Qvov †gvU µqg~j¨ = 2015 = 300UvKv|
‡gvU f¨vU = 300 Gi 4% = 12 UvKv|
f¨vU mn ‡gvU g~j¨ = 300+12 = 312UvKv|
500 UvKv w`‡j †diZ cv‡e = 500-312 = 188
67. wg. ivq GKRb miKvwi Kg©KZ©v| wZwb Zx_©¯’vb
cwi`k©‡bi Rb¨ fvi‡Z hv‡eb| hw` evsjv‡`kx 1 UvKv
mgvb fviZxq 0.63 iæwc nq, Z‡e fviZxq 3000
iæwci Rb¨ evsjv‡`‡ki KZ UvKv cÖ‡qvRb n‡e?
☼ mgvavb:
fviZxq 0.63 iæwci Rb¨ evsjv‡`‡ki cÖ‡qvRb 1 UvKv
 Ó 1 Ó Ó Ó Ó =
0.63
1
Ó
 Ó 3000 Ó Ó Ó Ó =
0.63
3000
Ó
= 4761.90 UvKv|
DËi: 4761.90 UvKv|
68. bxwjgv mv‡ne GKRb PvKwiRxwe| Zvi gvwmK g~j‡eZb
22,250 UvKv| evwl©K †gvU Av‡qi cÖ_g `yB jÿ cÂvk
nvRvi UvKvi AvqKi 0 (k~Y¨) UvKv| cieZ©x UvKvi
Dci AvqK‡ii nvi 10 UvKv n‡j bxwjg Ki eve` KZ
UvKv cwi‡kva K‡ib?

☼ mgvavb:
1 gv‡mi g~j †eZb = 22,250 UvKv
12 Ó Ó Ó = (2225012) UvKv
= 267000 UvKv
Ki‡hvM¨ UvKvi cwigvY = (267000-180000) UvKv
= 87,000 UvKv
GLb, ‡gvU AvqKi = 87000 Gi 10% = 8700 UvKv|
DËi: 8,700 UvKv|
Aóg †kÖwY
69. GKRb †`vKvb`vi cÖwZ nvwj wWg 25 UvKv `‡i µq
K‡i cÖwZ 2 nvwj 56 UvKv `‡i weµq Ki‡j Zuvi
kZKiv KZ jvf n‡e?
K. jvf 12% L. jvf 14%
M. jvf 14% N. jvf 16% DËi K
☼ mgvavb:
1 nvwj wW‡gi µqg~j¨ 25 UvKv
2 nwj Ó Ó 25 2 UvKv ev 50 UvKv|
jvf = ( 56-50) UvKv ev 6 UvKv|
50 UvKvq jvf =6 UvKv
 1 Ó Ó =
50
6
UvKv
 100 Ó Ó =
50
1006
= 12 UvKv
 jvf 12%
70. GKwU QvMj 8% ÿwZ‡Z weµq Kiv n‡jv| QvMjwU
AviI 800 UvKv †ewk g~‡j¨ weµq Ki‡j 8% jvf
n‡Zv| QvMjwUi µqg~j¨ wbY©q Ki? (8g †kÖwY, Aby:2.1
Gi D`v:2)
K.3000 UvKv L. 4000 UvKv
☼ mgvavb:
QvMjwUi µqg~j¨ 100 UvKv n‡j, 8% ÿwZ‡Z weµqg~j¨
(100-8) = 92 UvKv|
Avevi, 8% jv‡f weµqg~j¨ ( 100+8) = 108 UvKv|
 weµqg~j¨ †ewk nq 108- 92= 16 UvKv|
weµqg~j¨ 16 UvKv †ewk n‡j µqg~j¨ = 100 UvKv
Ó 1 Ó Ó Ó Ó =
16
100
Ó
Ó 800 ÕÕ Ó Ó Ó =
16
800100 
Ó
= 5000 UvKv  QvMjwUi µqg~j¨ 5000 UvKv|
71. GKwU cY¨`ªe¨ weµq K‡i cvBKvwi we‡µZvi 20% Ges
LyPiv we‡µZvi 20% jvf nq| hw` `ªe¨wUi LyPiv
weµqg~j¨ 576 UvKv nq Z‡e cvBKvwi we‡µZvi
µqg~j¨ KZ? [AvBwmwe-G¨vwm‡÷›U ‡cÖvMÖvgvi - 2019]
K.750 L.600
M.400 N.480 DËi M
mgvavb: (GB wbq‡g cÖPzi cÖkœ wcÖwj: I wjwLZ
cixÿvq Av‡m| wjwLZ As‡k we¯ÍvwiZ mgvavb Av‡Q )
awi, µqg~j¨ = x
Zvn‡j cÖkœg‡Z,xGi 120% Gi 120% = 576
x = 576
120
100
120
100
 = 400 UvKv|
(hZ evi B jvf ÿwZ ‡nvK †k‡li `vg †_‡K ïiæi `vg
†ei Kivi Rb¨ GB wbqgUv memgq g‡b ivLvi Rb¨
mnR|)
Z‡e `ªæZ DËi †ei Ki‡Z PvB‡j Gfv‡e fvev Av‡iv mnR:
`yevi 20% K‡i jvf A_© GKmv‡_ 44% jv‡fi mgvb|
Zvn‡j 144% = 576 n‡j 100% = 400
72. GKRb †`vKvb`vi wKQz Wvj 2375.00 UvKvq weµq
Kivq Zvi 5% ÿwZ n‡jv| H Wvj KZ UvKvq weµq
Ki‡j Zvi 6% jvf n‡Zv?
☼ mgvavb:
5% ÿwZ‡Z, µqg~‡j¨ 100 UvKv n‡j weµqg~j¨ (100 -
5) UvKv = 95 UvKv
GLb, weµqg~j¨ 95 UvKv n‡j µqg~j¨ = 100 UvKv
Ó 1 Ó Ó Ó =
95
100
Ó
 Ó2375 Ó Ó Ó =
95
2375100 
Ó
= 2500 UvKv
Wv‡ji µqg~j¨ 2500 UvKv|
Avevi, 6% jv‡f, µqg~j¨ 100 UvKv n‡j weµqg~j¨
(100+6) UvKv ev 106 UvKv
µqg~j¨ 100 UvKv n‡j weµqg~j¨ 106 UvKv
 Ó 1 Ó Ó Ó
100
106
Ó
kU©KvU© †UKwbK:
8% ÿwZ + 8% jvf = 16% cv_©K¨
UvKvq cv_©K¨ = 800 UvKv|
Zvn‡j 16% = 800 myZivs µqg~j¨ 100% = 5000

 Ó 2500 Ó Ó Ó
100
2500106 
Ó
= 2650 UvKv
73. 30 UvKvq 10wU `‡i I 15wU `‡i mgvb msL¨K Kjv
µq K‡i me¸‡jv 30 UvKvq 12wU `‡i weµq Ki‡j
kZKiv KZ jvf ev ÿwZ n‡e?
K. jvf n‡e 5 UvKv
L. jvf n‡e 15 UvKv
M. ÿwZ n‡e 15 UvKv
N. †Kvb jvf ev ÿwZ n‡e bv DËi N
☼ mgvavb:
10wU Kjvi µqg~j¨ 30 UvKv
 1 Ó Ó Ó =
10
30
Ó = 3 UvKv|
Avevi, 15wU Kjvi µqg~‡j¨ = 30 UvKv
 1 Ó Ó Ó =
15
30
Ó = 2 UvKv|
(1+1) ev, 2wU Kjvi µqg~j¨ = (3+2) = 5 UvKv
Aevi, 12wU Kjvi weµqg~j¨ = 30 UvKv
 1 Ó Ó Ó =
12
30
UvKv
 2 Ó Ó Ó =
12
230
= 5 UvKv|
GLv‡b µqg~j¨ I weµqg~j¨ DfqB mgvb| myZivs jvf ev
ÿwZ †KvbwUB n‡e bv|
DËi: jvf ev ÿwZ wKQzB n‡e bv|
mij I †h․wMK
gybvdv
Aóg †kÖwY 2.1, 2.2
beg-`kg †kÖwY 3.5 (AvswkK)
74. evwl©K kZKiv gybvdvi nvi 10.50 UvKv n‡j 2000
UvKvi 5 eQ‡ii gybvdv KZ n‡e?
☼ mgvavb:
100 UvKvi 1 eQ‡ii gybvdv = 10.50 UvKv
 1 Ó 1 Ó Ó =
100
10.50
Ó
2000 Ó 5 Ó Ó =
100
5200010.50 
= 1050 UvKv| (DËi)
75. evwl©K gybvdv kZKiv 10 UvKv †_‡K K‡g 8 UvKv n‡j,
3000 UvKvi 3 eQ‡ii gybvdv KZ Kg n‡e?
mgvavb:
evwl©K kZKiv gybvdvi nvi K‡g (10 - 8) = 2 UvKv
A_©vr, 100 UvKvi 1 eQ‡ii gybvdv K‡g 2 UvKv
 1 Ó 1 Ó Ó Ó =
100
2
Ó
 3000 Ó 3 Ó Ó Ó =
100
330002 
= 180 UvKv DËi: 180 UvKv K‡g
76. evwl©K kZKiv gybvdv KZ n‡j, 13000 UvKv 5 eQ‡i
gybvdv-Avm‡j 18850 UvKv n‡e?
K. 11% L. 10%
M. 9% N. 8% DËi M
☼ mgvavb:
gybvdv-Avmj = 18850 UvKv
Avmj = 13000 UvKv
gybvdv = 5850 UvKv
GLv‡b, 13000 UvKvi 5 eQ‡ii gybvdv = 5850 UvKv
1 Ó 1 Ó Ó =
513000
5850

Ó
 100 Ó 1 Ó Ó =
513000
1005850


= 9 UvKv
wb‡Y©q gybvdvi nvi 9%| DËi: 9%
77. evwl©K kZKiv KZ gybvdvq †Kv‡bv Avmj 8 eQ‡i
gybvdv-Avm‡j wØ¸Y n‡e? (8g †kÖwY, Aby: 2.1 Gi 7)
K.12.5% L. 13%
M. 14% N. 15% DËi K
mgvavb: (‡evW© eB‡qi wbq‡g)
kU©KvU© †UKwbK:
95% = 2375 UvKv
106% =
95
1062375
= 2650 UvKv|
mij ‡hŠwMK gybvdvI

Avgiv Rvwb, I =
100
pnr
ev, I =
100
81 r
ev, 8r = 100
ev, r =
8
100
ev, r = 12.5gybvdvi nvi 12.5%
r =?
n =8
Avmj p = 1
gybvdv I = 1
78. 6500 UvKv †h nvi gybvdvq 4 eQ‡i gybvdv-Avm‡j
8840 UvKv nq, H GKB nvi gybvdvq KZ UvKv 4
eQ‡i gybvdv-Avm‡j 10200 UvKv n‡e?
☼ mgvavb:
gybvdv-Avmj = 8840 UvKv
Avmj = 6500 UvKv
gybvdv = 2340 UvKv
A_©vr, 6500 UvKvi 4 eQ‡ii gybvdv 2340 UvKv
 1 Ó 1 Ó Ó =
46500
2340

Ó
 100 Ó 1 Ó Ó =
46500
1002340


= 9 UvKv
wb‡Y©q gybvdvi nvi 9%
Avevi, 100 UvKvi 1 eQ‡ii gybvdv 9 UvKv
100 Ó 4 Ó Ó =(94) = 36 UvKv
gybvdv-Avm‡j = (36+100) UvKv ev, 136 UvKv
A_©vr, gybvdv-Avm‡j 136 UvKv n‡j Avmj 100 UvKv
 Ó Ó 1 Ó Ó Ó =
136
100
Ó
 Ó 10200 Ó Ó Ó =
136
10200100 
= 7500 UvKv|
myZivs Avmj = 7500 UvKv|
79. wiqvR mv‡ne wKQz UvKv e¨vs‡K Rgv †i‡L 4 eQi ci 4760
UvKv gybvdv cvb| e¨vs‡Ki evwl„©K gybvdvi nvi 8.50 UvKv
n‡j wZwb e¨vs‡K KZ UvKv Rgv †i‡LwQ‡jb?
K. 14000 UvKv L. 12000 UvKv
☼ mgvavb:
e¨vs‡K 100 UvKvi 1 eQ‡ii gybvdv 8.50 UvKv
 Ó 100 Ó 4 Ó Ó = 8.504 = 34 UvKv
gybvdv 34 UvKv n‡j Avmj = 100 UvKv
 Ó 1 Ó Ó Ó =
34
100
Ó
Ó 4760 Ó Ó Ó =
34
4760100 
= 14000
Avmj = 14000 UvKv
80. kZKiv evwl©K †h nv‡i †Kv‡bv g~jab 6 eQ‡i gybvdv
g~ja‡b wØ¸b nq, †mB nv‡i KZ UvKv 4 eQ‡i gybvdv-
g~ja‡b 2050 UvKv n‡e?
☼ mgvavb:
awi, g~jab 100 UvKv
gybvdv-g~ja‡b wØ¸Y n‡j (100  2) = 200 UvKv nq
gybvdv n‡e = (200 - 100) UvKv = 100 UvKv
100 Ó 1 Ó Ó =
6
100
=
3
50
UvKv
gybvdvi nvi =
3
50
%
Avevi, 100 UvKvi 1 eQ‡ii gybvdv
3
50
UvKv
100 Ó 4 Ó Ó =
3
450
=
3
200
UvKv
gybvdv-g~jab =
3
200
100  =
3
200300
UvKv
=
3
500
UvKv
gybvdv-g~jab
3
500
UvKv n‡j g~jab = 100 UvKv
 Ó 1 Ó Ó Ó =
500
3100 
Ó
 Ó 2050 Ó Ó Ó =
500
20503100 
mnR wbq‡g:
Avmj 100 UvKv n‡j 8 eQi ci Zv 200 UvKv
n‡e | †hLv‡b Avmj 100
Avi my` = 200-100 = 100 UvKv|
GLb 8 eQ‡i my` 100 n‡j
1 eQ‡i my` n‡e =
8
100
=12.5%

= 1230 UvKv
myZivs 1230 UvKv| DËi: 1230 UvKv
81. evwl©K kZKiv 6 UvKv gybvdvq 500 UvKvi 4 eQ‡ii
gybvdv hZ nq, evwl©K kZKiv 5 UvKv gybvdvq KZ
UvKvi 2 eQi 6 gv‡mi gybvdv ZZ n‡e?
☼ mgvavb:
100 UvKvi 1 eQ‡ii gybvdv = 6 UvKv
 1 Ó 1 Ó Ó =
100
6
Ó
500 Ó 4 Ó Ó =
100
45006 
Ó = 120 UvKv
Avevi, 2 eQi 6 gvm = 2
2
1
eQi =
2
5
eQi
100 Ó
2
5
Ó Ó =
2
55
=
2
25
Ó
gybvdv
2
25
UvKv n‡j Avmj = 100 UvKv
Ó 1 Ó Ó Ó Ó
25
2100 
Ó
Ó 120 Ó Ó Ó
25
1202100 
Ó = 960 UvKv
wb‡Y©q g~jab 960 UvKv| DËi: 960 UvKv
82. evwl©K gybvdv 8% †_‡K ‡e‡o 10% nIqvq wZkv
gvigvi Avq 4 eQ‡i 128 UvKv †e‡o †Mj| Zvi g~jab
KZ wQj? (8g †kÖwY, Aby: 2.1 Gi 12)
☼ mgvavb: (‡evW© eB‡qi wbq‡g)
I =
100
pnr
ev, 128 =
100
24p
ev, 8p = 100128
ev, p =
8
128100 
ev, p = 1600
 g~jab 1600 UvKv
GLv‡b,
r =10-8=2
I =128
n= 4
p=?
mnR wbq‡g:
my‡`i nvi Kg‡jv 10%-8% = 2%
Avevi, 4 eQ‡i K‡g 128 UvKv
1 eQ‡i K‡g,
4
128
=32 UvKv(DfqwU 1 eQ‡i Avm‡jv)
2 UvKv Avq Kg‡j Avmj = 100UvKv
1 ÕÕ ÕÕ ÕÕ ÕÕ =
2
100
UvKv
32 ÕÕ ÕÕ ÕÕ ÕÕ =
2
32100 
=1600UvKv (2
UvKvi †_‡K 32, 16¸Y eo ZvB 100 Gi 16 ¸Y = 1600)
83. ‡Kv‡bv Avmj 3 eQ‡i gybvdv Avm‡j 1578 UvKv Ges
5 eQ‡i gybvdv-Avm‡j 1830 UvKv nq| Avmj I
gybvdvi nvi wbY©q Ki|
K. 1200 UvKv I 11.50%
L. 1200 UvKv I 10.50%
M. 1200 UvKv I 9.50%
N. 1200 UvKv I 12.50% DËi L
mgvavb:
Avmj + 5 eQ‡ii gybvdv = 1830 UvKv
2 eQ‡ii gybvdv = 252 UvKv
1 Ó Ó =
2
252
3 Ó Ó =
2
3252 
= 378
3 eQ‡ii gybvdv-Avm‡j = 1578 UvKv
3 eQ‡iigybvdv = 378 UvKv
 Avmj = 1200 UvKv
GLb, 1200 UvKvi 3 eQ‡ii gybvdv = 378 UvKv
 1 Ó 1 Ó Ó
31200
378

Ó
 100 Ó 1 Ó Ó
31200
100378


Ó =
2
21
UvKv
gybvdvi nvi =
2
21
% = 10
2
1
%
DËi: Avmj 1200 UvKv Ges gybvdvi nvi 10
2
1
%
gy‡L gy‡L: (`y cv‡k 1eQi) A_ev, (`y cv‡k 4 eQi)
2% = 32 (16 ¸Y)
100% = 1600
8% = 128 (16 ¸Y)
100% = 1600

kU©KvU :
Back solved Method Gi gva¨‡g GB AsKwU
mn‡R mgvavb Kiv hvq| 1200 UvKv Avmj n‡j 3
eQ‡ii jvf 378 UvKv| A_©vr 1eQ‡ii gybvdv
126UvKv| myZivs gybvdvi nvi =
126
12
1 =
10.50%|
84. evwl©K 10% gybvdvq 3000 UvKv Ges 8% gybvdvq
2000 UvKv wewb‡qvM Ki‡j †gvU g~ja‡bi Ici M‡o
kZKiv KZ UvKv nv‡i gybvdv cvIqv hv‡e?
K. 9.5% L. 9.2%
M. 9.7% N. 9.8% DËi L
mgvavb:
1 Ó 1 Ó Ó =
100
10
Ó
3000 Ó 1 Ó Ó =
100
300010
= 300 UvKv
Avevi, 100 UvKvi 1 eQ‡ii gybvdv 8 UvKv
 1 Ó 1 Ó Ó
100
8
Ó
 2000 Ó 1 Ó Ó
100
20008
= 160 UvKv
‡gvU gybvdv = (300 + 160) UvKv = 460 UvKv
‡gvU g~jab = (3000 + 2000) UvKv = 5000 UvKv
 1 Ó 1 Ó Ó
5000
460
Ó
100 Ó 1 Ó Ó
5000
100460
Ó
=
5
46
UvKv ev 9.2 UvKv
wb‡Y©q Mo gybvdvi nvi = 9.2 %| DËi: 9.2 %
kU©KvU:
3000 Gi 10%+2000 Gi 8%
= 300+160 = 460 UvKv|
GLb 5000 UvKvi gybvdv = 460 n‡j,
gybvdvi nvi n‡e
5000
100460
= 9.2%
85. iwWªK †Mv‡gR 3 eQ‡ii Rb¨ 10000 UvKv Ges 4
eQ‡ii Rb¨ 15000 UvKv e¨vsK †_‡K FY wb‡q e¨vsK
†K †gvU 9900 UvKv gybvdv †`b| Dfq‡ÿ‡Î gybvdvi
nvi mgvb n‡j gybvdvi nvi wbY©q Kiæb|
K. 10% L. 11%
M. 12% N. 13% DËi L
☼ mgvavb:
GLv‡b Dfq‡ÿ‡Î gybvdvi nvi mgvb
10000 UvKvi 3 eQ‡ii gybvdv = (100003)
ev, 30000 UvKvi 1 eQ‡ii gybvdv
Avevi, 15000 UvKvi 4 eQ‡ii gybvdv = (150004)
ev, 60000 UvKvi 1 eQ‡ii gybvdv
(30000 + 60000) ev, 90000 UvKvi 1 eQ‡ii gybvdv
GLv‡b, 90000 UvKvi 1 eQ‡ii gybvdv = 9900 UvKv
 1 Ó 1 Ó Ó
90000
9900
Ó
 100 Ó 1 Ó Ó
90000
1009900
Ó
= 11 UvKv
wb‡Y©q gybvdvi nvi = 11% DËi: 11%
kU©KvU :
Back solved Method Gi gva¨‡g GB AsKwU mn‡R
mgvavb Kiv hvq|
o (
1
1
11 3) (
15
1
11 4)
33 + 6600 = 99000 (DËi)
86. GKB nvi gybvdvq †Kv‡bv Avmj 6 eQ‡i gybvdv-Avm‡j
wØ¸bn‡j, KZ eQ‡i Zv gybvdv-Avm‡j wZb¸b n‡e?
K. 10 eQi L. 12 eQi
M. 11 eQi N. 13 eQi DËi L
mgvavb:
g‡b Kwi, Avmj 100 UvKv
gybvdv- Avm‡j wØ¸Y n‡j nq(100  2) = 200 UvKv
gybvdv n‡e = (200 - 100) UvKv = 100 UvKv
A_©vr, 100 UvKvi 6 eQ‡ii gybvdv = 100 UvKv
100 Ó 1 Ó Ó =
6
100
=
3
50
UvKv
Avevi, gybvdv-Avm‡j wZb¸‡Y n‡j n‡e (100  3)
= 300 UvKv
‡hLv‡b ïay gybvdv n‡e = (300 - 100) UvKv = 200
GLb,
3
50
UvKv gybvdv nq = 1 eQ‡i
 1 Ó Ó Ó =
50
31
Ó
 200Ó Ó Ó =
50
20031 
= 12 eQi|
DËi: 12 eQ‡i gybvdv-Avmj wZb¸b n‡e|

87. ‡Kv‡bv wbw`©ó mg‡qi gybvdv-Avmj 5600 UvKv Ges
gybvdv, Avm‡ji
5
2
Ask| gybvdv evwl©K kZKiv 8
UvKv n‡j, mgq wbY©q Ki| (8g †kÖwY, Aby:2.1 Gi 17)
K. 2 eQi L. 4 eQi
M. 5 eQi N. 6 eQi DËi M
☼ mgvavb:
Avmj 5 UvKv n‡j my`=2 UvKv,my`vmj 5+2 =7 UvKv
my`vmj 7 UvKv n‡j Avmj = 5 UvKv|
ÕÕ 1 ÕÕ ÕÕ ÕÕ =
7
5
ÕÕ
 ÕÕ 5600 ÕÕ ÕÕ ÕÕ =
7
56005
ÕÕ
= 4000 UvKv|
myZivs my` = 5600-4000 = 1600 UvKv|
Avevi,
100 UvKvi my` 8 UvKv nq = 1 eQ‡i
 1 ÕÕ ÕÕ 1 ÕÕ ÕÕ =
8
100
ÕÕ
 4000 ÕÕ ÕÕ 1600 ÕÕ ÕÕ =
40008
1600100


=5
DËi: 5 eQi
88. Rvwgj mv‡ne †cbk‡bi UvKv †c‡q 10 jvL UvKvi
wZbgvm AšÍi gybvdv wfwËK 5 eQi †gqvw` †cbk‡bi
mÂqcÎ wKb‡jb, evwl©K gybvdv 12% n‡j, wZwb 1g
wKw¯Í‡Z A_©vr cÖ_g 3 gvm ci KZ gybvdv cv‡eb?
K. 25,000 UvKv L. 30,000 UvKv
M. 35,000 UvKv N. 35,000 UvKv DËi L
☼ mgvavb:
Rvwgj mv‡n‡ei †cbk‡bi cwigvY ev Avmj,
P = 1000000 UvKv
gybvdvi nvi, r = 12% =
100
12
n =
12
3
eQi =
4
1
eQi
gybvdv, I = ‡ei Ki‡Z n‡e|
GLv‡b,I = pnr =
100
12
4
1
1000000  UvKv
= 30000 UvKv
DËi: cÖ_g 3 gvm ci 30,000 UvKv gybvdv cv‡eb
89. GKRb dj eëmvqx h‡kvi †_‡K 36 UvKvq 12wU `‡i
wKQz msL¨K Ges Kzwóqv †_‡K 36 UvKvq 18wU `‡i
mgvb msL¨K Kjv Lwi` Kij| eëmvqxi weµqKg©x
15wU `‡i Zv weµq Ki‡jb|
K) eëmvqx h‡kvi †_‡K cÖwZk Kjv wK `‡i µq K‡iwQj?
L) weµqKg©x me¸‡jv Kjv wewµ Ki‡j kZKiv KZ jvf ev
ÿwZ n‡e?
M) eëmvqx 25% jvf Ki‡Z PvB‡j cÖwZ nvwj Kjv wK `‡i
weµq Ki‡Z n‡e?
☼ mgvavb:
(K)h‡kvi †_‡K,12wU Kjv µq K‡i 36 UvKvq
1wU Ó Ó Ó
12
36
Ó
100wU Ó Ó Ó
12
10036
Ó
= 300 UvKvq
cÖwZk Kjvi µqg~j¨ = 300 UvKvq|
(L) h‡kvi †_‡K,
12wU Kjv µq K‡i 36 UvKvq
1wU Ó Ó Ó
12
36
= 3 UvKvq
Kzwóqv †_‡K, 18wU Kjv µq K‡i 36 UvKvq
1wU Ó Ó Ó
18
36
= 2 UvKvq
(1+1)ev, 2wU Kjvi µqg~j¨ (3+2) UvKv ev 5 UvKv
Avevi, 15wU Kjvi weµqg~j¨ 36 UvKv
1 Ó Ó Ó
15
36
Ó
2 Ó Ó Ó
15
236
=
5
24
UvKv
‡h‡nZz weµqg~j¨ µqg~j¨ A‡cÿv Kg| myZivs ÿwZ n‡e|
ÿwZ = 5 -
5
24
UvKv =
5
2425
UvKv =
5
1
UvKv
GLb, 5 UvKvq ÿwZ nq =
5
1
UvKv
1 Ó Ó Ó
55
1

Ó
100Ó Ó Ó
55
1001


= 4 UvKv
 me¸‡jv Kjv weµq Ki‡j kZKiv 4 UvKv ÿwZ n‡e|
(M) 25% jvf Ki‡Z PvB‡j,
µqg~j¨ 100 UvKv n‡j weµqg~j¨ n‡e (100 + 25) UvKv
= 125 UvKv

100 UvKvi Kjv weµq Ki‡Z n‡e 125 UvKvq
 1 Ó Ó Ó Ó Ó
100
125
Ó
 5 Ó Ó Ó Ó Ó
100
5125
=
4
25
UvKvq
A_©vr, 2wU Kjv weµq Ki‡Z n‡e
4
25
UvKvq
1 Ó Ó Ó Ó Ó
24
25

Ó
4 ÓÓ ÓÓ Ó
24
425


= 12
2
1
UvKvq
cÖwZ nvwj Kjv weµq Ki‡Z n‡e 12
2
1
UvKvq|
90. ‡Kv‡bv Avmj 3 eQ‡ii mij gybvdvmn 28000 UvKv
Ges 5 eQ‡ii mij gybvdv mn 30000 UvKv|
K) cÖZxK¸‡jvi eY©bv mn g~jab wbY©‡qi m~ÎwU †jL|
L) gybvdvi nvi wbY©q Ki|
M) GKB nv‡i e¨vs‡K KZ UvKv Rgv ivL‡j 5 eQ‡ii
gybvdv Avm‡j 48000 UvKv n‡e?
☼ mgvavb:
(K) Avgiv Rvwb, GLv‡b,
I = pnr P = g~jab
ev, p =
nr
I
I = mij gybvdv
n = mgq
r = gybvdvi nvi
BnvB g~jab wbY©‡qi m~Î|
(L) Avmj + 5 eQ‡ii gybvdv = 30000 UvKv
2 eQ‡ii gybvdv = 2000 UvKv
1 Ó Ó =
2
2000
Ó
3 Ó Ó =
2
32000 
Ó
= 3000 UvKv
 Avmj + 3000 UvKv = 28000 UvKv
ev, Avmj = (28000 - 3000) UvKv= 25000 UvKv
GLb, Avgiv Rvwb, gybvdvi nvi, r =
pn
I 100
Avmj, P = 25000 UvKv
gybvdv, I = 3000 UvKv
mgq, n = 3 eQi
gybvdvi nvi r =
pn
I 100
=
325000
1003000


= 4%
A_©vr gybvdvi nvi 4%
(M) gybvdvi nvi 4% n‡j
100 UvKvi 1 eQ‡ii gybvdv 4 UvKv
100 Ó 5 Ó Ó (45)Ó = 20 UvKv
gybvdv-Avmj = (20 + 100) UvKv = 120 UvKv
gybvdv-Avmj 120 UvKv n‡j Avmj 100 UvKv
 Ó Ó 1 Ó Ó Ó
120
100
Ó
 Ó Ó 48000Ó Ó Ó
120
48000100 
Ó
= 40000 UvKv
 GKB nv‡i e¨vs‡K 40000 UvKv Rgv ivL‡Z n‡e|
Abykxjbx 2.2 (Pµe„w× my`vmj)
91. evwl©K kZKiv 8 UvKv gybvdvq 62500 UvKvi 3
eQ‡ii Pµe„w× g~jab wbY©q Ki| (D`vniY- 01)
K. 78732 UvKv L. 77432 UvKv
mgvavb:
Avgiv Rvwb, C = P ( 1 + r)n
‡`Iqv Av‡Q, cÖviw¤¢K g~jab, P = 62500 UvKv
evwl©K gybvdvi nvi, r =8%
Ges mgq n= 3 eQi
C = 62500
3
100
8
1 





 UvKv,
ev 62500 
3
25
27






= 62500 (1.08)3
UvKv
= 62500 1.259712 UvKv = 78732 UvKv
 Pµe„w× g~jab 78732 UvKv|
92. evwl©K 10.50% gybvdvq 5000 UvKvi 2 eQ‡ii
Pµe„w× gybvdv wbY©q Ki|
K. 1105.13 UvKv L. 1100 UvKv
☼ mgvavb:
Avgiv Rvwb, Pµe„w× my`vmj C = P ( 1 + r)n
GLv‡b,Avmj P = 5000 UvKv,

gybvdvi nvi r =10.50% =
100
10.5
=
10100
105

=
200
21
mgq, n = 2 eQi
 C = P ( 1 + r)2
= 5000 
2
200
21
1 






= 5000 
2
200
221






= 5000 
200
221

200
221
=
8
48841
= 6105.13 UvKv (cÖvq)
Pµe„w× gybvdv = my`vmj-Avmj = 6105.13 - 5000
= 1105.13 UvKv (cÖvq)
93. GKwU d¬¨vU gvwjK Kj¨vY mwgwZ Av`vqK…Z mvwf©m PvR©
†_‡K D×…Ë 200000 UvKv e¨vs‡K Qq gvm AšÍi
Pµe„w× gybvdvwfwËK ¯’vqx AvgvbZ ivL‡jb| gybvdvi
nvi evwl©K 12 UvKv n‡j, Qq gvm ci H mwgwZi wnmv‡e
KZ UvKv gybvdv Rgv n‡e? GK eQi ci Pµe„w× g~jab
KZ n‡e?
K. 12000, 224720 UvKv L. 12000, 224620 UvKv
M. 12000, 224550 UvKv N. 12000, 225320 UvKv
☼ mgvavb: DËi : (K)
†`Iqv Av‡Q, g~jab P = 200000 UvKv,
gybvdvi nvi r = 12%,
mgq, n = 6 gvm ev
2
1
eQi
gybvdvI = Prn
= 200000 
100
12

2
1
= 12000 UvKv
 6 gvm ci gybvdv n‡e 12000 UvKv
1g Qq gvm ci Pµe„w×g~j = (200000+12000) UvKv
= 212000 UvKv
Avevi, cieZx© Qq gv‡mi gybvdv-Avmj
= 212000 (1 +
100
12

2
1
) UvKv
= 2120001.06 UvKv= 224720 UvKv
1 eQi ci Pµe„w× g~jab (gybvdv Avmj)=224720
94. †Kv‡bv kn‡ii eZ©gvb RbmsL¨v 80 jÿ| H kn‡ii
RbmsL¨v e„w×i nvi cÖwZ nvRv‡i 30 n‡j, 3 eQi ci
H kn‡ii RbmsL¨v KZ n‡e?
K. 8741000 UvKv L. 8741816 UvKv
M. 8741356 UvKv N. 8741256 UvKv D L
mgvavb:
kniwUi eZ©gvb RbmsL¨v, P = 8000000 UvKv,
RbmsL¨v e„w×i nvi =
1000
30
100% =3%
mgq, n = 3 eQi
GLv‡b RbmsL¨v e„w×i †ÿ‡Î Pµe„w× g~ja‡bi m~Î
cÖ‡hvR¨|
C = P(1+r)n
= 80,00,000
3
100
3
1 






=80,00,000
100
103

100
103

100
103
=8 103103103=8741816
myZivs 3 eQi ci kniwUi RbmsL¨v n‡e 87,41,816
95. g‡bvqviv †eMg Zvi cvwievwiK cÖ‡qvR‡b 6% nv‡i x
UvKv Ges 4% nv‡i y UvKv FY wbj| †m †gvU 56000
UvKv FY wbj| Ges eQi †k‡l 2840 UvKv gybvdv †kva
Kij|
K. m¤ú~Y© F‡Yi Dci 5% gybvdv cÖ‡hvR¨ n‡j evwl©K
gybvdv KZ?
L. x Ges y Gi gvb wbY©q Ki|
M. m¤ú~Y© F‡Yi Dci 5% Pµe„w× gybvdv cÖ‡hvR¨ n‡j
2 eQi ci g‡bvqviv †eMg‡K KZ UvKv gybvdv cwi‡kva
Ki‡Z n‡e?
☼ mgvavb:
(K) †gvU F‡Yi cwigvb, p = 56000 UvKv
gybvdvi nvi r =
100
5
mgqn = 1 eQi
GLb gybvdv I = pnr
= (560001
100
5
) = 2800 UvKv| (DËi)
(L) 6% nvi gybvdvq x UvKvi evwl©K gybvdv
= (x1
100
6
) UvKv =
100
6x
UvKv
Avevi 4% nvi gybvdvq y UvKvi evwl©K gybvdv

= (y1
100
4
) UvKv =
100
4y
UvKv
GLb, DÏxc‡Ki Z_¨vbymv‡i x+y = 56000 ......(i)
Ges,
100
6x
+
100
4y
= 2840ev, 6x + 4y = 284000
ev, 3x + 2y = 142000 ............ (ii)
GLb, (i)bs mgxKiY‡K 3 Øviv ¸Y K‡i ¸Ydj †_‡K
(ii)bs mgxKiY we‡qvM Kwi,
3x+3y= 168000
3x+2y = 142000
y = 26000
yGi gvb (i) mgxKi‡Y ewm‡q cvB, x = 30,000
 x = 30,000; y = 26000
(M) g‡bvqvivi F‡Yi cwigvb p = 56000 UvKv
gybvdvi nvi r =
100
5
mgqn = 2 eQi
GLb, Pµe„w×i †ÿ‡Î me„w×g~j = p(1+r)n
 2 eQi ci g‡bvqvivi F‡Yi me„w×g~j = 56000
(1+
100
5
)2
UvKv
= 56000  (1+.05)2
UvKv
= 56000  (1.05)2
UvKv
= 61740 UvKv
g‡bvqviv cwi‡kva Ki‡eb (61740 - 56000) UvKv
=5740 UvKv|
96. 1050 UvKvi 8% wb‡Pi †KvbwU? (8g †kÖwY, Aby: 2.2
Gi 01 b¤î AsK)
K. 80 L. 82
M. 84 N. 86 DËi M
[Hints: 1050 Gi 8% = 84]
97. evwl©K 10% mij gybvdvq 1200 UvKvi 4 eQ‡ii mij
gybvdv KZ?
K. 120 L.240
M.360 N.480 DËi N
[Hints: 1200 Gi 10%  4 = 1204 = 480]
98. UvKvq 5wU `‡i µq K‡i 4wU `‡i weµq Ki‡j kZKiv
KZ jvf ev ÿwZ n‡e?
K.25% jvf L.25% ÿwZ
M.20% jvf N.20% ÿwZ DËi K
[Hints: 4wU †Z 1wU jvf ev 25%]
99. gybvdv wnmv‡ei †ÿ‡Î-
i. gybvdv = gybvdv-Avmj - Avmj
ii.gybvdv =
Avmjgybvdvmgq
2
iii. Pµe„w× gybvdv = Pµe„w×g~j - gyjab
Dc‡ii Z‡_ï Av‡jv‡K wb‡Pi ‡KvbwU mwVK?
K. i I ii L. i I iii
M. ii I iii N. i, ii I iii DËi L
100.10% mij gybvdvq 2000 UvKvi-
i. 1 eQ‡ii gybvdv 200 UvKv
ii. 5eQ‡ii gybvdv-Avmj,Avm‡ji 1
3
2
¸Y
iii. 6 eQ‡ii gybvdv Avm‡ji mgvb n‡e
wb‡Pi ‡KvbwU mwVK?
K. i I ii L. i I iii
M. ii I iii N. i, ii I iii
DËi:‡bvU: Ack‡b mwVK DËi †bB| DËi (i) n‡e|
101. Rvwgj mv‡ne evwl©K 10% gybvdvq e¨vs‡K 2000 UvKv
Rgv ivL‡jb|
wb‡Pi cÖkœ¸‡jvi DËi `vI
(1) 1g eQiv‡šÍ gybvdv-Avmj KZ n‡e?
K. 2050 UvKv L.2100 UvKv
M.2200 UvKv N.2250 UvKv DËi M
(2) mij gybvdvq 2q eQiv‡šÍ gybvdv-Avmj KZ n‡e?
M.2440 UvKv N.2450 UvKv DËi K
(3) 1g eQiv‡šÍ Pµe„w× gyjab KZ n‡e?
M.2150 UvKv N.2200 UvKv DËi N
102.evwl©K 10% gybvdvq 8000 UvKvi 3 eQ‡ii Pµe„w×
g~jab (my`vmj) wbY©q Kiæb| [8g †kÖwY-2.2-Gi 7 bs
cÖkœ]
K. 10500 UvKv L. 10700 UvKv
☼ mgvavb:
Avgiv Rvwb, Pµe„w× g~jab C = P (1 + r)n
‡Ìqv Av‡Q, cÖviw¤¢K g~jab P = 8000
evwl©K gybvdvi nvi, r = 10% Ges mgq n = 3 eQi
C = 8000 
3
100
10
1 





 UvKv = 8000  (1 +
0.1)3
UvKv
= 8000  (1.1)3
UvKv = 8000  1.331 UvKv
= 10648 UvKv

 Pµe„w× g~jab 10648 UvKv|
103. evwl©K kZKiv 10 UvKv gybvdvq 5000 UvKvi 3
eQ‡ii mij gybvdv I Pµe„w× gybvdvi cv_©K¨ KZ n‡e?
[RAKUB officer-2015]+(8g †kÖwY-2.2-Gi 8 bs
cÖkœ)
☼ mgvavb:
Pµe„w× gybvdvAvmj = P(1+r)n
5000
3
100
10
1 






= 5000
3
100
110






= 5000
10
11

10
11

10
11
= 6655
myZivs ïay Pµe„w× gybvdv = 6655-5000 = 1655
Avevi, mij gybvdv =
100
mgqnvimy‡ìAvmj 
=
100
3105000 
=1500
myZivs Pµe„w× gybvdv Ges mij gybvdvi cv_©K¨
= 1655 - 1500 = 155 UvKv|
me‡_‡K mnR wbq‡g †evSvi Rb¨ e·wU †`Lyb+e¨vL¨v co–b
e‡·ªi K_v¸‡jv †f‡½ wjL‡j Ggb n‡Zv:
10%nv‡i 5000 UvKvi 1 eQ‡ii mij my` 500 UvKv
n‡j 3 eQ‡ii †gvU my` =5003=1500 UvKv|
wKš‘ Pµe„w× my` n‡e cÖ_g eQ‡i 500 UvKv wØZxq
eQ‡i 500 (gyjab 5000 UvKvi my`)+cÖ_g eQ‡ii
500UvKvi my‡ì Dci AwZwi³ 50 UvKv A_©vr 500+50=
550UvKv, †Zgwbfv‡e 3q eQ‡ii my` n‡e 500+50+55
= 615UvKv ZvB 3 eQ‡ii ‡gvU Pµe„w× my`
500+550+615 = 1655 UvKv|
mij my‡ì mv‡_ Pµe„w× my‡ì cv_©K¨= 1655-1500
= 155 UvKv| DËi: 155 UvKv|
104.GKB nvi gybvdvq †Kv‡bv g~ja‡bi GK eQiv‡šÍ g~jab
6500 I `yB eQiv‡šÍ Pµe„w× g~jab 6760 UvKv n‡j
g~jab KZ?
☼ mgvavb:
g~jab p Ges r gybvdvi nvi n‡j, GKeQi ci Pµe„w×
g~jab C = 6500 n‡j,
6500 = p (1 + r) …………(i)
2 eQ‡i Pµe„w× g~jab
6760 = p (1 + r)2
…………(ii)
GLb, (ii) ‡K (i) Øviv fvM K‡i cvB,
)(
)(
rp
rp 2


1
1
=
6500
6760
ev,
)(
)(
rp
rp 2


1
1
=
25
26
ev,
)(
))((
rp
rrp


1
11
=
25
26
ev, )( r1 =
25
26
GLb, )( rp 1 = 6500 ev, P
25
26
= 6500
ev, 26p = 6500  25 ev, p =
26
256500
P = 6250  wb‡Y©q g~jab 6250 UvKv|
105.evwl©K kZKiv 8.50 UvKv Pµe„w× gybvdvq 10,000
UvKvi 2 eQ‡ii me„w×g~j I Pµe„w× gybvdv wbY©q Ki|
K. 11772.25 UvKv I 1925.25 UvKv
L.11772.25 UvKv I 1935.35 UvKv
M.11772.25 UvKv I 1772.25 UvKv
N. 11892.75 UvKv I 1750.25 UvKv DËi M
☼ mgvavb:
Avgiv Rvwb, Pµe„w× g~jab C = P (I + r)n
GLv‡b, p = 10000, r = 8.50% Ges n = 2
C = 10000 
2
100
8.50
1 





 UvKv
= 10000(1+.085)2
UvKv
Avmj 5000 my‡ì my`
1g eQ‡ii my` 500
2q eQ‡ii my` 500 50
3q eQ‡ii my` 500 50 50+5
‡gvU my` 1500 100 55
ïay Avm‡ji
3 eQ‡ii
my`
1g eQ‡ii
500Gi c‡ii
2 eQ‡ii my`
2q eQ‡ii
500+50 Gi
Rb¨ 50+5 = 55
GUvB mij
my`
100+55 = 155 UvKv n‡jv
Pµe„w×i Rb¨ cÖvß AwZwi³ my`|
‡gvU Pµe„w× my`
1500+155
= 1655
‡gvU mij
my` 1500
myZivs Pµe„w× my` I mij my‡ì
cv_©K¨: 1655-1500 =155UvKv
kU©KvU:
my‡ì my` (155) B n‡jv cv_©K¨|

= 100001.177225 UvKv = 11772.25 UvKv
wb‡Y©q me„w×g~j 11772.25 UvKv
 Pµe„w× gybvdv = (11772.25 - 10000) =
1772.25 UvKv
DËi: me„w×g~j 11772.25 UvKv
Ges Pµe„w× gybvdv 1772.25 UvKv
106. ‡Kv‡bv kn‡ii eZ©gvb RbmsL¨v 64 jÿ| kniwUi
RbmsL¨v e„w×i nvi cÖwZnvRv‡i 25 Rb n‡j, 2 eQi
ci H kn‡ii RbmsL¨v KZ n‡e?
K. 6721000 Rb L. 6700000 Rb
M. 6724000 Rb N. 6735000 Rb
☼ mgvavb: (DËi M)
kniwUi eZ©gvb RbmsL¨v p = 6400000
RbmsL¨v e„w×i nvi, r =
1000
25
100% = 2.5%
mgq, n = 2 eQi
GLv‡b, RbmsL¨v e„w×i ‡ÿ‡Î Pµe„w×i g~ja‡bi m~Î
cÖ‡hvR¨|
 C = P(1 + r)n
= 6400000 
2
100
2.5
1 






= 6400000  (1 + .025)2
= 6400000  (1.025)2
= 64000001.050625 =
6724000
 2 eQi ci kniwUi RbmsL¨v n‡e 6724000 Rb|
DËi: 6724000 Rb
107.GKe¨w³ GKwU FY`vb ms¯’v †_‡K evwl©K 8% Pµe„w×
gybvdvq 5000 UvKv Fb wb‡jb| cÖwZeQi †k‡l wZwb
2000 UvKv K‡i cwi‡kva Ki‡b| 2q wKw¯Í cwi‡kv‡ai
ci Zvi Avi KZ UvKv Fb _vK‡e?
mgvavb:
GLv‡b, P = 5000 Ges r = 8%
1g eQiv‡šÍ Pµe„w× g~jab = p(1 + r)1
= 5000 







100
8
1 UvKv
= 5000(1+.08) UvKv
= 50001.08 UvKv = 5400 UvKv
cÖ_g eQi 2000 UvKv wKw¯Í cwi‡kva Kivq Zvi FY
_v‡K= (5400-2000) UvKv = 3400 UvKv|
Avevi, Pµe„w× g~jab = p(1 + r)
= 3400  






100
8
1 UvKv
= 3400(1+.08) UvKv = 34001.08 UvKv
= 3672 UvKv
wØZxq eQi 2000 UvKv wKw¯Í cwi‡kva Kivq Zvi FY
_vK‡e = (3672 - 2000) UvKv = 1672 UvKv|
DËi: 1672 UvKv
108. GKB nvi Pµe„w× gybvdvq †Kv‡bv g~jab GKeQiv‡šÍ
me„w×g~j 19500 UvKv Ges `yB eQiv‡šÍ me„w×g~j
20280 UvKv|
K. gybvdv wbY©‡qi m~Î wjL|
L. g~jab wbY©q Ki|
M. GKB nv‡i D³ g~ja‡bi Rb¨ 3 eQi ci mij gybvdv
I Pµe„w× gybvdvi cv_©K¨ wbY©q Ki|
mgvavb:
(K)Avgiv Rvwb,
Pµe„w× gybvdv = C – p = P (1 + r)n
– p
GLv‡b, P = Avmj; r = gybvdvi nvi Ges n = mgq
(L)Pµe„w× gybvdvq GK eQiv‡šÍ me„w×g~j, C = 19500
UvKv
Ó Ó `yB Ó Ó C = 20280
UvKv
C = P (1 + r)n
m~Î n‡Z cvB,
19500 = P(1 + r)1
(mgq, n = 1 eQi)
P(1 + r)1
= 19500 .............(1)
Avevi, 20280 = (1 + r)2
(mgq, n = 2 eQi)
P(1 + r)2
= 20280.............(2)
(2) bs mgxKiY‡K (1) Øviv fvM K‡i cvB,
1 + r =
19500
20280
ev, 19500 + 19500r = 20280
ev, 19500r = 20280 - 19500
ev, r =
19500
780
=
25
1
rGi gvb (1) bs mgxKi‡Y ewm‡q cvB,
p 






25
1
1 = 19500
ev, p = 





25
26
= 19500
ev, p =
26
2519500 
= 18750
 g~jab 18750 UvKv

(M)ÔLÕ n‡Z cÖvß g~jab, P = 18750 UvKv Ges gybvdvi
nvi, r =
25
1
GLv‡b, mgq n = 3 eQi
Pµe„w× g~jab, C = P (1 + r)3
= 18750
3
25
1
1 





 UvKv
= 18750 
3
25
26






UvKv = 21091.2
Pµe„w× gybvdv = C – P = (21091.2 - 18750)
UvKv = 2341.2 UvKv
mij gybvdv, I = prn = 





 3
25
1
18750 = 2250
 Pµe„w× gybvdv I mij gybvdvi cv_©K¨ = (2341.2 -
2250) UvKv = 91.2 UvKv
109.wkcÖv eoyqv †Kv‡bv e¨vs‡K 3000 UvKv Rgv †i‡L 2
eQi ci gybvdv mn 3600 UvKv †c‡q‡Qb|
K. mij gybvdvi nvi wbY©q Ki|
L. AviI 3 eQi ci gybvdv-Avmj KZ n‡e?
M. 3000 UvKv GKB nvi Pµe„w× gybvdvq Rgv ivL‡j 2
eQi ci Pµe„w× g~jab KZ n‡Zv?
☼ mgvavb:
(K)gybvdv = gybvdv-Avmj - Avmj = (3600 - 3000)
UvKv = 600 UvKv
 1 Ó 1 Ó Ó
23000
600

Ó
 100 Ó 1 Ó Ó
23000
100600


= 10 UvKv
mij gybvdvi nvi 10%|
(L)K-n‡Z cÖvß,
 1 Ó 1 Ó Ó
100
10
Ó
 3000 Ó 3 Ó Ó
100
3300010 
=
900 UvKv
GLv‡b, wkcÖv eo–qv 2 eQi ci gybvdvmn cvb 3600 UvKv
 AviI 3 eQi ci gybvdv-Avmj n‡e = (3600 +
900) UvKv = 4500 UvKv
 AviI 3 eQi ci gybvdv-Avmj n‡e = 4500 UvKv
(M)GLv‡b, g~jab P = 3000 UvKv
mgq, n = 2 eQi
K-n‡Z cÖvß,
gybvdvi nvi, r = 10% =
100
10
Avgiv Rvwb,
Pµe„w× g~jab, C = P (1 + r)n
C = 3000 
2
100
10
1 





 UvKv = 3000 
2
100
10100





 
UvKv
= 3000 
2
100
110






UvKv = 3000 
2
10
11






UvKv
= 3000 
100
121
= 3630 UvKv
 2 eQi ci Pµe„w× g~jab n‡Zv 3630 UvKv|
AbycvZ I
mgvbycvZ
lô †kÖwY 2.1
mßg †kÖwY 2.1
beg-`kg †kÖwY 11.1, 11.2
6ô †kÖwY (Abykxjbx 2.1)
110. †Rmwgb I Avwe`vi eZ©gvb eq‡mi AbycvZ 3:2 Ges
Avwe`v I AvwbKvi eZ©gvb eq‡mi AbycvZ 5:1|
AvwbKvi eZ©gvb eqm 3 eQi 6 gvm|
(K) DÏxc‡Ki cÖ_g AbycvZ‡K kZKivq cÖKvk Ki|
(L) 5 eQi ci Avwe`vi eqm KZ n‡e ?
(M) AvwbKvi eZ©gvb eqm †Rmwg‡bi eZ©gvb eq‡mi
kZKiv KZ fvM ?
mgvavb:
(K) DÏxc‡Ki cÖ_g AbycvZ = 3:2 =
2
3
=
1002
1003


=





 
2
1003
% =150%
AbycvZ mgvbycvZI

(L) Avwe`vi eZ©gvb eqm : AvwbKvi eZ©gvb eqm = 5:1
A_©vr, Avwe`vi eZ©gvb eqm, AvwbKvi eZ©gvb eq‡mi 5
¸Y
AvwbKvi eZ©gvb eqm = 3 eQi 6 gvm = (312+6)
gvm = 42 gvm [‡h‡nZz 1 eQi = 12 gvm]
myZivs Avwe`vi eZ©gvb eqm = (425) gvm = 210 gvm
=
12
210
eQi =
2
35
eQi ev 17
2
1
eQi
 5 eQi ci Avwe`vi eqm n‡e = (17
2
1
+5) = 22
2
1
eQi|
(M) ‡Rmwgb I Avwe`vi eZ©gvb eq‡mi AbycvZ = 3:2
A_v©r, †Rmwg‡bi eZ©gvb eqm, Avwe`vi eZ©gvb eq‡mi
2
3
¸Y
(LÕ) n‡Z Avwe`vi eZ©gvb eqm = 17
2
1
eQi
 †Rmwg‡bi eZ©gvb eqm
=17
2
1

2
3
=
2
35

2
3
=
4
105
= 26
4
1
eQi
AvwbKvi eZ©gvb eqm = 3 eQi 6 gvm
= 3
12
6
eQi = 3
2
1
=
2
7
eQi
AvwbKvi eZ©gvb eqm †Rmwg‡bi eZ©gvb eq‡mi
= (
2
7
26
4
1
) Ask = (
2
7

105
4
) =
15
2
Ask
= (
15
1002 
)% =
3
40
% ev 13
3
1
%
 AvwbKvi eZ©gvb eqm †Rmwg‡bi eZ©gvb eq‡mi
13
3
1
%
111. 500 UvKv `yBRb kÖwg‡Ki gv‡S 2:3 Abycv‡Z fvM K‡i
w`‡Z n‡e| (6ô †kÖwY 2.1 Gi D`v:2)
mgvavb:
Abycv‡Zi c~e© ivwk 2 Ges DËi ivwk 3
ivwk `yBwUi mgwó = (2+3) = 5
 1g Rb kÖwgK cv‡e, 500 UvKvi
5
2
Ask
= 500 UvKv 
5
2
= 200 UvKv
Ges 2q kÖwgK cv‡e, 500 UvKvi
5
3
Ask
= 500 
5
3
= 300 UvKv |
 `yBRb kÖwg‡Ki GKRb cv‡e 200 UvKv I AciRb
cv‡e 300 UvKv|
112.cÖ`Ë mij AbycvZ¸‡jvi wgkÖ AbycvZ wbY©q Ki: 5:7,
4:9, 3:2| (6ô †kÖwY 2.1 Gi D`v:3)
K. 10 : 21 L. 10 : 126
M. 60 : 21 N. 11 : 31 DËi K
☼ mgvavb :
AbycvZ wZbwUi c~e© ivwk¸‡jvi ¸Ydj 543 = 60
Ges DËi ivwk¸‡jvvi ¸Ydj = 792 = 126
wb‡Y©q wgkÖ AbycvZ = 60:126 ev 10:21 [Dfq‡K
6 Øviv fvM K‡i]
113.`yBwU msL¨vqi †hvMdj 360| msL¨v `yBwUi AbycvZ
4:5 n‡j, msL¨v `yBwU wbY©q Ki| (6ô †kÖwY 2.1 Gi
D`v:4)
K. 160 I 180 L. 160 I 200
M. 180 I 210 N. 160 I 220 DËi L
☼ mgvavb:
msL¨v `yBwUi AbycvZ 4 : 5
AbycvZwUi c~e© I DËi ivwki †hvMdj = 4 + 5 = 9
cÖ_g msL¨vwU = 360 Gi
9
4
Ask
= 360 
9
4
= 160
wØZxq msL¨vwU = 360 Gi
9
5
Ask
= 360 
9
5
= 200
wb‡Y©q msL¨v `yBwU n‡jv 160 I 200| (DËi)
114. 40 †KwR wgkÖ‡Y evwj I wm‡g‡›Ui cwigv‡Yi AbycvZ
4:1| wgkÖYwUi evwj I wm‡g‡›Ui cwigvY wbY©q Ki| (6ô
†kÖwY 2.1 Gi D`v:5)
K. 8 †KwR L. 9 †KwR
M. 10 †KwR N. 11 †KwR DËi K
☼ mgvavb:
wgkÖ‡Yi cwigvY 40 †KwR|

evwj I wm‡g‡›Ui AbycvZ 4 : 1
GLv‡b, AbycvZwUi c~e© I DËiivwki †hvMdj
= 4 + 1 = 5
evwji cwigvY = 40 †KwRi
5
4
Ask
= 40 
5
4
†KwR = 32 †KwR|
wm‡g‡›Ui cwigvY = 40 †KwRi
5
1
Ask
= 40 
5
1
†KwR = 8 †KwR| DËi: 8 †KwR |
115.GKwU we`¨vj‡q QvÎ I QvÎxi msL¨vi AbycvZ 5:7| H
we`¨vj‡q QvÎxmsL¨v 350 Rb n‡j, Qv‡Îi msL¨v KZ?
(6ô †kÖwY 2.1 Gi D`v:6)
K. 250 Rb L. 220 Rb
☼ mgvavb:
QvÎmsL¨v : QvÎxmsL¨v = 5 : 7
A_©vr, Qv‡Îi msL¨v QvÎxi msL¨vi
7
5
¸Y|
‡`Iqv Av‡Q, QvÎx msL¨v 350 Rb|
 Qv‡Îi msL¨v = 350 
7
5
Rb
wb‡Y©q Qv‡Îi msL¨v 250 Rb| DËi: 250|
Abykxjbx 2.1
116.wb‡Pi mij AbycvZ¸‡jv wgkª Abycv‡Z cÖKvk Ki :
(K) 3:5, 5:7, 7:9
☼ mgvavb:
3 : 5, 5 : 7 I 7 : 9
AbycvZ¸‡jvi c~e© ivwki ¸Ydj = 3  5  7 = 105
AbycvZ¸‡jvi DËi ivwki ¸Ydj = 5  7  9 = 315
wb‡Y©q wgkÖ AbycvZ = 105 : 315 : = 1 : 3 [Dfq
ivwk‡K 105 w`‡q fvM K‡i]
(L) 5:3, 7:5, 9:7
☼ mgvavb:
5 : 3, 7 : 5 I 9 : 7
AbycvZ¸‡jvi c~e© ivwki ¸Ydj =5  7 9 = 315
AbycvZ¸‡jvi DËi ivwki ¸Ydj = 3  5  7 = 105
wb‡Y©q wgkÖ AbycvZ = 315 : 105 : = 3 : 1 [Dfq
ivwk‡K 105 w`‡q fvM K‡i]
117. 9 : 16 AbycvZwU‡K e¨¯Í Abycv‡Z cÖKvk Ki|
☼ mgvavb:
cÖ`Ë AbycvZ 9 : 16
c~e© ivwk I DËi ivwki ¯’vb wewbgq K‡i cvB, 16 : 9
wb‡Y©q e¨¯Í AbycvZ = 16 : 9
118.wb‡¤œi AbyccvZ¸‡jvi ‡KvbwU GKK AbycvZ
(K) 16:13
AbycvZwUi c~e© ivwk I DËi ivwk mgvb bq|
ZvB GwU GKK AbycvZ bq|
(L) 13:17
AbycvZwUi c~e© ivwk I DËi ivwk mgvb bq|
ZvB GwU GKK AbycvZ bq|
(M) 21:21
AbycvZwUi c~e© ivwk I DËi ivwk mgvb|
ZvB GwU GKK AbycvZ| DËi: (M) 21 : 21
119.550 UvKv‡K 5:6 I 4:7 Abycv‡Z fvM Ki| (6ô †kÖwY
2.1 Gi 9)
☼ mgvavb:
5 : 6 AbycvZwUi c~e© I DËi ivwki †hvMdj
= 5 + 6 = 11
cÖ_g fvM = 550 UvKvi
11
5
Ask
= 550
11
5
UvKv
= 250 UvKv
Ges wØZxq fvM = 550 UvKvi
11
6
Ask
= 550
11
6
UvKv = 300 UvKv
Avevi, 4 : 7 AbycvZwUi c~e© I DËi ivwki †hvMdj
= 4 + 7 = 11
 550 UvKv‡K UvKv‡K 4 : 7 Abycv‡Z fvM Ki‡j
cÖ_g fvM = 550 UvKvi
11
4
Ask
= 550
11
4
UvKv = 200 UvKv
Ges wØZxq fvM = 550 UvKvi
11
7
Ask = 550
11
7
UvKv = 350 UvKv
DËi: 250 UvKv I 300 UvKv; 200 UvKv I 350 UvKv|
120.wcZv I cy‡Îi eq‡mi AbycvZ 14:3 wcZvi eqm 56
eQi n‡j cy‡Îi eqm KZ?(6ô †kÖwY 2.1 Gi 1 )

K. 15 eQi L. 13 eQi
☼ mgvavb:
wcZv I cy‡Îi eq‡mi AbycvZ 14 : 3|
cy‡Îi eqm wcZvi eq‡mi
14
3
 56 eQi = 12 eQi|
DËi: 12 eQi
121.`yBwU msL¨vi †hvMdj 630| G‡ì AbycvZ 10 : 11
n‡j, msL¨v `yBwU wbY©q Ki|(6ô †kÖwY 2.1 Gi 11)
K. 300 I 330 L. 300 I 350
M. 350 I 340 N. †KvbwUB bq DËi K
☼ mgvavb:
`yBwU msL¨vi †hvMdj = 630
msL¨v `yBwUi = 10 : 11
GLv‡b, AbycvZwUi c~e© I DËi ivwki †hvMdj
= 10 + 11 = 21
 cÖ_g msL¨v = 630 Gi
21
10
Ask = 300
Ges wØZxq msL¨v = 630 Gi
21
11
Ask = 330
DËi: 300 I 330
122.`yBwU eB‡qi g~‡jï AbycvZ 5:7| wØZxqwUi g~j¨ 84
UvKv n‡j, cÖ_gwUi g~j¨ KZ?(6ô †kÖwY 2.1 Gi 12)
☼ mgvavb:
2q eBwUi g~j¨ = 7 Ask = 84 n‡j 1 Ask
= 847 = 12 |
Zvn‡j cÖ_g eBwUi g~j¨ n‡e = 5 Ask = 512 = 60
UvKv| DËi: 60UvKv|
A_ev,
`yBwU eB‡qi g~jï AbycvZ 5 : 7
cÖ_gwUi g~j¨ wØZxqwUi g~j¨
7
5
Ask|
GLv‡b, wØZxqwUi g~j¨ = 84 UvKv
cÖ_gwUi g~j¨ =
7
5
 wØZxqwUi g~j¨
=
7
5
Gi 84 UvKv
= 60 UvKv|
123. 18 K¨v‡i‡Ui 20 MÖvg IR‡bi †mvbvi Mnbvq †mvbv I
Lv‡ì AbycvZ 3:1 n‡j, H Mnbvq †mvbv I Lv‡ì
cwigvY wbY©q Ki|(6ô †kÖwY 2.1 Gi 13)
K. 15 MÖvg I 5 MÖvg
L. 20 MÖvg I 10 MÖvg
M. 15 MÖvg I 10 MÖvg
N. 12 MÖvg I 13 MÖvg DËi K
☼ mgvavb:
‡mvbvvi Mnbvi IRb = 20 MÖvg
MnbvwU‡Z †mvbv I Lv‡ì AbycvZ 3 : 1| AbycvZwUi c~e©
I DËi ivwki ‡hvMdj = 3+1 = 4
 MnbvwU‡Z †mvbvi cwigvY = 20 MÖv‡gi
4
3
Ask
= 20 
4
3
MÖvg = 15 MÖvg
Ges Lv‡ì cwigvY = 20 MÖv‡gi
4
1
Ask
= 20 
4
1
MÖvg = 5 MÖvg
DËi: †mvbvi cwigvb= 15 MÖvg, Lv‡ì cwigvb 5 MÖvg|
124.`yB eÜzi evwo n‡Z ¯‥z‡j Avmv I hvIqvi mg‡qi AbycvZ
2:3, 1g eÜzi evwo n‡Z ¯‥z‡ji `~iZ¡ 5 wK.wg n‡j,
wØZxq eÜzi evwo n‡Z ¯‥z‡ji `~iZ¡ KZ?(6ô †kÖwY 2.1 Gi
14)
K. 7.50 wK.wg. L. 6.50 wK.wg.
M. 8.00 wK.wg. N. 8.50 wK.wg DËi K
☼ mgvavb:
`yB eÜzi evwo n‡Z ¯‥z‡j Avmv-hvIqvi mg‡qi AbycvZ
2 : 3| myZivs Zv‡ì evwo n‡Z ¯‥z‡ji `~i‡Z¡i AbycvZ
2 : 3|
 2q eÜzi evwo n‡Z ¯‥z‡ji `~iZ¡ 1g eÜzi evwo n‡Z
¯‥z‡ji `~i‡Z¡i
2
3
¸Y|
GLv‡b, 1g eÜzi evwo n‡Z ¯‥z‡ji `~iZ¡ 5 wK.wg.|
 2q eÜzi evwo n‡Z ¯‥z‡ji `~iZ¡ =
2
3
 1g eÜzi evwo
n‡Z ¯‥z‡ji =
2
3
 5 wK.wg.
=
2
15
wK.wg.
= 7
2
1
wK.wg.| DËi:

125.cv‡q‡m `ya I wPwbi AbycvZ 7:2| H cv‡q‡m wPwbi
cwigvY 4 ‡KwR n‡j, `y‡ai cwigvY KZ?
K. 12 †KwR L. 14 †KwR
M. 13 †KwR N. 15 †KwR DËi L
☼ mgvavb:
cv‡q‡m `ya I wPwbi AbycvZ 7 : 2| myZivs `y‡ai cwigvY
wPwbi cwigv‡Yi
2
7
¸Y|
GLv‡b, wPwbi cwigvY = 4 †KwR
 `y‡ai cwigvb =
2
7
 wPwbi cwigvY =
2
7
 2
= 14 †KwR| (DËi)
126. `yBwU Kw¤úDUv‡ii `v‡gi AbycvZ 5:6| cÖ_gwUi
`vg 25,000 UvKv n‡j, wØZxqwUi `vg KZ? g~j¨ e„w×i
d‡j hw` cÖ_gwUi `vg 5000 UvKv †e‡o hvq ZLb
Zv‡`i `v‡gi AbycvZwU Kx ai‡bi AbycvZ?(6ô
†kÖwY2.1 Gi 15)
☼ mgvavb:
`yBwU Kw¤úDUv‡ii `v‡gi AbycvZ 5 : 6|
myZivs wØZxqwUi `vg cÖ_gwUi `v‡gi
5
6
¸Y|
GLv‡b, cÖ_gwUi `vg = 25000 UvKv
wØZxqwUi `vg =
5
6
 cÖ_gwUi `vg
=
5
6
 25000 UvKv = 30000 UvKv
Avevi, g~j¨ e„w×i d‡j cÖ_gwUi `vg †e‡o nq
(25000 + 5000) UvKv = 30000 UvKv
 ZLb Kw¤úDUvi `yBwUi `v‡gi AbycvZ nq
30000 : 30000 ev 1 : 1| GwU GKwU GKK AbycvZ|
DËi: 1:1, GwU GKwU GKK AbycvZ
mßg †kÖwY
Abykxjbx: 2.1
127.7:5 Ges 8:9 `yBwU AbycvZ| G‡`i‡K avivevwnK
Abycv‡Z cÖKvk Ki|
☼ mgvavb: 1g AbycvZ = 7 : 5
=
5
7
=
85
87


=
40
56
= 56 : 40
2q AbycvZ = 8 : 9 =
9
8
=
59
58


=
45
40
= 40 : 45
AbycvZ `yBwUi avivevwnK AbycvZ 56 : 40 : 45
DËi: 56 : 40 : 45
128. 3, 6, 7 Gi 4_© mgvbycvZx wbY©q Ki|
K. 12 L. 14
M. 16 N. 18 DËi L
☼ mgvavb:
GLv‡b 1g ivwk 3, 2q ivwk 6, 3q ivwk 7
Avgiv Rvwb, 1givwk  4_© ivwk = 2q ivwk  3q ivwk
ev, 3  4_© ivwk = 6  7
ev, 4_© ivwk =
3
42
ev,4_© ivwk = 14
 4_© ivwk = 14
wb‡Y©q 4_© mgvbycvwZK 14
129.8, 7 Ges 14 Gi 3q ivwk wbY©q Ki|
K. 12 L. 16
M. 10 N. 8 DËi L
☼ mgvavb:
GLv‡b 1g ivwk 8, 2q ivwk 7 Ges 4_© ivwk 14
Avgiv Rvwb, 1givwk 4_© ivwk = 2q ivwk3q ivwk
ev, 8  14 = 7 3q ivwk
ev, 3q ivwk =
7
148
3q ivwk = 16 DËi: 16
130. GKwU µwgK mgvbycv‡Zi 1g I 3q ivwk h_vµ‡g 4 I
16 n‡j, ga¨ mgvbycvZx I µwgK mgvbycvZ wbY©q Ki|
☼ mgvavb:
Avgiv Rvwb, 1g ivwk3q ivwk = ( 2q ivwk)2
GLv‡b, 1g ivwk = 4 Ges 3q ivwk = 16
ev, 4  16 = ( ga¨ ivwk)2
A_ev, (ga¨ ivwk)2
= 64
ga¨ ivwk = 64 = 8
wb‡Y©q µwgK mgvbycvZ 4 : 8 :: 8 : 16 Ges wb‡Y©q
ga¨ mgvbycvZx 8
DËi: 4:8::8:16 Ges ga¨ mgvbycvZx 8
131.5wU LvZvi `vg 200 UvKv n‡j, 7wU LvZvi `vg KZ?

☼ mgvavb:
GLv‡b LvZvi msL¨v evo‡j `vgI evo‡e|
A_©vr, LvZvi msL¨vi AbycvZ = LvZvi `v‡gi AbycvZ
5 : 7 = 200 UvKv : 7wU LvZvi `vg
ev,
7
5
=
`vgLvZvi7wU
UvKv200
ev, 7wU LvZvi `vg =
5
2007 
UvKv = 280 UvKv|
DËi: 280
132. 12 Rb †jvK GKwU KvR 9 w`‡b Ki‡Z cv‡i| GKB
nv‡i KvR Ki‡j 18 R‡b KvRwU KZ w`‡b Ki‡Z
cvi‡e?
K. 12 w`b L. 9 w`b
M. 6 w`b N. 11 w`b DËi M
☼ mgvavb:
jÿKwi, †jvK msL¨v evo‡j mgq Kg jvM‡e, Avevi
†jvKmsL¨v Kg‡j mgq †ewkjvM‡e| †jvKmsL¨vi mij
AbycvZ mg‡qi e¨¯Í Abycv‡Zi mgvb n‡e|
12 : 18 = wb‡Y©q mgq : 9 w`b
ev,
18
12
=
wb‡Y©q mgq
9 w`b
ev,
3
2
=
wb‡Y©q mgq
9 w`b
ev, wb‡Y©q mgq =
3
92 
w`b = 6 w`b
DËi: 6 w`b
133. 20 wgUvi Kvco‡K wZb fvB‡evb AwgZ, mywgZ I
‣PwZi g‡a¨ 5 : 3 : 2 Abycv‡Z fvM Ki‡j cÖ‡Z¨‡Ki
Kvc‡oi cwigvY KZ?[7g †kÖwY-2.1-D`vniY-7]
☼ mgvavb:
Kvc‡oicwigvY = 20 wgUvi
cÖ`Ë AbycvZ = 5 : 3 : 2
Abycv‡Zi msL¨v¸‡jvi †hvMdj = 5 + 3 + 2 = 10
Awg‡Zi Ask = 20 wgUv‡ii
10
5
Ask = 10 wgUvi
mywg‡Zi Ask = 20 wgUv‡ii
10
3
Ask = 6 wgUvi
Ges •PwZi Ask = 20 wgUv‡ii
10
2
Ask = 4 wgUvi
DËi:AwgZ, mywgZ I ‣PwZi Kvc‡oi cwigvY h_vµ‡g 10
wgUvi, 6wgUvi I 4 wgUvi|
134.cwbi I Zc‡bi Av‡qi AbycvZ 4:3| Zcb I iwe‡bi
Av‡qi AbycvZ 5:4| cwb‡ii Avq 120 UvKv n‡j,
iwe‡bi Avq KZ?
☼ mgvavb:
cwbi : Zcb = 4:3 = 45:35 = 20:15
Zcb : iweb = 5:4 = 53:43 = 15:12 [Dfq Abycv‡Z
Zc‡bi gvb mgvb Kivi Rb¨ Dc‡i 5 Ges wb‡P 3w`‡q ¸Y]
myZivs, cwbi:Zcb:iweb = 20:15:12
awi, cwbi, Zcb I iwe‡bi Avq h_vµ‡g, 20K, 15K
Ges 12K
GLv‡b, cwb‡ii Avq, 20K = 120 myZivs K = 6
Zvn‡j iwe‡bi Avq = 12K = 126 = 72 UvKv|
135. wb‡Pi µwgK mgvbycv‡Zi cÖvšÍxq ivwk `yBwU †`Iqv
Av‡Q| mgvbycvZ •Zix Ki:
K) 6, 24
mgvavb:
Avgiv Rvwb, µwgK mgvbycv‡Z
1g ivwk  3q ivwk = (2q ivwk)2
 6  24 = (ga¨ ivwk)2
ev, (ga¨ ivwk)2
= 144
ga¨ ivwk = 144 = 12
wb‡Y©q µwgK mgvbycvZ = 6 : 12 :: 12 : 24
L) 25, 81
mgvavb:
 25  81 = (ga¨ ivwk)2
ev, (ga¨ ivwk)2
= 2025
ga¨ ivwk = 2025 = 45
M) 16, 49
mgvavb:
 16 49 = (ga¨ ivwk)2
ev, (ga¨ ivwk)2
= 784
ga¨ ivwk = 784 = 28

N)
7
5
, 1
5
2
O) 1.5, 13.5
mgvavb:
GLv‡b, 1g ivwk =
7
5
Ges 3q ivwk = 1
5
2
=
5
7

7
5

5
7
= (ga¨ ivwk)2
ev, (ga¨ ivwk)2
= 1
 ga¨ ivwk = 1 = 1
wb‡Y©q µwgK mgvbycvZ =
7
5
: 1 :: 1 : 1
5
2
136. wb‡Pi ivwk¸‡jvi 4_© mgvbycvZx wbY©q Ki:
K) 5, 7, 10
mgvavb:
Avgiv Rvwb, 4wU ivwk mgvbycvZx n‡j,
1g ivwk  4_© ivwk = 2q ivwk  3q ivwk
GLv‡b, 1g ivwk = 5, 2q ivwk = 7, 3q ivwk = 10
 5  4_© ivwk = 7  10 ev, 4_© ivwk =
5
107 
ev, 4_© ivwk = 14  4_© mgvbycvZx = 14
L) 15, 25, 33
mgvavb:
 15  4_© ivwk = 25  33 ev, 4_© ivwk =
15
3325
M) 16, 24, 32
mgvavb:
 16  4_© ivwk = 24  32 ev, 4_© ivwk =
16
3224
N) 8, 8
2
1
, 4
mgvavb:
GLv‡b, 1g ivwk = 8, 2q ivwk = 8
2
1
ev,
2
17
,
3q ivwk = 4
 8  4_© ivwk =
2
17
 4 ev, 4_© ivwk
=
2
17
4
8
1
ev, 4_© ivwk =
4
17
 4_© mgvbycvZx =
4
17
O) 5, 4.5, 7
☼ mgvavb:
GLv‡b, 1g ivwk = 5, 2q ivwk = 4.5, 3q ivwk = 7
 5  4_© ivwk = 4.5 7 ev, 4_© ivwk =
5
31.5
ev, 4_© ivwk = 6.3 4_© mgvbycvZx = 6.3
137.15 †KwR Pv‡ji `vg 600 UvKv n‡j, Giƒc 25 ‡KwR
Pv‡ji `vg KZ?
☼ mgvavb:
GLv‡b, Pv‡ji cwigvY evo‡j `vgI evo‡e|
A_©vr, Pv‡ji cwigv‡Yi AbycvZ = Pv‡ji `v‡gi AbycvZ
15 : 25 = 600 UvKv : 25 †KwR Pv‡ji `vg
ev,
25
15
=
600 UvKv
25 †KwRi Pv‡ji `vg
ev, 25 †KwRi Pv‡ji `vg =
15
60025
UvKv = 1000
UvKv| (DËi)
138. GKwU Mv‡g©›Um d¨v±wi‡Z •`wbK 550 wU kvU© •Zix
nq| H d¨v±wi‡Z GKB nv‡i 1 mßv‡n KZwU kvU© •Zwi
nq|
K. 3750wU L. 3800wU
M. 3850wU N. 3900wU DËi M

☼ mgvavb:
GLv‡b w`‡bi msL¨v evo‡j •Zix kv‡U©i msL¨vI evo‡e|
A_©vr, w`‡bi msL¨vi AbycvZ = •Zix kv‡U©i msL¨vi
AbycvZ
Avevi, 1 mßvn = 7 w`b
1 : 7 = 550wU : 7 w`‡b •Zix kv‡U©i msL¨v
ev,
7
1
=
550wU
7 w`‡b •Zix kv‡U©i msL¨v
 7 w`‡b •Zix kv‡U©i msL¨v = (7  550) wU
= 3850 wU (DËi)
gy‡Lgy‡L mgvavb :
7 550 = 3850wU
139. Kwei mv‡n‡ei wZb cy‡Îi eqm h_vµ‡g 5 eQi,
7 eQi I 9 eQi| wZwb 4200 UvKv wZb cyÎ‡K Zv‡ì
eqm Abycv‡Z fvM K‡i w`‡jb, †K KZ UvKv cv‡e?
☼ mgvavb:
GLv‡b, wZb cy‡Îi eqm h_vµ‡g 5, 7 I 9 eQi|
wZb cy‡Îi eq‡mi AbycvZ = 5 : 7 : 9
Abycv‡Zi msL¨v¸‡jvi ‡hvMdj = 5 + 7 + 9 = 21
 5 eQi eqmx cyÎ cv‡e 4200 UvKvi
21
5
Ask
= 






21
5
4200 UvKv = 1000 UvKv
7 eQi eqmx cyÎ cv‡e 4200 UvKvi
21
7
Ask
= 






21
7
4200 UvKv = 1400 UvKv
9 eQi eqmx cyÎ cv‡e 4200 UvKvi
21
9
Ask
= 






21
9
4200 UvKv = 1800 UvKv
DËi: 1800 UvKv, 1400 UvKv, 1000 UvKv|
140.2160 UvKv iƒwg, ‡Rmwgb I KvKwji g‡a¨ 1:2:3
Abycv‡Z fvM K‡i w`‡j †K KZ UvKv cv‡e?
☼ mgvavb:
iæwg,†Rmwgb I KvKwji cÖvß UvKvi AbycvZ = 1 : 2 : 3
Abycv‡Zi msL¨v¸‡jvi †hvMdj = 1 + 2+ 3 = 6
iæwg cv‡e 2160 UvKvi
6
1
Ask = 






6
1
2160 UvKv =
360 UvKv
‡Rmwgb cv‡e 2160 UvKvi
6
2
Ask = 






6
2
2160
UvKv = 720 UvKv
KvKwj cv‡e 2160 UvKvi
6
3
Ask = 






6
3
2160 UvKv
= 1080 UvKv
DËi: iæwg=360, †Rmwgb=720, KvKwj=1080
141. wKQz UvKv jvwee, mvwg I wmqv‡gi g‡a¨ 5 : 4 : 2
Abycv‡Z fvM K‡i †Ìqv nj | wmqvg 180 UvKv †c‡j
jvwee I mvwg KZ UvKv cv‡e wbY©q Ki| [7g †kÖwY-
(Aby:2.1)]
K. 450 UvKv I 350 UvKv
L. 450 UvKv I 400 UvKv
M. 450 UvKv I 360 UvKv
N. 460 UvKv I 380 UvKv DËi M
☼ wjwLZ mgvavb:
awi, jvwee , mvwg I wmqv‡gi Kv‡Q h_vµ‡g 5x, 4x Ges
2x UvKv Av‡Q|
cÖkœg‡Z, wmqv‡gi UvKv 2x = 180
x = 1802 = 90 UvKv|
myZivs jvwee cv‡e, 5x, = 590 = 450 UvKv
Ges mvwg cv‡e, = 4x = 490 = 360UvKv
142.meyR, Wvwjg I wjsKb wZb fvB| Zv‡ì wcZv 6300
UvKv Zv‡ì g‡a¨ fvM K‡i w`‡jb| G‡Z meyR Wvwj‡gi
5
3
Ask Ges Wvwjg wjsK‡bi wØ¸Y UvKv cvq|
cÖ‡Z¨‡Ki UvKvi cwigvY †ei Ki| DËi: meyR=1800,
Wvwjg=3000, wjsKb=1500
143.Zvgv, `¯Ív I iƒcv wgwj‡q GK iK‡gi Mnbv •Zwi Kiv
n‡jv| H Mnbvq Zvgv I `¯Ívi AbycvZ 1:2 Ges `¯Ív I
iƒcvi AbycvZ 3:5| 19 MÖvg IR‡bi Mnbvq KZ MÖvg
iƒcv Av‡Q?
K. 10 MÖvg L. 20 MÖvg
M. 15 MÖvg N. 25 MÖvg DËi K
☼ mgvavb:
Zvgv : `¯Ív = 1 : 2 = 3 : 6 [Dfq ivwk‡K 3 Øviv ¸Y
K‡i]
gy‡L gy‡L: x a‡i cv‡ki wbq‡gi GB wjwLZ mgvavbwU‡KB
Avgiv 1 Ask a‡i gy‡L gy‡L DËi †ei Ki‡Z cvwi| Abycv‡Z
wmqv‡gi Ask 2 Avevi ‡h‡nZz cÖ‡kœ wmqv‡gi UvKv 180 Zvn‡j
2 Ask = 180 n‡j 1 Ask = 90, jvwee= 5 Ask =
590= 450 I mvwg 4 Ask = 490= 360

`¯Ív : iƒcv = 3 : 5 = 6 : 10 [Dfq ivwk‡K 2 Øviv ¸Y
K‡i]
myZivs, Zvgv : `¯Ív : iƒcv = 3 : 6 : 10
 Abycv‡Zi msL¨v¸‡jvi †hvMdj = 3 + 6 + 10 = 19
 19 MÖvg IR‡bi Mnbvq iƒcvi cwigvY 19 MÖv‡gi
19
10
Ask = 






19
10
19 MÖvg = 10 MÖvg| (DËi)
144. `yBwU mgvb gv‡ci Møvm kie‡Z cwic~Y© Av‡Q| H
kie‡Z cvwb I wmiv‡ci AbycvZ h_vµ‡g cÖ_g Møv‡m 3
: 2 I wØZxq Møv‡m 5 : 4| H `yBwU Møv‡mi kieZ GK‡Î
wgkÖY Ki‡j cvwb I wmiv‡ci AbycvZ KZ n‡e? [7g
†kÖwY-2.1]
K. 13:9 L. 26:19
M. 26:15 N. 12:7 DËi L
☼ mgvavb:
cÖ_g Møv‡m cvwb I wmivc = 3:2
Abycv‡Zi †hvMdj = 2+3 = 5
myZivs cvwb Av‡Q =
5
3
Ask Ges
wmivc Av‡Q,
5
2
Ask [Gfv‡e fMœvsk AvKv‡i Avjv`v
cwigvb †ei K‡i wb‡Z nq|]
Avevi,
2q Møv‡m, cvwb I wmivc = 5:4 ,
Abycv‡Zi †hvMdj = 5+4 = 9
myZivs, 2q Møv‡m cvwb Av‡Q, =
9
5
Ask Ges
wmivc Av‡Q,
9
4
Ask
`ywU Møv‡m GK‡Î †gkv‡j bZzb wgkÖ‡Y cvwbi cwigvb n‡e
=
5
3
+
9
5
=
45
52
45
2527


[cÖ_g Møv‡mi cvwb+2q
Møv‡mi cvwb]
bZzb wgkÖ‡Y wmiv‡ci †gvU cwigvb =
5
2
+
9
4
=
45
38
45
2018


[cÖ_g Møv‡mi wmivc+2q Møv‡mi wmivc]
myZivs bZzb wgkÖ‡Y cvwb I wmiv‡ci AbycvZ =
45
52
:
45
38
ev
45
52
45:
45
38
45 = 52:38 = 26:19
DËi: 26:19
`y‡qi AwaK †QvU cvÎ †_‡K eo cv‡Î ivL‡jI wbqg
GKB: cÖ_‡g AbycvZ †_‡K fMœvsk evbv‡Z nq Zvici
GKB `ªe‡Yi fMœvsk¸‡jv †hvM K‡i Avevi AbycvZ mvRv‡Z
nq|
145.K:L = 4:7, L:M = 10:7 n‡j, K:L:M wbY©q Ki|
K. 40:70:49
L. 40 : 70 : 50
M. 40 : 60 : 49
N. 40 : 65 : 70 DËi K
☼ mgvavb:
K : L = 4 : 7 = 4  10 : 7  10 = 40 : 70
L : M = 10 : 7 = 10  7 : 7  7 = 70 : 49
K : L : M = 40 : 70 : 49
146.9600 UvKv mviv, gvBgybv I ivBmvi g‡a¨ 4 : 3 : 1
Abycv‡Z fvM K‡i w`‡j †K KZ UvKv cv‡e? [7g †kÖwY-
(Aby:2.1)]
☼ mgvavb:
mviv : gvBgybv : ivBmv = 4 : 3 : 1
Abycv‡Zi ivwk¸‡jvi †hvMdj = 4 + 3 + 1 = 8 8
As‡ki gvb 9600 n‡j 1 As‡ki gvb =96008=
1200
myZivs: mviv cv‡e 41200 = 4800,
gvBgybv cv‡e 31200 = 3600 Ges
ivBmv cv‡e 11200 = 1200
147. wZbRb Qv‡Îi g‡a¨ 4200 UvKv Zv‡`i †kÖwY Abycv‡Z
fvM K‡i †`Iqv nj | Zviv hw` h_vµ‡g 6ô, 7g I
8g †kÖwYi wkÿv_x© nq, Z‡e †K KZ UvKv cv‡e? [7g
†kÖwY-(Aby:2.1)]
☼ mgvavb:
GLv‡b, wZbRb Qv‡Îi †kÖwY 6ô, 7g I 8g
wZbRb Qv‡Îi †kÖwYi AbycvZ = 6 : 7 : 8
 Abycv‡Zi msL¨v¸‡jvi †hvMdj = 6 + 7 + 8 = 21
6ô †kÖwYi QvÎ cv‡e 4200 UvKvi
21
6
Ask

= 






21
6
4200 UvKv = 1200 UvKv
7g †kÖwYi QvÎ cv‡e 4200 UvKvi
21
7
Ask
= 






21
7
4200 UvKv = 1400 UvKv
8g †kÖwYi QvÎ cv‡e 4200 UvKvi
21
8
Ask
= 






21
8
4200 UvKv = 1600 UvKv
DËi: 6ô †kÖwYi wkÿv_x© cv‡e 1200 UvKv,7g †kÖwYi
wkÿv_x© cv‡e 1400 UvKv,8g †kÖwYi wkÿv_x© cv‡e
1600 UvKv |
148.†mvjvqgvb I mvjgv‡bi Av‡qi AbycvZ 5 : 7 mvjgvb
I BDmy‡di AbycvZ 4 : 5, †mvjvqgv‡bi Avq 120
UvKv n‡j BDmy‡di Avq KZ? [7g †kÖwY-(Aby:2.1)]
☼ mgvavb:
†mvjvqgvb : mvjgvb = 5 : 7 = 20 : 28 [Dfq‡K 4
Øviv ¸b ]
mvjgvb : BDmyd = 4 : 5 = 28 : 35 [7 w`‡q ¸Y
K‡i]
 †mvjvqgvb : mvjgvb : BDmyd = 20 : 28 : 35
[mvjvgv‡bi `ywU 28 †_‡K GKwU †bqv n‡jv|]
‡mvjvqgv‡bi Avq = 20 Ask = 120 UvKv n‡j 1 Ask =
12020 = 6 UvKv|
myZivs BDmy‡di Avq = 356 = 210 UvKv|
DËi: 210 UvKv|
149.wcZv I cy‡Îi eZ©gvb eq‡mi AbycvZ 7:2 Ges 5 eQi
c‡i Zv‡ì eq‡mi AbycvZ 8:3 n‡e| Zv‡ì eZ©gvb
eqm KZ? [9g-10g †kÖwY(Aby.11)-Gi D`v:2]
K. wcZv 35 eQi Ges cyÎ 15 eQi
L. wcZv 40 eQi Ges cyÎ 10 eQi
M. wcZv 50 eQi Ges cyÎ 15 eQi
N. wcZv 35 eQi Ges cyÎ 10 eQi DËi L
awi, wcZv I cy‡Îi eZ©gvb eq‡mi AbycvZ = 7x Ges
2x
cÖkœg‡Z,
3
8
52
57



x
x
(Dfq cv‡k 5 eQi c‡ii eq‡mi AbycvZ)
ev, 21x+15= 16x + 40
ev, 21x- 16x = 40-15
ev, 5x = 25ev, x =
5
25
ev, x = 5
myZivs Zv‡ì eZ©gvb eqm: wcZv = 75 = 35 eQi Ges
cyÎ 25 = 10 eQi| DËi: wcZv 35 I cyÎ 10|
150.GKwU `ªe¨ µq K‡i 28% ÿwZ‡Z weµq Kiv nj|
weµqg~j¨ I µqg~‡jï AbycvZ wbY©q Ki| [9g-10g
†kÖwY-(Aby:11.1)]
DËi; 18:25 [Help: weµqg~j¨ : µqg~j¨ = 72:100 =
18:25 ]
151.wcZv I cy‡Îi eZ©gvb eq‡mi mgwó 70 eQi| Zv‡ì
eq‡mi AbycvZ 7 eQi c~‡e© wQj 5:2| 5 eQi ci
Zv‡ì eq‡mi AbycvZ KZ n‡e? [9g-10g †kÖwYAby.11.1]
mgvavb:
wcZv I cy‡Îi eZ©gvb eq‡mi mgwó 70 eQi 7 eQi c~‡e©
wcZv I cy‡Îi eqm wQj = 70-(7+7) = (70-14)=56
eQi
7 eQi c~‡e© wcZv I cy‡Îi eq‡mi AbycvZ wQj 5 : 2
Abycv‡Zi ivwkØ‡qi †hvMdj = 5 + 2 = 7
 7 eQi c~‡e© wcZvi eqm wQj
=
7
5
56 = 40 eQi
 7 eQi c~‡e© cy‡Îi eqm wQj
=
7
2
56 = 16 eQi
eZ©gv‡b wcZvi eqm = (40+7) eQi = 47 eQi
eZ©gv‡b cy‡Îi eqm=(16+7) eQi = 23 eQi
5 eQi ci wcZvi eqm = (47+5) eQi = 52 eQi
5 eQi ci wcZvi eqm = (23+5) eQi = 28 eQi
 5 eQi ci wcZv I cy‡Îi eq‡mi AbycvZ
52:28 = 26 : 14= 13: 7 DËi: 13:7
152.K, L, I M GK RvZxq ivwk Ges K:L = 3:4, L:M =
6:7 n‡j, K:L:M KZ?[9g-10g †kÖwYi (Aby:11.1)
D`v:12]
K. 9:11:14 L. 9:12:14
M. 9:13:14 N. 9:10:14 DËi L
☼ mgvavb:
K:L =
12
9
34
33
4
3




Ges L:M =
14
12
27
26
7
6




[GLv‡b 4 I 6 Gi j.mv.¸. 12]

 K:L:M = 9:12:14
153. GKwU wÎfz‡Ri wZbwU †Kv‡Yi AbycvZ 3:4:5; †KvY
wZbwU wWwMÖ‡Z cÖKvk Ki| [9g-10g †kÖwY(Aby.11.2)-Gi
D`v:13]
mgvavb:
g‡b Kwi,
cÖ`Ë AbycvZ Abymv‡i †KvY wZbwU h_vµ‡g 3x, 4x Ges
5x|
wÎf~‡Ri wZb †Kv‡Yi mgwó = 1800
|
cÖkœvbymv‡i, 3x +4x+5x = 1800
ev, 12x = 1800
ev, x = 150
AZGe †KvY wZbwU n‡jv, 3x = 3150
=450
4x = 4150
=600
Ges 5x = 5150
= 750
DËi: 450
, 600
I 750
154.Avwid I AvwK‡ei eq‡mi AbycvZ 5:3; Avwi‡di eqm
20 eQi n‡j, KZ eQi ci Zv‡ì eq‡mi AbycvZ 7:5
n‡e? [9g-10g †kÖwYAby.11.2]
K. 5 eQi L. 6 eQi
mgvavb:
Avwid I AvwK‡ei eq‡mi AbycvZ 5x Ges 3x
cÖkœg‡Z, 5x = 20
ev, x = 4 [5 Øviv fvM K‡i]
 x = 5
AvwK‡ei eqm 3x = 3  4 = 12 eQi
awi, p eQi ci Zv‡ì eq‡mi AbycvZ 7 : 5 n‡e|
(20 + p) : (12 + p) = 7 : 5
ev,
p12
p20


=
5
7
ev, 5 (20 + p) = 7(12 + p)
ev, 100 + 5p = 84 + 7p
ev, 100 – 84 = 7p – 5p
ev, 16 = 2p
ev, p = 8
 8 eQi ci AbycvZ n‡e 7 : 5
155. x : y = 7 : 5, y : z = 5 : 7 n‡j x : z = KZ?
[9g-10g †kÖwYAby.11.2]
a. 35 : 49 b. 35 : 35
c. 25 : 49 d. 49 : 25 Ans b
mgvavb:
x : y = 7 : 5
y : z = 5 : 7
x : y : z = 7 : 5 : 7
x : z = 7 : 7
= (7  5) : (7  5) [Dfq ivwk‡K 5 Øviv ¸Y]
= 35 : 35
156. K, L, M, N Gi g‡a¨ 300 UvKv Ggb fv‡e fvM
K‡i `vI †hb, K Gi Ask : L Gi Ask = 2:3; L Gi
Ask : M Gi Ask = 1:2 Ges M Ask : N Gi Ask =
3:2 nq| [9g-10g †kÖwYAby.11.2]
mgvavb:
K Gi Ask : L Gi Ask = 2 : 3
L Gi Ask : M Gi Ask = 1 : 2= 3:6 [ Dfq ivwk‡K 3
Øviv ¸Y K‡i ] [`y RvqMvq L mgvb n‡jv]
 L Gi Ask : M Gi Ask = 3 : 6
M Gi Ask : N Gi Ask = 3 : 2 = = 6:4 [ Dfq
ivwk‡K 2 Øviv ¸b ] [M Gi Ask mgvb n‡jv]
 K Gi Ask : L Gi Ask : M Gi Ask : N Gi Ask =
2: 3 : 6 : 4
Abycv‡Zi ivwk¸‡jvi †hvMdj = 2+3+6+4 = 15 Ges
†gvU UvKvi cwigvY = 300 UvKv|
 K Gi Ask = UvKv
15
2
300
= 40 UvKv
 L Gi Ask = UvKv
15
3
300
= 60 UvKv
 M Gi Ask = UvKv
15
6
300
= 120 UvKv
 N Gi Ask = UvKv
15
4
300
= 80 UvKv
myZivs K, L, M Ges N cv‡e h_vµ‡g, 40, 60, 120 I
80 UvKv| [DËi:]
157.wZbRb †R‡j 690 wU gvQ ai‡Q| Zv‡ì As‡ki
AbycvZ
5
4
,
3
2
Ges
6
5
n‡j, †K KqwU gvQ ‡cj? [9g-
10g †kÖwYAby.11.2]

mgvavb:
wZbRb †R‡ji As‡ki AbycvZ =
3
2
:
5
4
:
6
5
= 20 : 24 : 25 [30 Øviv ¸Y K‡i]
AbycvZ ivwk¸‡jvi †hvMdj = (20 + 24 + 25)
= 69
 1g ‡R‡j cvq = 690 Gi
69
20
wU
= 200 wU
2q ‡R‡j cvq = 690 Gi
69
24
wU
= 240 wU
3q ‡R‡j cvq = 690 Gi
69
25
wU
= 250 wU
Ans: 200 wU, 240 wU, 250 wU
158. GKwU wÎfz‡Ri cwimxgv 45 †m.wg.| evû¸‡jvi •`‡N©ï
AbycvZ 3:5:7 n‡j, cÖ‡Z¨K evûi cwigvb wbY©q Ki|
☼ mgvavb:
g‡b Kwi, Abycv‡Zi mvaviY ivwk x
wÎfz‡Ri evû¸‡jvi •`N©¨ 3x, 5x Ges 7x
cÖkœg‡Z, 3x + 5x + 7x = 45 ev, 15x = 45 ev, x =
15
45
 x = 3
wÎfz‡Ri evûi •`N©¨ h_vµ‡g,
3x = 3  3 = 9 ‡m.wg. ; 5x = 5  3
= 15 ‡m.wg. ; 7x = 7  3 = 21 ‡m.wg.|
Ans: 9 ‡m.wg. ; 15 ‡m.wg. ; 21 ‡m.wg.|
159. `yBwU msL¨vi AbycvZ 5:7 Ges Zv‡ì M.mv.¸ 4 n‡j
msL¨v `yBwUi j.mv.¸ KZ? [9g-10g †kÖwYAby.11.2]
K. 6 L. 9
M. 12 N. 4 DËi N
☼ mgvavb:
awi, msL¨v `ywU h_vµ‡g, 5x Ges 7x Ges G‡ì j.mv.¸
= 35x (msL¨v 5 I 7 Gi j.mv.¸ = 35 Ges Dfq ivwk
†_‡K x wb‡Z n‡e)
Ges msL¨v `ywUi M.mv.¸ = x
GLv‡b, x = 4
myZivs msL¨v `ywUi j.mv.¸ = 354 =140 DËi: 140
160. wµ‡KU †Ljvq mvwKe, gykwdKzi I gvkivdx 171 ivb
Ki‡jv| mvwKe I gykwdK Ges gykwdKzi I gvkivwdi
iv‡bi AbycvZ 3:2 n‡j †K KZ ivb K‡i‡Q? [9g-10g
†kÖwYAby.11.2]
☼ mgvavb:
mvwKe : gykwdK = 3 : 2 = 9:6
[Dfq ivwk‡K 3 Øviv ¸b]
Avevi, gykwdK : gvkivwd = 3 : 2 = 6 : 4
[2 w`‡q ¸Y K‡i Dfq ‡ÿ‡Î gykwd‡Ki gvb 6 Ask]
myZivs mvwKe:gykwdK :gvkivwd = 9:6:4
Abycv‡Zi ivwk¸‡jvi †hvMdj = 9+6+4=19 Ges
‡gvU iv‡bi cwigvb 171
mvwKe K‡i‡Q =
19
9
171 ivb = 81 ivb
gykwdK K‡i‡Q =
19
6
171 ivb = 54 ivb
gvkivwd K‡i‡Q =
9
4
171 ivb = 36 ivb
DËi: 81, 54 Ges 36 ivb|
161.GKwU Awd‡m 2 Rb Kg©KZ©v, 7 Rb KiwYK Ges 3 Rb
wcqb Av‡Q| GKRb wcqb 1 UvKv †c‡j GKRb KiwYK
cvq 2 UvKv, GKRb Kg©KZ©v cvq 4 UvKv| Zv‡ì
mK‡ji †gvU †eZb 150,000 UvKv n‡j, †K KZ †eZb
cvq? [9g-10g †kÖwYAby.11.2]
☼ mgvavb:
awi, 1 Rb wcqb cvq x UvKv
3 " " " 3x "
1 Rb KiwYK cvq 2x UvKv
7 " " " = (7 2x)" = 14x UvKv
1 Rb Kg©KZ©v cvq 4x UvKv
2 " " " (2 4x)" = 8x UvKv
cÖkœg‡Z, 3x+14x+8x = 150000
ev, 25x = 150000
ev, x =
25
150000
 x = 6000
1 Rb wcqb cvq x =6000 UvKv
1 Rb KiwYK cvq = 2x =(26000)=12000 UvKv
1 Rb Kg©KZ©v cvq 4x = (46000)=24000 UvKv
DËi: 1 Rb KiwYK cvq=12000 UvKv, 1 Rb Kg©KZ©v cvq=
24000 UvKv Ges wcqb cvq 6000 UvKv|
162. hw` †Kvb eM©‡ÿ‡Îi evûi cwigvb 20% e„w× cvq,
Z‡e Zvi †ÿÎdj kZKiv KZ e„w× cv‡e? [9g-10g
†kÖwYAby.11.2]
K. 40% L. 30%
M. 44% N. 45% DËi M

☼ mgvavb:
cÖ_‡g ‣`N©¨ 20% ‡e‡o n‡e 120% Gici Avevi 20%
evo‡j Zv 120 Gi Dci 20% ev 24 †e‡o n‡e 144|
myZivs †gv‡Ui Dci †ÿÎdj evo‡e = 44%|
kU©KvU :
=
= 40 + 4 = 44%
eM©‡ÿ‡Îi cÖ‡Z¨K evû mgvb nIqvq ‣`N©¨ I cÖ¯’ mgvb|
ZvB cÖwZ‡ÿ‡Î 20% e„w× cv‡e| †h‡nZz •`N¨© I cÖ¯’ e„w×
‡c‡q‡Q ZvB abvZ¥K wPý MÖnY Ki‡Z n‡e|
163. GKwU AvqZ‡ÿ‡Îi ‣`N©¨ 10% e„w× cvq Ges cÖ¯’
10% n«vm †c‡j AvqZ‡ÿ‡Îi †ÿÎdj kZKiv KZ e„w×
ev n«vm cv‡e? [9g-10g †kÖwYAby.11.2]
K. 1% n«vm L. 2% e„w×
M. 1% e„w× N. 2% n«vm DËi K
☼ mgvavb:
g‡b Kwi, AvqZ‡ÿ‡Îi •`N©¨ = x GKK Ges
cÖ¯’ = y GKK
 ‡ÿÎdj = xy eM© GKK
10% e„w×‡Z AvqZ‡ÿ‡Îi ‣`N©¨
= (x + xGi
100
10
) GKK
= (x +
10
x
) GKK
=
10
x11
GKK
Ges 10% n«v‡m AvqZ‡ÿ‡Îi cÖ¯’ = (y - y Gi
100
10
)
GKK = (y -
10
y
) GKK
=
10
y9
GKK
 ‡ÿÎdj
10
x11

10
y9
eM© GKK
=
100
xy99
eM© GKK
‡ÿÎdj n«vm cvq = 






100
xy99
xy eM© GKK
=
100
xy
eM© GKK
xyeM© GK‡K ‡ÿÎdj n«vm cvq =
100
xy
eM© GKK
1 Ó Ó Ó Ó Ó
xy100
xy

Ó Ó
100 Ó Ó Ó Ó Ó
xy100
100xy


Ó Ó
= 1 eM© GKK
Ans: 1% n«vm cvq|
kU©KvU :
= = (-1)
e„w× n‡j abvZ¥K Ges n«vm n‡j FYvZ¥K | ‣`N©¨ e„w×
cvIqvq Ô+Õ n‡q‡Q Ges cÖ¯’ n«vm cvIqvq Ô-Õ n‡q‡Q|
164.GKwU gv‡Vi Rwg‡Z †m‡Pi my‡hvM Avmvi Av‡Mi I
c‡ii dj‡bi AbycvZ 4:7 H gv‡V †h Rwg‡Z Av‡M
304 KzB›Uvj avb dj‡Zv, †mP cvIqvi c‡i Zvi djb
KZ n‡e? [9g-10g †kÖwYAby.11.2]
K. 530 KzB›Uvj L. 532 KzB›Uvj
M. 535 KzB›Uvj N. 540 KzB›Uvj DËi L
☼ mgvavb:
Av‡Mi Ask = 4Ask = 304 Zvn‡j 1 Ask
= 3044 = 76 KzB›Uvj|
bZzb Ask = 7 Ask = 767 = 532 KzB›Uvj |
DËi: 532 KzB›Uvj|
165. avb I avb †_‡K Drcbœ Pv‡ji AbycvZ 3:2 Ges Mg I
Mg †_‡K Drcbœ mywRi AbycvZ 4:3 n‡j, mgvb
cwigv‡bi avb I Mg †_‡K Drcbœ Pvj I mywRi AbycvZ
†ei Ki? [9g-10g †kÖwYAby.11.2]
K. 9 : 8 L. 8 : 9
M. 7 : 9 N. 9 : 7 DËi L
☼ mgvavb:
awi, avb I avb †_‡K Drcbœ Pv‡ji cwigvY h_vµ‡g
= 3x I 2x
Ges Mg I Mg †_‡K Drcbœ mywRi AbycvZ = 4y I 3y
[`y‡Uv Avjv`v AbycvZ ZvB Avjv`v ivwk ai‡Z n‡e]
GLb avb I M‡gi mg cwigvY †ei Kivi Rb¨ avb 3x
Ges Mg 4y G‡`i j.mv.¸ = 12xy
3x cwigvY av‡b Pvj nq =2x
 1 ‘’ ‘’ ‘’ ‘’ =
x3
x2

 12xy ‘’ ‘’ ‘’ ‘’ =
x3
xy12x2 
= 8xy
Avevi,
4y cwigvY M‡g mywR nq = 3y
 1 ‘’ ‘’ ‘’ ‘’ =
y4
y3
 12xy ‘’ ‘’ ‘’ ‘’ =
y4
xy12y3 
= 9xy
myZivs Drcbœ Pvj I mywRi AbycvZ
= 8xy: 9xy = 8:9
166. GKwU Rwgi †ÿÎdj 432 eM© wgUvi| H Rwgi •`N©¨
I cÖ‡¯’i m‡½ Aci GKwU Rwgi •`N©¨ I cÖ‡¯’i AbycvZ
h_vµ‡g 3:4 Ges 2:5 n‡j, Aci Rwgi †ÿÎdj KZ?
K. 1450 eM©wgUvi L. 1440 eM©wgUvi
M. 1460 eM©wgUvi N. 1470 eM©wgUvi DËi L
☼ mgvavb:
awi, GKwU Rwgi •`N©¨ 3x wgUvi I Aci Rwgi ‣`N©¨ 4x
wgUvi
GKwU Rwgi cÖ¯’ 2y wgUvi I Aci Rwgi cÖ¯’ 5y wgUvi
 GKwU Rwgi †ÿÎdj
= (3x 2y )eM© wg. =6xy eM© wg.
cÖkœg‡Z,
6xy = 432 ev,
6
432
xy 
ev, xy = 72...............(i)
Aci Rwgi †ÿÎdj = (4x 5y ) eM© wg.
=20xy eM© wg.
=2072 eM© wg [†h‡nZz xy = 72]
=1440 eM© wg.
myZivs Aci Rwgi †ÿÎdj = 1440 eM© wgUvi (DËi)
167.‡Rwg I wmwg GKB e¨vsK †_‡K GKB w`‡b 10% mij
gybvdvq Avjv`v Avjv`v cwigvY A_© FY †bq| †Rwg 2
eQi ci gybvdv Avm‡j hZ UvKv †kva K‡i 3 eQi ci
wmwg gybvdv Avm‡j ZZ UvKv †kva K‡i| Zv‡`i F‡Yi
AbycvZ wbY©q Ki| [9g-10g †kÖwYAby.11.2]
K. 12 : 13 L. 12 : 14
M. 13: 14 N. 13: 12 DËi N
☼ mgvavb:
awi, †Rwgi F‡Yi cwigvb = x UvKv
wmwgi F‡Yi cwigvb= y UvKv
10%nv‡i †Rwgi 2 eQ‡ii gybvdv
= UvKv





 2
100
10
x = UvKv





5
x
10%nv‡i wmwgi 3 eQ‡ii gybvdv
= UvKv





 3
100
10
y = UvKv





10
3y
cÖkœg‡Z,
5
x
x  =
10
y3
y 
[ ‡Rwgi gybvdv Avmj = wmwgi gybvdv Avmj]

5
xx5 
=
10
y3y10 

5
x6
=
10
y13
 60x = 65 y

60
65
y
x


12
13
y
x

x : y = 13 : 12
myZivs †Rwg I wmwgi F‡Yi AbycvZ 13 : 12(DËi)
gy‡L gy‡L: 10 †m‡K‡Û mgvavb|
avb:Pvj = 3:2 = 12:8
Mg:mywR = 4:3 =12 :9
`yB Abycv‡Z avb I Mg‡K mgvb Kivi Rb¨
cÖ_gwU‡Z 4 w`‡q Ges wb‡PiwU‡Z 3 w`‡q ¸Y|
GLb: Pvj:mywR = 8:9
wcGmwm KZ©„K wba©vwiZ wm‡jevm Abymv‡i.
cvwUMwYZ †_‡K gvÎ 03wU AsK cixÿvq
Avmvi K_v| Z‡e gv‡S gv‡S 1-2wU AsK †ewk
Avm‡Z †`Lv hvq| Avgv‡`i g‡Z, wcÖwjwgbvwi
cixÿvi Rb¨ ‡evW© eB‡qi GB AsK¸‡jv
h‡_ô| ZvB GB 167wU AsK †ewk †ewk K‡i
Abykxjb Kivi civgk© †`qv n‡jv|

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