04. lcm (math tutor by kabial noor [www.onlinebcs.com]

1. Math Tutor  1
CHAPTER- 04
j.mv.¸ I M.mv.¸
j.mv.¸ (LCM)
 04.01 ¸wYZK, mvaviY ¸wYZK I jwNô mvaviY ¸wYZK Kx?
(K) MywYZK (Multiple)t ¸wYZK wK? ¸wYZK n‡”Q †Kvb msL¨v‡K bvgZv AvKv‡i G‡Mv‡j †h msL¨v¸‡jv cvIqv hvq
†m¸‡jv‡K H msL¨vi ¸wYZK e‡j| †hgb- 3 I 6 Gi ¸wYZK wK wK †`Lv hvK|
3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 ...
6 = 6, 12, 18, 24, 30...
Dc‡i 3 I 6 Gi bvgZv w`‡q †h msL¨v¸‡jv †cjvg G¸‡jvB n‡”Q 3 I 6 Gi ¸wYZK| GLv‡b ... (WU) w`‡q eySv‡bv
n‡q‡Q 3 I 6 ¸wYZK Amxg Ni ch©šÍ Pj‡Z _vK‡e A_©vr †Kvb msL¨vi ¸wYZK Amxg msL¨K nq|
 g‡b ivLyb, †Kvb msL¨vi me‡P‡q †QvU ¸wYZK H msL¨vi mgvb| †hgb- 3 Gi me‡P‡q †QvU ¸wYZK 3 ev Zvi
mgvb, GKBfv‡e 6 Gi me‡P‡q †QvU ¸wYZK 6 ev Zvi mgvb|
(L) mvaviY ¸wYZK (Common Multiple)t Common ev mvaviY kãwUi gv‡b n‡”Q GKvwaK wRwb‡mi gv‡S
hZUzKz mv`„k¨ ev wgj _v‡K †mUzKz| mvaviY ¸wYZK Gi gv‡b n‡”Q GKvwaK msL¨vi ¸wYZK¸‡jvi gv‡S †h †h
¸wYZK mv`„k¨ ev wgj Av‡Q †mme ¸wYZK‡K eySvq| 3 I 6 Gi ¸wYZK¸‡jv gv‡S †Kvb ¸wYZK ¸‡jv common
ev mvaviY ev mv`„k¨ ev wgj Av‡Q †`Lv hvK|
3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 ...
6 = 6, 12, 18, 24, 30...
Avgiv †`L‡Z cvw”Q 3 I 6 Gi ¸wYZK ¸‡jvi gv‡S 6, 12, 18, 24, 30 ¸wYZK¸‡jv `yB RvqMv‡ZB Av‡Q, GB
¸wYZK¸‡jv‡KB ejv nq mvaviY ¸wYZK|
(M) jwNô mvaviY ¸wYZK (Lowest Common Multiple/ Least Common Multiple)t Avgiv †h‡Kvb
msL¨vi me‡P‡q eo ¸wYZK wbY©q Ki‡Z cvwi bv, KviY †h‡Kvb msL¨vi ¸wYZK Amxg Ni ch©šÍ n‡q _v‡K| †h‡nZz
†Kvb msL¨vi me‡P‡q eo ¸wYZK wbY©q Kiv hvq bv, †m‡nZz GKvwaK msL¨vi gv‡S me‡P‡q eo mv`„k¨ ev mvaviY
¸wYZKI wbY©q Kiv m¤¢e bq, Z‡e me‡P‡q †QvU mvaviY ¸wYZK wbY©q Kiv m¤¢e| Dc‡ii 3 I 6 Gi ¸wYZK¸‡jvi
gv‡S me‡P‡q †QvU ev jwNô mvaviY ¸wYZK n‡”Q 6| A_v©r, 3 I 6 Gi jwNô mvaviY ¸wbZK ev jmv¸ n‡”Q 6|
A_©vr, j.mv.¸ gv‡b n‡”Q jwNô mvaviY ¸wYZK (Least Common Multiple ev LCM)!
 jÿYxq-
(1) Avgiv 3 I 6 Gi hZ¸‡jv mvaviY ¸wYZK †c‡qwQ, †m¸‡jvi gv‡S j.mv.¸ ev jwNô mvaviY ¸wYZK n‡”Q 6|
gRvi welq n‡”Q evKx mvaviY ¸wYZK¸‡jv jmv¸ 6-Gi ¸wYZK! †`Lyb, 6  2 = 12, 6  3 = 18, 6 
4 = 24, 6  5 = 30 A_v©r, 6 Gi ¸wYZK- 12, 18, 24, 30 ... |
(2) †Kvb msL¨vi ¸wYZK‡K Zv‡K w`‡q fvM Ki‡j wbt‡k‡l wefvR¨ n‡e| †hgb- 3 Gi ¸wYZK 3, 6, 9, 12, 15,
18, 21, 24, 27, 30 †K 3 Øviv fvM Ki‡j wbt‡k‡l wefvR¨ n‡e| GKBfv‡e 6 Gi ¸wYZK¸‡jv‡K 6 Øviv
fvM Ki‡j wbt‡k‡l wefvR¨ n‡e|
(3) GKvwaK msL¨vi mKj mvaviY ¸wYZK‡K H GKvwaK msL¨v Øviv fvM Ki‡j wbt‡k‡l wefvR¨ n‡e| †hgb- 3 I
6 Gi mvaviY ¸wYZK 6, 12, 18, 24, 30 †K 3 I 6 Øviv fvM Ki‡j wbt‡k‡l wefvR¨ n‡e| GB K_vi A_©
`uvovj, jwNô mvaviY ¸wYZK (j.mv.¸) 6 †K 3 I 6 Øviv fvM Ki‡j wbt‡k‡l wefvR¨ n‡e Ges 6 Gi mKj
¸wYZK 12,18, 24, 30 ‡K 3 I 6 Øviv fvM Ki‡jI wbt‡k‡l wefvR¨ n‡e|

2. 2  Math Tutor
 04.02 j.mv.¸ wbY©q Kivi wbqgvewj
(K) ¸wYZK wbY©‡qi mvnv‡h¨ j.mv.¸ wbY©q: GwU j.mv.¸
wbY©‡qi GKwU †ewmK c×wZ| j.mv.¸ Gi msÁv
eyS‡Z GB c×wZwU mvnvh¨ K‡i| G c×wZi gva¨‡g
Avgiv Ô¸wYZK, mvaviY ¸wYZK I jwNô mvaviY
¸wYZKÕ msµvšÍ †gŠwjK aviYv¸‡jv eyS‡Z cvwi
(Dc‡i Av‡jvPbv co–b)| wKš‘ mivmwi j.mv.¸ wbY©q
Kivi Rb¨ GB c×wZwU Kvh©Kix bq, KviY G
c×wZ‡Z mgq †ewk jv‡M Ges A‡bK¸‡jv msL¨vi
†ÿ‡Î j.mv.¸ wbY©q Kiv RwUj jv‡M|
01. 6 I 12 Gi j.mv.¸ wbY©q Kiæb|
6 Gi ¸wYZKmg~n: 6, 12, 18, 24, 30, 36 ...
12 Gi ¸wYZKmg~n : 12, 24, 36, 48...
6 I 12 Gi ¸wYZK¸‡jvi gv‡S me‡P‡q †QvU/jwNô
mvaviY ¸wYZK n‡”Q 12| myZivs, 6 I 12 Gi
j.mv.¸ n‡jv 12|
(L) †gŠwjK ¸YbxqK ev Drcv`‡Ki mvnv‡h¨ j.mv.¸ wbY©q:
G c×wZwU cvwUMwY‡Zi Kvh©Kix bq, Z‡e exRMwY‡Zi
Rb¨ GKwU Kvh©Kix c×wZ|
02. 12, 24 I 30 Gi j.mv.¸ wbY©q Kiæb|
cÖ_‡g msL¨v¸‡jvi j.mv.¸ †gŠwjK Drcv`K ev
¸YbxqKmg~n †ei K‡i wb‡Z n‡e|
12 Gi †gŠwjK Drcv`K ev ¸YbxqKmg~n : 223
24 Gi †gŠwjK Drcv`K ev ¸YbxqKmg~n : 2223
30 Gi †gŠwjK Drcv`K ev ¸YbxqKmg~n : 235
 wPšÍb cÖwµqv: Kgb ev AvbKgb me Drcv`K wb‡Z
n‡e| †Kvb Drcv`‡Ki me©vwaK msL¨vwU wb‡Z n‡e|
†hgb- msL¨v¸‡jvi †gŠwjK Drcv`Kmg~‡ni gv‡S 2
Av‡Q mev©waK 3 evi, 3 Av‡Q 1 evi, 5 Av‡Q 1 evi|
Kv‡RB 2 wZbevi (2  2  2), 3 GKevi (3), 5
GKevi (5) wb‡q avivevwnK ¸Y Ki‡j cÖvß ¸YdjB
n‡e j.mv.¸|
∴ wb‡Y©q j. mv. ¸ = 22235 = 120|
(M) mswÿß c×wZ‡Z j.mv.¸ wbY©q: GB c×wZ KwVb I
mnR mKj msL¨vi †ÿ‡ÎB †ek Kvh©Kix GKwU
c×wZ| `ye©j‡`i GB c×wZwU e¨envi Kiv DwPZ|
03. 12, 24 I 30 Gi j.mv.¸ wbY©q Kiæb|
2 12, 24, 30
2 6, 12, 15
3 3, 6, 15
1, 2, 5
∴ wb‡Y©q j.mv.¸ = 22325 = 120
(N) †KŠkjwfwËK j.mv.¸ wbY©qt G c×wZwU Zv‡`i Rb¨
hviv bvgZvi †ÿ‡Î †ek `ÿ, Zv‡`i Rb¨ A‡bK †ewk
Kvh©Kix| j.mv.¸ QvovI KvR I mgq, bj-‡PŠev”Pv
Aa¨vq¸‡jv‡Z GB †KŠkj g¨vwR‡Ki gZ KvR K‡i|
04. 2, 4, 6 I 12 Gi j.mv.¸ wbY©q Kiæb|
GLv‡b me‡P‡q eo Drcv`KwU n‡jv 12| Gevi
†`Lyb 12 †K evKx †QvU Drcv`K 2, 4, 6 Øviv fvM
Kiv hvq wKbv? n¨vu, fvM Kiv hvq| Zvn‡j 2, 4, 6 I
12 Gi j.mv.¸ n‡jv 12| g‡b ivLyb, eo Drcv`KwU‡K
evKx meKqwU Øviv fvM Kiv †M‡j †mwUB n‡e j.mv.¸|
05. 4, 6, 12 I 30 Gi j.mv.¸ wbY©q Kiæb|
me‡P‡q eo Drcv`K 30 †K ïay 6 Øviv fvM Kiv
hvq, wKš‘ j.mv.¸ n‡Z n‡j meKqwU ØvivB fvM †h‡Z
nq| Gevi 30 †K 2 Øviv ¸Y Kiæb, 302 = 60,
GLb 60†K 4, 6, 12 Øviv fvM Kiv hvq|
∴ wb‡Y©q j.mv.¸ n‡e 60|
06. 3, 6, 12, 24 I 30 Gi j.mv.¸ wbY©q Kiæb|
me‡P‡q eo Drcv`K 30†K ïay 3 I 6 Øviv fvM
Kiv hvq, wKš‘ j.mv.¸ n‡Z n‡j me KqwU ØvivB fvM
†h‡Z nq|
 Gevi 30 †K 2 Øviv ¸Y Kiæb, 302 = 60, 60
†K 3, 6, 12 Øviv fvM †M‡jI 24 Øviv fvM hvq bv,
ZvB 60 msL¨v¸‡jvi j.mv.¸ n‡e bv|
 Gevi 30 †K 3 Øviv ¸Y Kiæb, 303 = 90,
G‡ÿ‡ÎI GKB K_v, 90 †K 24 Øviv fvM hvq bv,
ZvB 90 msL¨v¸‡jvi j.mv.¸ n‡e bv|
 Gevi 30 †K 4 Øviv ¸Y Kiæb, 304 = 120, GB
120 †K GLb me KqwU msL¨v Øviv fvM Kiv hvq|
∴ 3, 6, 12, 24, 30 Gi j.mv.¸ n‡jv 120|
Tips: eo Drcv`KwU‡K ZZÿY ch©šÍ ¸Y K‡i
G‡Mv‡Z _vKzb, hZÿY ch©šÍ evKx meKqwU msL¨v
Øviv fvM bv hvq| gy‡L gy‡L K‡qKw`b P”©v Ki‡j
welqwU cvwbi gZ mnR jvM‡e|
07. †Kvb ÿz`ªZg msL¨v‡K 4, 5 I 6 Øviv fvM Ki‡j
cÖwZ‡ÿ‡Î wbt‡k‡l wefvR¨ nq?
2 4, 5, 6
2, 5, 3
∴ wb‡Y©q j.mv.¸ = 2 × 2 × 5 × 3 = 60
 gy‡L gy‡L: 6 †K 2 Øviv fvM Kiv †M‡jI 4 I 5 Øviv
fvM Kiv hvq bv, ZvB 6 †K ZZÿY ch©šÍ ¸Y Ki‡Z
_vKzb hZÿY bv ch©šÍ 6 Gi ¸Ydj‡K 4 I 5 Dfq
mgvavb
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3. 2  Math Tutor
Øviv fvM Kiv hvq: 6 × 2 = 12, 6 × 3 = 18 ...
6 × 10 = 60 (j.mv.¸)
 cÖ‡kœ Ôÿz`ªZg msL¨vÕ ej‡Z wK eySv‡bv n‡q‡Q?
DËit 4, 5, 6 Gi me‡P‡q †QvU ev jwNô mvaviY
¸wYZK n‡”Q 60, hv‡K 4, 5, 6 Øviv fvM Ki‡j
wbt‡k‡l wefvR¨ nq| GB 60 e¨ZxZ Av‡iv mvaviY/
mv`„k¨ ¸wYZK Av‡Q †h¸‡jv 4, 5, 6 Øviv wbt‡k‡l
wefvR¨ nq| †hgb- 120, 180, 240 BZ¨vw`| G‡`i
gv‡S 60 n‡”Q me‡P‡q ÿz`ªZg mvaviY ¸wYZK| Avi
ÿz`ªZg msL¨v ej‡Z g~jZ GB 60 †KB eywS‡q‡Q,
hv‡K Avgiv ewj j.mv.¸| wet`ªt we¯ÍvwiZ Rvb‡Z Ô¸wYZK,
mvaviY ¸wYZK I jwNô mvaviY ¸wYZK wK?Õ co–b
08. 5, 7 I 9 Gi j.mv.¸ KZ?
5, 7, 9 Gi j.mv.¸ = 579 = 315|
 g‡b ivLyb: hw` GKvwaK msL¨vi meKqwUi g‡a¨
AšÍZ `ywU‡ZI †Kvb mvaviY Drcv`K bv _v‡K,
Zvn‡j H msL¨v¸‡jv‡K mivmwi ¸Y Ki‡jB j.mv.¸
cvIqv hvq|
09. †Kvb ÿy`ªZg msL¨v‡K 3, 4 I 5 Øviv fvM Ki‡j
wbt‡kl wefvR¨ ? K‡›Uªvjvi †Rbv‡ij wW‡dÝ dvBbvÝ-Gi Aaxb
AwWUi : 14
160 90
120 60 DËi: N
3, 4 I 5 Gi j.mv.¸ = 345 = 60 |
myZivs, wb‡Y©q ÿz`ªZg msL¨v = 60|
10. K GKwU †gŠwjK msL¨v Ges K, L Øviv wefvR¨ bq|
K Ges L Gi j.mv.¸ KZ? mgevq Awa`߇ii wØZxq †kÖwYi
†M‡R‡UW Awdmvi : 97
1 1K
KL 1L DËi: M
†h‡nZz K, L Øviv wefvR¨ bq, †m‡nZz K I L
mivmwi ¸Y Ki‡jB j.mv.¸ cvIqv hv‡e|
 K I L Gi j.mv.¸ = KL = KL|
11. 5, 6, 10 I 15 Gi j.mv.¸ KZ? cÖv_wgK we`¨vjq
mnKvix wkÿK : 90; cÖv_wgK we`¨vjq mnKvix wkÿK : 89
60 30
50 90 DËi: L
2 5, 6, 10, 15
3 5, 3, 5, 15
5 5, 1, 5, 5
1, 1, 1, 1
∴ wb‡Y©q j.mv.¸ = 235 = 30
 wefvR¨Zvi bxwZ cÖ‡qv‡M: 5, 6, 10 I 15 msL¨v-
¸‡jvi gv‡S 6 I 15 msL¨v `ywU‡Z Drcv`K 3 Av‡Q|
, I Ackb¸‡jv 3 Øviv wefvR¨, G‡`i gv‡S
ÿz`ªZg msL¨v n‡”Q 30|
12. †Kvb ÿz`ªZg c~Y©eM© msL¨v 9, 15 Ges 25 Øviv
wefvR¨? Z_¨gš¿Yvj‡qi mnKvix Awdmvi-2013
75 225
1125 900 DËi: L
3 9, 15, 25
5 3, 5, 25
3, 1, 5
∴ j.mv.¸ = 3355 = 225, hv GKwU ÿz`ªZg
c~Y©eM© msL¨v|
13. 2002 msL¨vwU †Kvb msL¨v¸‡”Qi j.mv.¸ bq? 24Zg
wewmGm (evwZjK…Z)
13, 77, 91, 143 7, 22, 26, 91
26, 77, 143, 154 2, 7, 11, 13 DËi: K
†h msL¨v¸‡”Qi j.mv.¸ 2002 n‡e bv, †mwUB DËi
n‡e|
13, 77, 91, 143 Gi j.mv.¸ = 1001
7, 22, 26, 91 Gi j.mv.¸ = 2002
26, 77, 143, 154 Gi j.mv.¸ = 2002
2, 7, 11, 13 Gi j.mv.¸ = 2002
14. b~¨bZg KZwU Kgjv‡K 4, 6, 10 ev 18 Rb wkïi
g‡a¨ fvM K‡i †`qv hvq? c~evjx e¨vsK wmwbqi Awdmvi
(K¨vk)-2012
16 60
180 240 DËi: M
Ôb~¨bZg KZwU Kgjv 4, 6, 10 ev 18 Rb wkïi
g‡a¨ fvM K‡i †`qvÕ Avi Ô†Kvb ÿz`ªZg msL¨v‡K 4,
6, 10, 18 w`‡q wbt‡kl wefvR¨ nIqvÕ GKB K_v|
ZvB j.mv.¸ Ki‡Z n‡e|
15. GKwU ¯‹z‡j c¨v‡iW Kivi mgq Qv·`i 10, 12 ev 16
mvwi‡Z mvRv‡bv nq| H b~¨bZg KZRb QvÎ Av‡Q?
mve-‡iwR÷vi wb‡qvM cixÿv-2016
120 180
220 240 DËi: N
b~¨bZg KZRb QvÎ hv‡`i‡K 10, 12 ev 16
mvwi‡Z mvRv‡bv hvq Avi †Kvb ÿz`ªZg msL¨v‡K 10,
12, 16 w`‡q wbt‡kl wefvR¨ nIqvÕ GKB K_v, ZvB
j.mv.¸ Ki‡Z n‡e|
16. me©‡gvU KZ msL¨K MvQ n‡j GKwU evMvb 7, 14,
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4. Math Tutor  3
21, 35 I 42 mvwi‡Z MvQ jvMv‡j GKwUI Kg ev
†ewk n‡e bv? Avbmvi I wfwWwc Awa`߇ii mv‡K©j
A¨vWRyU¨v›U : 2010
210 220
230 260 DËi: K
me©‡gvU MvQ‡K 7, 14, 21, 35 I 42 mvwi‡Z
jvMv‡j GKwUI Kg ev †ewk bv nIqv gv‡b n‡”Q
wbt‡kl wefvR¨ nIqv, Avi msL¨v¸‡jv †_‡K cÖvß
j.mv.¸ n‡e me©‡gvU Mv‡Qi msL¨v|
†R‡b wbb - 24
cÖ_‡g cÖkœwU fv‡jvfv‡e co–b|
17. 3wU N›Uv GK‡Î †e‡R h_vµ‡g 5, 6 I 10 wgwbU AšÍi evR‡Z jvMj| KZÿY ci N›Uv¸‡jv cybivq GK‡Î evR‡e?
20 25 30 60 DËi: M
5, 6 I 10 Gi j.mv.¸-B n‡e DËi|
2 5, 6, 10
5 5, 3, 5
1, 3, 1
∴ 5, 6 I 10 Gi j.mv.¸ = 253 = 30|
myZivs, N›Uv 3wU c~bivq 30 wgwbU ci GK‡Î evR‡e|
 ÔGKvwaK N›Uv GK‡Î evRvi ci GKwU wbw`©ó mgq AšÍi N›Uv¸‡jv evR‡Z _vK‡j KZÿY ci N›Uv¸‡jv cybivq
GK‡Î evR‡e?Õ Giƒc cÖ‡kœi †ÿ‡Î Avgiv †Kb j.mv.¸ Kwi?
DËi: Dc‡ii cÖ‡kœ Avgiv 5, 6 I 10 Gi j.mv.¸ †c‡qwQ 30| MvwYwZK wbqgvbymv‡i, N›Uv¸‡jv 30wgwbU ci cybivq
GK‡Î evR‡e| welqwU wPÎwfwËK wPšÍv Kiv hvK| wP‡Î †`Lyb, cÖ_‡g ÔïiæÕi RvqMvq 3wU N›Uv GK‡Î †e‡RwQj,
Zvici 1g N›UvwU 5 wgwbU AšÍi, 2qwU 6 wgwbU AšÍi Ges 3qwU 10 wgwbU AšÍi evR‡Z _v‡K Ges 30 wgwbU ci
Zviv Avevi GKwU RvqMvq wM‡q wgwjZ nq| gRvi welq n‡”Q 30 wgwb‡Ui Av‡M Avi †Kv_vI Zviv wgwjZ nqwb!
Avgiv welqwU Ab¨fv‡eI †`L‡Z cvwi| Pjyb 5, 6 I 10 Gi ¸wYZK¸‡jv †ei Kiv hvK|
5 Gi ¸wYZKmg~n : 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95...
6 Gi ¸wYZKmg~n : 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96...
10 Gi ¸wYZKmg~n : 10, 20, 30, 40, 50, 60, 70, 80, 90, 100...
Dc‡ii 5, 6 I 10 Gi ¸wYZKmg~‡ni gv‡S mvaviY ¸wYZKmg~n n‡”Q 30, 60, 90...| G‡`i gv‡S me‡P‡q †QvU
mvaviY ¸wYZK ev j.mv.¸ n‡”Q 30| MwY‡Zi ïiæ‡Z N›Uv¸‡jv 1gevi GK‡Î †e‡RwQj (Nwoi wPÎ †`Lyb), 30
wgwb‡Ui gv_vq N›Uv 3wU cybivq 2qev‡ii gZ GK‡Î evR‡e, 60 wgwb‡Ui gv_vq N›Uv¸‡jv 3qev‡ii gZ GK‡Î
evR‡e Ges 90 wgwb‡Ui gv_vq N›Uv¸‡jv 4_©ev‡ii gZ GK‡Î evR‡e|
 cÖ‡kœ ejv n‡q‡Q, Ô3wU N›Uv GK‡Î †e‡R h_vµ‡g 5, 6 I 10 wgwbU AšÍi evR‡Z jvMj| KZÿY ci N›Uv¸‡jv
cyYivq GK‡Î evR‡e?Õ Gi Øviv eySv‡bv n‡q‡Q 2qev‡ii gZ N›Uv 3wU KZÿY ci cybivq GK‡Î evR‡e| DËi n‡e,
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5. 2  Math Tutor
5, 6 I 10 Gi j.mv.¸|
 cÖ‡kœ hw` ejv nZ Ô3wU N›Uv GK‡Î †e‡R h_vµ‡g 5, 6 I 10 wgwbU AšÍi evR‡Z jvMj| 30 wgwb‡U, Zviv
KZevi GKmv‡_ evR‡e?Õ Zvn‡j DËi n‡e 2 evi| ïiæ‡Z GK‡Î ev‡R 1 evi Ges 30 wgwb‡Ui gv_vq GK‡Î
evR‡e AviI 1 evi A_©vr, †gvU 2 evi|
 cÖ‡kœ hw` ejv nZ Ô3wU N›Uv GK‡Î †e‡R h_vµ‡g 5, 6 I 10 wgwbU AšÍi evR‡Z jvMj| 60 wgwb‡U, Zviv
KZevi GKmv‡_ evR‡e?Õ Zvn‡j DËi n‡e 3 evi| MvwYwZKfv‡e mgvavb Ki‡eb †hfv‡e-
5, 6 I 10 Gi j.mv.¸ 30 w`‡q 60 †K fvM w`b- 60 ÷ 30 = 2| cÖvß fvMdj 2 Gi mv‡_ cÖ_gevi GKÎ _vKvi
1 †hvM Kiæb- 2 + 1 = 3 evi| A_©vr, (60 ÷ 30) + 1 = 2 + 1 = 3 evi|
Gfv‡e 90 wgwb‡U KZevi GK‡Î evR‡e Rvb‡Z PvB‡j DËi n‡e, (90 ÷ 30) + 1 = 3 + 1 = 4 evi|
 cÖ‡kœ hw` ejv nZ Ô3wU N›Uv GK‡Î †e‡R h_vµ‡g 5, 6 I 10 wgwbU AšÍi evR‡Z jvMj| hw` Zviv GK‡Î `ycyi
12 Uvq †e‡R _v‡K, Zvn‡j cybivq Avevi KLb GK‡Î evR‡e?
DËi: `ycyi 12 Uvq GK‡Î †e‡R 30 wgwbU ci Avevi GK‡Î evR‡e| A_©vr, 12pm + 30minutes = 12.30 pm
G N›Uv 3wU cybivq Avevi GK‡Î evR‡e|
 GLv‡b 3wU wbqg Av‡jvPbv Kiv n‡q‡Q, Gi evwni †_‡K cÖkœ Avm‡e bv|
18. cuvPwU N›Uv GK‡Î †e‡R h_vµ‡g 3, 5, 7, 8 I 10
†m‡KÛ AšÍi AšÍi evR‡Z jvMj| KZÿY c‡i N›Uv
¸‡jv cybivq GK‡Î evR‡e? _vbv mn. wkÿv wdmvi : 95
10 wgwbU 90 †m‡KÛ
14 wgwbU 240 †m‡KÛ DËi: M
2 3, 5, 7, 8, 10
5 3, 5, 7, 4, 5
3. 1, 7, 4, 1
∴ 3, 5, 7, 8 I 10 Gi j.mv.¸ = 25374
= 840
myZivs, N›Uv¸‡jv cybivq GK‡Î evR‡e = 840 †m‡KÛ ci
=
60
840
wgwbU ci
= 14 wgwbU ci|
 †m‡KÛ‡K 60 w`‡q fvM K‡i wgwb‡U iƒcvšÍi Kiv nj
19. KZ¸‡jv N›Uv GKmv‡_ evRvi 10 †mt, 15 †mt, 20
†mt Ges 25 †mt ci ci evR‡Z jvMj| Dnviv Avevi
KZÿY ci GK‡Î evR‡e? _vbvwkÿvAwdmvit96,evsjv‡`kcjøx
we`y¨Zvqb†ev‡W©imnKvixcwiPvjK(cÖkvmb)-16
1wgwbU 20 †m‡KÛ 1 wgwbU 30 †m‡KÛ
3 wgwbU 5 wgwbU DËi: N
10, 15, 20 I 25 Gi j.mv.¸ = 300
 N›Uv¸‡jv Avevi GK‡Î evR‡e = 300 †m‡KÛ ci
=
60
300
= 5 wgwbU ci|
20. 5wU N›Uv GK‡Î †e‡R h_vµ‡g 5, 10, 15, 20 I 25
†m‡KÛ AšÍi Avevi evR‡Z jvMj, KZÿY ci N›Uv
¸‡jv Avevi GK‡Î evR‡e? 12Zg†emiKvwiwkÿKwbeÜb
(¯‹zj/mgch©vq)2015
5 wgwbU 6 wgwbU
10 wgwbU 5 N›Uv DËi: K
21. 2wU Nwo h_vµ‡g 10 I 25 wgwbU AšÍi ev‡R| GKevi
GK‡Î evRvi ci Avevi KLb Nwo `yÕwU GK‡Î evR‡e?
K‡›Uªvjvi†Rbv‡ijwW‡dÝdvBbvÝ-GiKvh©vj‡qAaxbAwWUi-2014
20 wgwbU ci 30 wgwbU ci
50 wgwbU ci 100 wgwbU ci DËi: M
22. eykiv, Glv I wcÖZzB 5 wgwbU, 10 wgwbU, 15 wgwbU AšÍi
AšÍi GKwU K‡i PK‡jU Lvq| KZÿY ci Zviv GK‡Î
PK‡jU Lvq? 7gcÖfvlKwbeÜbIcÖZ¨qbcixÿv-2011
25 wgwbU 50 wgwbU
40 wgwbU 30 wgwbU DËi: N
5, 10 I 15 Gi j.mv.¸ = 30| myZivs, eykiv,
Glv I wcÖZzB 30 wgwbU ci GK‡Î PK‡jU Lv‡e|
23. wZb Rb †g‡q GKwU e„ËvKvi gv‡Vi Pvwiw`K eivei
GKwU wbw`©ó ¯’vb †_‡K †`Šov‡bv ïiæ Kij Ges
cÖ‡Z¨‡K GKwU cvK h_vµ‡g 24 †m‡KÛ, 36 †m‡KÛ
Ges 48 †m‡K‡Û c~Y© K‡i| KZ mgq ci Zviv GKB
¯’v‡b wgwjZ n‡e? †Kvqvw›U‡UwUf A¨vcwUwUDU, Wt Avi,
Gm AvMviIqvj; (evsjv) wiwcÖ›U-2020
2 wgwbU 20 †m‡KÛ 2 wgwbU 24 †m‡KÛ
3 wgwbU 36 †m‡KÛ 4 wgwbU 12 †m‡KÛ D: L
24, 36 I 48 Gi j.mv.¸ = 144
wZbRb †g‡q GKB ¯’v‡b wgwjZ n‡e = 144 †m‡KÛ ci
mgvavb
N
M
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mgvavb
N
M
L
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N
M
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K
N
M
L
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mgvavb
N
M
L
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mgvavb
N
M
L
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6. 2  Math Tutor
=
60
144
= 2
60
24
= 2
5
2
wgwbU ci
= 2wgwbU
5
2
60 †m‡KÛ = 2wgwbU 24†m‡KÛ ci|
24. A, B Ges C GKB mg‡q, GKB w`‡K GKwU e„ËvKvi
†÷wWqv‡gi Pvicv‡k †`Šov‡Z ïiæ K‡i| A GKwU
cvK 252 †m‡K‡Û, B GKwU cvK 308 †m‡K‡Û Ges
C GKwU cvK 198 †m‡K‡Û c~Y© K‡i| cÖ‡Z¨‡K GKB
RvqMv †_‡K †`Šo ïiæ K‡i| cÖ‡Z¨‡K cybivq KLb
ïiæi ¯’v‡b GKmv‡_ n‡e? †Kvqvw›U‡UwUf A¨vcwUwUDU, Wt
Avi, Gm AvMviIqvj; (evsjv) wiwcÖ›U-2020
26 wgwbU 18 †m‡KÛ 42 wgwbU 36 †m‡KÛ
45 wgwbU 46 wgwbU 12 †m‡KÛ D: N
25. mxgv, wgbv Ges wigv GKwU e„ËvKvi †÷wWqv‡gi Pvicv‡k
†`Šov‡Z ïiæ Kij| Zviv Zv‡`i GKcvK h_vµ‡g 54
†m‡KÐ, 42 †m‡KÐ Ges 63 †m‡KÐ mg‡q c~Y© Kij| KZ
mgq ci Zviv ïiæi ¯’v‡b Avevi GKmv‡_ n‡e? †Kvqvw›U‡UwUf
A¨vcwUwUDU,WtAvi,GmAvMviIqvj;(evsjv)wiwcÖ›U-2020
54, 42 I 63 Gi j.mv.¸ = 378|
 mxgv, wgbv Ges wigv ïiæi mv‡_ GKmv‡_ n‡e =
378 †m‡KÛ ci | DËi: 378 †m‡KÛ ci
26. 5wU NÈv cÖ_‡g GK‡Î †e‡R c‡i h_vµ‡g 6, 12,
24, 30 I 40 †m‡KÐ AšÍi AšÍi evR‡Z jvMj|
KZÿY c‡i N›Uv¸‡jv cybivq GK‡Î evR‡e? wb¤œ gva¨wgK
MwYZ,lô†kÖwY(1996wkÿvel©),cÖkœgvjv1.4Gi9bscÖkœ
6, 12, 24, 30 I 40 Gi j.mv.¸ = 120
 5wU N›Uv cybivq GK‡Î evR‡e 120 wgwbU ci
=
60
120
= 2 wgwbU ci| DËi: 2 wgwbU
27. †Kv‡bv evm÷¨vÛ †_‡K 4wU evm GKwU wbw`©ó mgq ci
h_vµ‡g 10 wK.wg., 20 wK.wg., 24 wK.wg. I 32 wK.wg.
c_ AwZµg K‡i| Kgc‡ÿ KZ`~i c_ AwZµg Kivi
ci evm PviwU GK‡Î wgwjZ n‡e? wb¤œ gva¨wgKMwYZ,lô†kÖwY
(2013wkÿvel©),Abykxjbx1.3Gi13bscÖkœ
10, 20, 24 I 32 Gi j.mv.¸ = 480
 4wU evm 480 wK.wg. c_ AwZµg Kivi ci
Avevi GK‡Î wgwjZ n‡e| DËi: 480 wK.wg.
28. 4 Rb †jvK mv‡q`vev` †_‡K PµKvi iv¯Ívq mKvj 6Uvq
GKB w`‡K hvÎv ïiæ K‡ib| Zvuiv cÖwZ N›Uvq h_vµ‡g
10, 20, 24 I 32 wK‡jvwgUvi c_ AwZµg K‡ib|
Kgc‡ÿ KZ `~i c_ AwZµg Kivi c‡i Zvuiv Avevi
GK‡Î wgwjZ n‡eb? wb¤œ gva¨wgKMwYZ,lô†kÖwY(1996wkÿvel©),
cÖkœgvjv1.4Gi10bscÖkœ
10, 20, 24 I 32 Gi j.mv.¸ = 480
 4 Rb †jvK Avevi GK‡Î wgwjZ n‡eb 480
wK.wg. ci| DËi: 480 wK.wg.
29. GKwU •e`y¨wZK hš¿ cÖwZ 60 †m‡KÛ AšÍi AvIqvR
K‡i| Aci GKwU hš¿ 62 †m‡KÐ AšÍi AvIqvR
K‡i| hw` Zviv GKmv‡_ mKvj 10Uvq AvIqvR K‡i
_v‡K, Zvn‡j Zviv cybivq Avevi KLb GKmv‡_
AvIqvR Ki‡e? †Kvqvw›U‡UwUfA¨vcwUwUDU,WtAvi,GmAvMviIqvj;
(evsjv)wiwcÖ›U-2020
10.30 a.m. 10.31 a.m.
10.59 a.m. 11 a.m. DËi: L
2 60, 62
30, 31
60 I 62 Gi j.mv.¸ = 23031 = 1860
∴ •e`y¨wZK hš¿ `ywU cybivq Avevi GKmv‡_ AvIqvR
Ki‡e 1860 †m‡KÐ ci =
60
1860
= 31 wgwbU ci|
myZivs, •e`y¨wZK hš¿ `ywU mKvj 10 Uvq GK‡Î
AvIqvR Kivi ci 31 wgwbU ci Avevi AvIqvR
Ki‡e 10.31 a.m. G|
30. QqwU N›Uv GKmv‡_ evRv ïiæ K‡i Ges h_vµ‡g 2,
4, 6, 8, 10 Ges 12 †m‡KÛ AšÍi ev‡R| 30 wgwb‡U,
Zviv KZevi GKmv‡_ evR‡e? †Kvqvw›U‡UwUfA¨vcwUwUDU,WtAvi,
GmAvMviIqvj;(evsjv)wiwcÖ›U-2020
4 10
15 16 DËi: N
2, 4, 6, 8, 10 I 12 Gi j.mv.¸ = 120
∴ QqwU N›Uv GKmv‡_ evR‡e 120 †m‡KÛ ci
=
60
120
= 2 wgwbU ci|
cÖ_gevi 1 mv‡_ evRvi ci,
2 wgwb‡U GK‡Î ev‡R AviI 1 evi
∴ 1 Ó Ó Ó Ó
2
1
Ó
mgvavb
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mgvavb
N
M
L
K
mgvavb
mgvavb
mgvavb
mgvavb
N
M
L
K

7. Math Tutor  3
∴ 30Ó Ó Ó Ó
2
1
30 = 15 evi|
myZivs, cÖ_‡g 1 I c‡i 15 evi †gvU 16 evi evR‡e|
 kU©KvU: 1
j.mv.¸
mgq
gvU
†

= 1
2
30
 = 15 + 1 = 16 evi|
31. wZbwU N›Uv GK‡Î evRvi ci Zviv 2 N›Uv, 3 NÈv I
4 N›Uv cici evR‡Z _vKj| 1 w`‡b Zviv KZevi
GK‡Î evR‡e? cÖv_wgKwe`¨vjqmnKvixwkÿK(PZz_© avc)2019;11Zg
†emKvwiwkÿKwbeÜb(¯‹zj/mgch©vq)-2014
12 evi 6 evi
4 evi 3 evi DËi: N
2, 3 I 4 Gi j.mv.¸ = 12|
∴ wZbwU N›Uv GK‡Î evR‡e 12 N›Uv ci|
myZivs, 1 w`‡b ev 24 NÈvq †gvU evR‡e = 1
12
24

3
1
2 

 evi|
32. GKwU †Nvovi Mvwoi mvg‡bi PvKvi cwiwa 3 wgUvi,
wcQ‡bi PvKvi cwiwa 4 wgUvi| MvwowU KZ c_ †M‡j
mvg‡bi PvKv wcQ‡bi PvKvi †P‡q 100 evi †ewk
Nyi‡e? WvKAwa`߇iDc‡Rjv†cv÷gv÷vi:10;gnvwnmvewbixÿKI
wbqš¿‡Ki Kvh©vj‡qmnKvixcwimsL¨vbKg©KZ©v(2q†kÖwY):98,beg-`kg†kÖwY
MwYZ,(1983ms¯‹iY)Abykxjbx1.1Gi44bscÖkœ
1 wK.wg. 1.2 wK.wg.
1.6 wK.wg. 1.8 wK.wg. DËi: L
3 I 4 Gi j.mv.¸ = 34 = 12
(PvKv `ywUi cwiwai j.mv.¸ n‡”Q PvKv `ywU‡K GKwU mgvb
`~i‡Z¡i c_ Pj‡Z 12 wgUvi AwZµg Ki‡Z n‡e)
12 wgUvi c_ Pj‡Z m¤§yL PvKv‡K Nyi‡Z n‡e
3
12
= 4 evi
12 Ó Ó Ó wcQ‡bi Ó Ó Ó
4
12
= 3 evi
(mvg‡bi PvKv A‡cÿv wcQ‡bi PvKv 1evi †ewk Ny‡i 12
wgUvi c_ Pj‡Z, Zvn‡j 100 evi †ewk Nyi‡j KZUzKz c_
Pj‡e?)
mvg‡bi PvKv wcQ‡bi PvKv A‡cÿv 1evi †ewk Ny‡i 12 wgUv‡i
Ó Ó Ó Ó Ó 100 Ó Ó Ó 12100Ó
= 1200 wgUvi
=
1000
1200
wK.wg.
= 1.2 wK.wg.|
 kU©KvUt mvg‡bi PvKv I wcQ‡bi PvKvi j.mv.¸
Gi mv‡_ Ô100 evi †ewk Nyi‡QÕ †mwU ¸Y Ki‡Z
n‡e| Zvici wgUvi‡K wK‡jvwgUvi Ki‡Z 1000
Øviv fvM Ki‡Z n‡e| A_©vr, 34100 = 1200
 1000 = 1.2 wKwg|
33. GKwU †Nvovi Mvwoi mvg‡bi PvKvi cwiwa 4 wgUvi,
†cQ‡bi PvKvi cwiwa 5 wgUvi| MvwowU KZ c_ †M‡j
mvg‡bi PvKv †cQ‡bi PvKvi †P‡q 200 evi †ewk
Nyi‡e? cjøx Dbœqb I mgevq gš¿Yvj‡qi mnKvix cÖ‡KŠkjx
(wmwfj) : 17; ciivóª gš¿Yvj‡qi Aax‡b cÖkvmwbK Kg©KZ©v : 01
1.2 wK.wg. 2.5 wK.wg.
4 wK.wg. 6 wK.wg. DËi: M
4 I 5 Gi j.mv.¸ = 45 = 20
20 wgUvi c_ Pj‡Z m¤§yL PvKv‡K †Nvi‡Z n‡e
4
20
= 5 evi
20 Ó Ó Ó wcQ‡bi Ó Ó Ó
5
20
= 4 evi
mvg‡biPvKvwcQ‡biPvKvA‡cÿv1evi†ewk†Nv‡i20wgUv‡i
Ó Ó Ó Ó Ó 200 Ó Ó Ó 20200Ó
= 4000 wgUvi
=
1000
4000
wK.wg.
= 4 wK.wg.|
 kU©KvUt 45200 = 4000  1000 = 4 wK.wg.
34. GKwU †Nvovi Mvwoi mvg‡bi PvKvi cwiwa 2 wgUvi
Ges †cQ‡bi PvKvi cwiwa 3 wgUvi| Kgc‡ÿ KZ
`~iZ¡ AwZµg Ki‡j mvg‡bi PvKv †cQ‡bi PvKv
A‡cÿv 10 evi †ewk Nyi‡e? evsjv‡`kcjøxDbœqb†ev‡W©iDc‡Rjv
cjøxDbœqbKg©KZ©v:13; cÖv_wgKwe`¨vjqmnKvixwkÿK(XvKvwefvM):02
60 wgUvi 20 wgUvi
25 wgUvi 40 wgUvi DËi: K
2310 = 60 wgUvi|
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
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c‡i 15 evi
ïiæ‡Z 1evi +

8. 2  Math Tutor
 04.03 fvM‡kl _vK‡e
†R‡b wbb- 25
 j.mv.¸ mgm¨v mgvav‡b Avcbv‡K †h KvRwU mevi Av‡M Ki‡Z n‡e
j.mv.¸ mgm¨v mgvav‡b ïiæ‡Z cÖ‡kœi kZ© wb‡q wPšÍv Kivi `iKvi †bB| cÖ_‡g cÖ‡kœ cÖ`Ë Drcv`K/fvRK ¸‡jvi
j.mv.¸ wbY©q Kiæb| Zvici cÖ‡kœi kZ©vbymv‡i KvR Kiæb| GLv‡b Drcv`K/fvRK ej‡Z Zv‡`i‡K eySv‡bv n‡”Q-
hv‡`i Øviv fvM Kivi K_v ejv nq| †hgb- (1) me‡P‡q †QvU †Kvb msL¨v‡K 7, 8 I 9 Øviv fvM Ki‡j 5 Aewkó _v‡K?
(2) GKwU c~Y© msL¨v wbY©q Kiæb hv‡K 3, 4, 5 Ges 6 fvM Ki‡j h_vµ‡g 2, 3, 4 Ges 5 Aewkó _v‡K? cÖkœ
`ywUi AvÛvijvBbK…Z msL¨v¸‡jvB n‡”Q cÖ‡kœ cÖ`Ë Drcv`K/fvRKmg~n|
 †Kvb ÿz`ªZg msL¨v‡K 3, 5 I 6 Øviv fvM Ki‡j fvM‡kl n‡e 1? [17Zg wewmGm]
71 41 31 39 DËi: M
ïiæ‡Z 3, 5, 6 Gi j.mv.¸ †ei Kiv hvK|
3 3, 5, 6
1, 5, 2
∴ 3, 5 I 6 Gi jmv¸ = 3 × 5 × 2 = 30|
AZGe, ÿz`ªZg msL¨vwU n‡”Q = 30 + 1 = 31|
 fvM‡kl hZ _vK‡e j.mv.¸Õi mv‡_ ZZ †hvM Ki‡Z n‡e †Kb?
Avgiv c~‡e© †R‡bwQ- GKvwaK fvRK †_‡K cÖvß j.mv.¸ H fvRK¸‡jv Øviv wbt‡k‡l wefvR¨| A_©vr, Avgiv
wbwðZfv‡e ej‡Z cvwi, 3, 5 I 6 fvRK¸‡jv †_‡K cÖvß j.mv.¸ 30, hv fvRK 3/5/6 Øviv wbt‡k‡l wefvR¨|
Zvi gv‡b j.mv.¸ 30 †K Avgiv ÿz`ªZg msL¨v ej‡Z cvwi| wKš‘ 30 †K ÿz`ªZg msL¨v ejv hv‡e bv| KviY cÖ‡kœ ejv
n‡q‡Q- ÿz`ªZg msL¨vwU‡K 3/5/6 Øviv fvM Ki‡j wbt‡k‡l wefvR¨ n‡Z cvi‡e bv, fvM‡kl 1 _vK‡Z n‡e| GRb¨
wK Ki‡Z n‡e? ejwQ- wbt‡k‡l wefvR¨ msL¨vwUi mv‡_ AwZwi³ hv †hvM Ki‡eb ZvB fvM‡kl wn‡m‡e _vK‡e|
A_©vr, AwZwi³ 1 †hvM Ki‡j, fvM‡kl 1 _vK‡e; AwZwi³ 2 †hvM Ki‡j fvM‡kl 2 _vK‡e| Pjyb †`Lv hvK-
hLb 1 †hvM Ki‡eb- 30 + 1 = 31 3 ) 31 ( 10 5) 31 ( 6 6) 31 ( 5
30 30 30
1 1 1
hLb 2 †hvM Ki‡eb- 30 + 2 = 32
3 ) 32 ( 10 5) 32 ( 6 6) 32 ( 5
30 30 30
2 2 2
GLb Avgiv wbwØ©avq ej‡Z cvwi, G ai‡Yi mgm¨vq hZ fvM‡kl Rvb‡Z PvB‡e, ZZ j.mv.¸i mv‡_ †hvM K‡i
w`‡jB n‡e| cÖ`Ë cÖ‡kœ fvM‡kl 1 Av‡Q ejvq, cÖvß j.mv.¸i mv‡_ 1 †hvM Kiv n‡q‡Q|
 kU©KvU : hZ fvM‡kl _vK‡e, ZZ j.mv.¸i mv‡_ †hvM Ki‡Z n‡e|
35. †Kvb ÿy`ªZg msL¨v‡K 4, 5 I 6 Øviv fvM Ki‡j
cÖwZ‡ÿ‡Î 1 Aewkó _v‡K? gva¨wgK mnKvix wkÿK-06
121 169
61 111 DËi: M
cÖ_‡g jmv¸ †ei Kiæb, Zvici D³ jmv¸i mv‡_
1 †hvM Kiæb| 4, 5 I 6 Gi jmv¸ = 60|
∴ wb‡Y©q ÿz`ªZg msL¨v = 60 + 1 = 61|
36. †Kvb& msL¨v‡K 4 I 6 Øviv fvM Ki‡j cÖwZ‡ÿ‡Î
fvM‡kl 2 _v‡K? Dc‡Rjv cwimsL¨vb Kg©KZv©: 2010
8 10
12 14 DËi: N
4 I 6 Gi jmv¸ = 12|
∴ wb‡Y©q ÿz`ªZg msL¨v = 12 + 2 = 14|
37. me‡P‡q †QvU †Kvb msL¨v‡K 7, 8 I 9 Øviv fvM
Ki‡j 5 Aewkó _v‡K? ivóªvqË e¨vsK wmwbqi Awdmvi: 00
499 599
549 509 DËi: N
7, 8 I 9 Gi jmv¸ = 504|
∴ wb‡Y©q ÿz`ªZg msL¨v = 504 + 5 = 509|
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K

9. 2  Math Tutor
 04.04 h_vµ‡g fvM‡kl _vK‡e
†R‡b wbb -26
 avc -G Drcv`K/fvRK¸‡jvi j.mv.¸ †ei Kiæb| (GB KvRwU jmv¸i me A‡¼B Ki‡Z nq)
avc -G cÖwZwU fvRK I fvM‡k‡li cv_©K¨ †ei Kiæb| Gevi ÔcÖvß j.mv.¸Õ †_‡K ÔfvRK I fvM‡k‡li cv_©K¨
we‡qvM Kiæb|
 kU©KvUt wb‡Y©q ÿz`ªZg msL¨v = fvRK¸‡jvi jmv¸ - fvRK I fvM‡k‡li cv_©K¨
†Kb we‡qvM Ki‡Z n‡e, GB KviYwU GLv‡b e¨vL¨v m¤¢e nj bv, KviY †m‡ÿ‡Î j¤^v e¨vL¨v wjL‡Z n‡e| hw` KLbI my‡hvM nq
jvBf K¬v‡m e¨vL¨v Kivi †Póv Kie Bbkvjøvn|
38. †Kvb ÿy`ªZg msL¨v‡K 20, 25, 30, 36 I 48 Øviv
fvM Ki‡j h_vµ‡g 15, 20, 25, 31 I 43 fvM‡kl
_v‡K? KvwiMix wkÿv Awa`߇ii Aax‡b Pxd BÝUªv±i: 03
3425 3478
3595 3565 DËi: M
avc : 20, 25, 30, 36 I 48 GB fvRK¸‡jvi
j.mv.¸ wbY©q Kiæb|
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
∴ 20, 25, 30, 36 I 48 Gi j.mv.¸ = 2 × 2 ×
3 × 5 × 5 × 3 × 4 = 3600
avc : fvRK I fvM‡k‡li cv_©K¨ †ei Kiæb-
20 25 30 36 48
15 20 25 31 43
5 5 5 5 5
∴ wb‡Y©q ÿz`ªZg msL¨v = 3600 - 5 = 3595|
39. GKwU c~Y© msL¨v wbY©q Kiæb hv‡K 3, 4, 5 Ges 6
fvM Ki‡j h_vµ‡g 2, 3, 4 Ges 5 Aewkó _v‡K?
_vbv mnKvix wkÿv Awdmvi: 05
47 49
57 59 DËi: N
3, 4, 5 Ges 6 Gi jmv¸ = 60 Ges cÖwZ‡ÿ‡Î
Aewkó _v‡K 1|
∴ wb‡Y©q c~Y© msL¨v = 60 - 1 = 59|
40. †Kvb jwNó msL¨v‡K 24 I 36 Øviv fvM Ki‡j
h_vµ‡g 14 I 26 Aewkó _vK‡e? hye Dbœqb Awa`߇ii
mnKvix cwiPvjK: 94
48 62
72 84 DËi: L
24 I 36 Gi jmv¸ = 72 Ges cÖwZ‡ÿ‡Î Aewkó
_v‡K 10 |
∴ wb‡Y©q jwNó msL¨v = 72 - 10 = 62|
 04.05 ÿy`ªZg msL¨vi mv‡_ †hvM ev we‡qvM K‡i
†R‡b wbb - 27
 Ô†Kvb ÿz`ªZg msL¨v + = †hvMdjÕ Gi †ÿ‡Î fvRK¸‡jvi j.mv.¸ n‡”Q Ô†hvMdjÕ, †hvMd‡ji mv‡_ e‡·i
†h msL¨vwU †hvM (+) Kiv n‡e †mB msL¨vwU we‡qvM (  ) Ki‡j Ôÿy`ªZg msL¨vÕ wU cvIqv hv‡e| 41, 42,
43 I 44 bs cÖkœ †`Lyb|
 Ô†Kvb ÿz`ªZg msL¨v  = we‡qvMdjÕ Gi †ÿ‡Î fvRK¸‡jvi j.mv.¸ n‡”Q Ôwe‡qvMdjÕ, we‡qvMdj †_‡K
e‡·i †h msL¨vwU we‡qvM (  ) Kiv n‡e †mB msL¨vwU †hvM (+) Ki‡j Ôÿy`ªZg msL¨vÕ wU cvIqv hv‡e| 45
I 46 bs cÖkœ †`Lyb|
 kU©KvU: †hvM _vK‡j we‡qvM Ges we‡qvM _vK‡j †hvM Ki‡Z n‡e|
41. †Kvb ÿy`ªZg msL¨vi mv‡_ 2 †hvM Ki‡j †hvMdj 3,
6, 9, 12 Ges 15 Øviv wbt‡k‡l wefvR¨ n‡e? `ybx©wZ
`gb ey¨‡iv cwi`k©K: 04
178 358
368 718 DËi: K
3 3, 6, 9, 12, 15
2 1, 2, 3, 4, 5
1, 1, 3, 2, 5
j.mv.¸ = 32325 = 180
 ÿz`ªZg msL¨v = †hvMdj 2 = 180  2 = 178|
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10. 2  Math Tutor
42. †Kvb ÿz`ªZg msL¨vi mv‡_ 1 †hvM Ki‡j †hvMdj 3,
6, 9, 12, 15 Øviv wbt‡k‡l wefvR¨ n‡e? cvewjK r
mvwf©m Kwgkb KZ…©K wbav©wiZ c`: 01
179 361
359 721 DËi: K
3, 6, 9, 12, 15 Gi jmv¸ = 180|
∴ ÿz`ªZg msL¨v = 180 - 1 = 179|
43. †Kvb ÿz`ªZg msL¨vi m‡½ 5 †hvM Ki‡j †hvMdj 16,
24 I 32 w`‡q wbt‡k‡l wefvR¨ n‡e? wb¤œ gva¨wgK
MwYZ: lô †kÖwY (wkÿvel©-13) : D`vniY 9
96 101
91 19 DËi: M
j.mv.¸ = 96| ∴ ÿz`ªZg msL¨v = 96 - 5=91|
44. †Kvb ÿz`ªZg msL¨vi mv‡_ 3 †hvM Ki‡j †hvMdj 21,
25, 27 I 35 Øviv wefvR¨ nq? beg-`kg †kÖwY MwYZ,
(1983 ms¯‹iY) Abykxjbx 1.1 Gi 26 bs cÖkœ
4725 4728
4722 †Kv‡bvwUB bq DËi: M
j.mv.¸ = 4725| ∴ ÿz`ªZg msL¨v = 4725 -
3 = 4722|
45. †Kvb ÿz`ªZg msL¨v n‡Z 1 we‡qvM Ki‡j we‡qvMdj
9, 12 I 15 Øviv wbt‡k‡l wefvR¨ n‡e? cvewjK mvwf©m
Kwgk‡b mnKvix cwiPvjK: 04
121 181
241 361 DËi: L
3 9, 12, 15
3, 4, 5
j.mv.¸ = 3345 = 180
 ÿz`ªZg msL¨v = 180 + 1 = 181|
46. †Kvb ÿz`ªZg msL¨v n‡Z 5 we‡qvM Ki‡j we‡qvMdj
5, 7, 21 I 35 Øviv wbt‡k‡l wefvR¨ n‡e?
105 110
115 120 DËi: L
5, 7, 21 I 35 Gi j.mv.¸ = 105|
∴ ÿz`ªZg msL¨v = 105 + 5 = 110 |
 04.06 A‡¼i ÿz`ªZg ev e„nËg msL¨v †_‡K †Kvb jwNô msL¨v †hvM ev we‡qvM K‡i
†R‡b ivLyb- 28
 avc  : GKvwaK fvRK¸‡jvi j.mv.¸ †ei K‡i cÖ‡kœ cÖ`Ë wZb/Pvi/cuvP As‡Ki ÿz`ªZg msL¨v/e„nËg msL¨v‡K D³
j.mv.¸ Øviv fvM Ki‡Z n‡e Ges fvM †k‡l GKwU ÔfvM‡klÕ cvIqv hv‡e| g‡b ivLyb: j.mv.¸ Øviv fvM Kiv gv‡b
j.mv.¸ n‡”Q D³ ÿz`ªZg/e„nËg msL¨vi fvRK|
 avc  : cÖ‡kœ we‡qvMdj ejv _vK‡j ÔfvM‡klÕ-B DËi| 47 I 48 bs cÖkœ †`Lyb|
Avi †hvMdj ejv _vK‡j ÔfvRK I fvM‡k‡li cv_©K¨Õ DËi| 49 I 50 bs cÖkœ †`Lyb|
(K) we‡qvM _vK‡j
47. wZb A‡¼i ÿz`ªZg msL¨v n‡Z †Kvb jwNô msL¨v
we‡qvM Ki‡j we‡qvMdj 5, 10, 15 Øviv wefvR¨
n‡e? cÖv_wgK we`¨vjq mnKvix wkÿK: 98
5 15
10 20 DËi: M
5 5, 10, 15
1, 2, 3
5, 10, 15 Gi j.mv.¸ = 523 = 30
wZb A‡¼i ÿz`ªZg msL¨v = 100
30) 100 ( 3
90
10
 wb‡Y©q jwNó msL¨v 10|
 g‡b ivLyb: wZb A‡¼i ÿz`ªZg msL¨v‡K 5, 10, 15 Øviv
fvM Kiv †h K_v, G‡`i j.mv.¸ 30 Øviv fvM KivI
GKB K_v| GRb¨ wZb A‡¼i ÿz`ªZg msL¨v‡K 5, 10,
15 Øviv fvM bv K‡i 30 Øviv fvM Kiv n‡q‡Q|
 Avgiv †Kb fvRKmg~‡ni j.mv.¸ Øviv cÖ`Ë msL¨v‡K
fvM Kwi? †KD PvB‡j e¨vL¨vwU c‡o wb‡Z cv‡ib-
Reve: 5, 10, 15 Gi j.mv.¸ 30 †K GB fvRK¸‡jv Øviv
fvM Ki‡j wbt‡k‡l wefvR¨ n‡e| GKBfv‡e 30 Gi MywYZK
30, 60, 90, 120 BZ¨vw`‡K fvM Ki‡jI 5, 10, 15 Øviv
wbt‡k‡l wefvR¨ n‡e| cÖ‡kœ ejv n‡q‡Q wZb A‡¼i ÿz`ªZg
msL¨v (100) †_‡K GKwU jwNô (ÿz`ªZg) msL¨v we‡qvM
Kivi ci cÖvß we‡qvMdj †h‡nZz wbt‡k‡l wefvR¨ n‡e, †m‡nZz
ÿz`ªZg msL¨vwU 100 Gi †P‡q †QvU n‡e Ges wbwðZfv‡e
msL¨vwU 30 A_ev 30 Gi ¸wYZK 60 wKsev 90 n‡e|
100 - 10 = 90
100 - 40 = 60
100 - 70 = 30
Dc‡ii ZvwjKv †_‡K eySv hv‡”Q 100 †_‡K 10/40/70
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11. 2  Math Tutor
†h‡Kvb GKwU msL¨v we‡qvM Ki‡jB Avgiv 30 Gi ¸wYZK
30, 60 I 90 cvw”Q| wKš‘ Avcwb 10, 40, 70 Gi gv‡S
†h‡Kvb msL¨v we‡qvM Ki‡Z cvi‡eb bv, Avcbv‡K we‡qvM
Ki‡Z n‡e G‡`i gv‡S me‡P‡q jwNô msL¨vwU‡K| G‡`i
gv‡S jwNô ( †QvU) msL¨v †KvbwU? wbwðZfv‡e 10| Gfv‡e
cixÿvq mgvavb Ki‡Z †M‡j A‡bK †ewk mgq jvM‡e|
MvwYwZK fvlvq GB mgm¨vwU AviI Kg mg‡q I mn‡R
mgvavb Kiv hvq| Pjyb †`Lv hvK 5, 10 I 15 Gi ¸wYZK
30 †K w`‡q 100 †K fvM Ki‡j wK N‡U!
30) 100( 1 30) 100 ( 2 30) 100 ( 3
30 60 90
70 40 10
GKwU we¯§qKi welq jÿ¨ K‡i‡Qb?
30 w`‡q hLb 1 evi fvM w`jvg ZLb we‡qvM Kivi Rb¨ 10,
40, 70 Gi gv‡S eo msL¨v 70 †cjvg! GKBfv‡e hLb 2
evi fvM w`jvg ZLb 40 Ges hLb 3 evi w`jvg me‡P‡q
†QvU msL¨v 10 †cjvg! A_©vr, m‡ev©”P msL¨K evi fvM w`‡j
me‡P‡q †QvU fvM‡kl P‡j Av‡m| cÖ‡kœ PvIqv jwNô
msL¨vwUB n‡”Q GB †QvU fvM‡klwU| GRb¨B Avgiv
fvRKmg~‡ni ¸wYZK w`‡q cÖ`Ë msL¨v‡K fvM K‡i _vwK|
48. 5 A‡¼i ÿz`ªZg msL¨v n‡Z ‡Kvb jwNô msL¨v we‡qvM
Ki‡j we‡qvMdj 5, 10, 15 Øviv wefvR¨ n‡e?
cÖv_wgK we`¨vjq mnKvix wkÿK, PÆMÖvg wefvM:02
5 10
15 20 DËi: L
(L) †hvM _vK‡j
49. Qq A‡¼i ÿz`ªZg msL¨vi mv‡_ †Kvb ÿz`ªZg msL¨v
†hvM Ki‡j mgwó 2, 4, 6, 8, 10 I 12 Øviv wefvR¨
n‡e? beg-`kg †kÖwY MwYZ, (1983 ms¯‹iY) Abykxjbx 1.1 Gi
42 bs cÖkœ
2 2, 4, 6, 8, 10, 12
2 1, 2, 3, 4, 5, 6
3 1, 1, 3, 2, 5, 3
1, 1, 1, 2, 5, 1
j.mv.¸ = 22325 = 120
Qq A‡¼i ÿz`ªZg msL¨v = 100000
120) 100000 ( 833
960
400
360
40
AZGe, wb‡Y©q ÿz`ªZg msL¨v = 120 - 40 = 80|
50. 999999 -Gi m‡½ †Kvb ÿz`ªZg msL¨v †hvM Ki‡j
†hvMdj 2,3,4,5 Ges 6 Øviv wbt‡k‡l wefvR¨ n‡e?
21Zg wewmGm
21 39
33 29 DËi: K
2, 3, 4, 5 I 6 Gi j.mv.¸ = 60
60) 999999 ( 16666
60
399
360
399
360
399
360
399
360
39
AZGe, wb‡Y©q j.mv.¸ = 60 - 39
= 21|
(M) †hvM ev we‡qvM ejv bv _vK‡j
 e„nËg msL¨vi †ÿ‡Î fvM‡kl we‡qvM K‡i wbt‡k‡l
wefvR¨ Ki‡Z nq|
 ÿz`ªZg msL¨vi †ÿ‡Î ÔfvRK I fvM‡klÕ Gi cv_©K¨
†hvM K‡i wbt‡k‡l wefvR¨ Ki‡Z nq|
51. Qq A‡¼i †Kvb ÿz`ªZg msL¨v 25, 50, 75 I 125
w`‡q wbt‡k‡l wefvR¨?
5 25, 50, 75, 125
5 5, 10, 15, 25
1, 2, 3, 5
j.mv.¸ = 55235 = 750
Qq A‡¼i ÿz`ªZg msL¨v = 100000
750) 100000 ( 133
750
2500
2250
2500
2250
250
(GLv‡b, wbt‡k‡l wefvR¨ msL¨vwU n‡e 100000 n‡Z
250 Kg A_ev 100000 n‡Z (750 - 250) ev
500 †ewk| wKš‘ 100000 n‡Z 250 Kg n‡j
msL¨vwU (100000 - 250) ev 99750, hv cuvP A‡¼i
weavq MÖnY‡hvM¨ bq|)
 wb‡Y©q ÿy`ªZg msL¨v = 100000+ (750 - 250)
= 100500| (DËi)
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12. Math Tutor  3
52. Qq A‡¼i †Kvb e„nËg msL¨v 27, 45, 60, 72 I 96
Øviv wefvR¨? beg-`kg †kÖwY MwYZ, (1983 ms¯‹iY)
Abykxjbx 1.1 Gi 38 bs cÖkœ
2 27, 45, 60, 72, 96
2 27, 45, 30, 36, 48
2 27, 45, 15, 18, 24
3 27, 45, 15, 9, 12
3 9, 15, 5, 3, 4
5 3, 5, 5, 1, 4
3, 1, 1, 1, 4
j.mv.¸ = 22233534
= 4320
Qq A‡¼i e„nËg msL¨v = 999999
4320 ) 999999 ( 231
8640
13599
12960
6399
4320
2079
(GLv‡b, wbt‡k‡l wefvR¨ msLvwU n‡e 999999 n‡Z
2079 Kg A_ev 999999 n‡Z ( 4320 - 2079) ev
2,241 †ewk| wKš‘ 999999 †_‡K 2,241 †ewk n‡j
msL¨vwU (999999 + 2241) ev 1,002,240, hv mvZ
A‡¼i weavq MÖnY †hvM¨ bq|)
 wb‡Y©q e„nËg msL¨v = 999999 - 2079
= 997920
53. cuvP A‡¼i †Kvb e„nËg msL¨v‡K 16, 24, 30 I 36
w`‡q fvM Ki‡j cÖ‡Z¨Kevi fvM‡kl 10 _v‡K?
(06 bs Gi gZ cvuP A‡¼I e„nËg msL¨vwU wbY©q
K‡i, Zvici 10 †hvM Ki‡jB DËi P‡j Avm‡e|)
4 16, 24, 30, 36
2 4, 6, 30, 9
3 2, 3, 15, 9
2, 1, 5, 3
j.mv.¸ = 4232353 = 720
Qq A‡¼i e„nËg msL¨v = 99999
720 ) 99999 ( 138
720
2799
2160
6399
5760
639
 wb‡Y©q e„nËg msL¨v = 99999 - 639
= 99360
wKš‘ cÖkœg‡Z, cÖ‡Z¨Kevi fvM‡kl 10 we`¨gvb _v‡K|
 e„nËg msL¨v = 99360 + 10 = 99370 (DËi)
M.mv.¸ (H.C.M)
(K) ¸YbxqK (Factor)t ¸YbxqK Kx? ¸YbxqK n‡”Q
†Kvb msL¨vi fvRKmg~n ev Drcv`K mg~n| †hgb-
20 ¸YbxqKmg~n Kx Kx?
20 Gi ¸YbxqKmg~n = 1, 2, 4, 5, 10, 20
20 Gi me‡P‡q †QvU ¸YbxqK 1 Ges me‡P‡q eo
Drcv`K 20|
 g‡b ivLyb: 1 †h‡Kvb msL¨vi Drcv`K ev ¸YbxqK
Ges †Kvb msL¨vi me‡P‡q eo Drcv`K msL¨vwUi
mgvb| mnR Ki‡j ej‡j, †Kvb msL¨vi Drcv`K
KLbB Zvi †P‡q eo n‡Z cv‡i bv|
(L) mvaviY ¸YbxqK (Common Factor)t Common
kãwUi gv‡b †h‡nZz GKvwaK wRwb‡mi gv‡S mv`„k¨ ev
wgj Ask, †m‡nZz mvaviY ¸YbxqK (Common
Factor) ej‡Z GKvwaK msL¨vi gv‡S †h ¸YbxqK ev
Drcv`Kmg~n wgj ev mv`„k¨ Av‡Q †m¸‡jv‡K eySvq|
†hgb- 24 I 36 Gi mvaviY ¸YbxqK mg~n Kx Kx?
24 = 1, 2, 3, 4, 6, 8, 12, 24
36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
28 I 36 Gi ¸YbxqK¸‡jvi gv‡S mvaviY
¸YbxqKmg~n n‡”Q 1, 2, 3, 4, 6, 12|
(M) Mwiô mvaviY ¸YbxqK (Highest Common
Factor) t Mwiô mvaviY ¸YbxqK ev M.mv.¸ n‡”Q
mvaviY ¸YYxqK¸‡jvi gv‡S me‡P‡q eo ¸YYxqK|
†hgb- Dc‡ii 24 I 36 Gi mvaviY ¸YbxqK 1, 2,
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13. 2  Math Tutor
3, 4, 6, 12 Gi gv‡S me‡P‡q eo ev Mwiô mvaviY
¸YbxqK ev M.mv.¸ n‡”Q 12|
 g‡b ivLyb- M.mv.¸ wbY©‡qi gvÎ ewY©Z †KŠkjwU
msÁv eyS‡Z mvnvh¨ Ki‡jI GB c×wZwU mgq
mv‡cÿ, ZvB M.mv.¸ wbY©‡qi wKQz †KŠkj Avgiv wb‡P
Zz‡j aijvg|
 04.07 M.mv.¸ wbY©‡qi †KŠkjmg~n
(K) fvM c×wZt †h `ywU msL¨vi M.mv.¸ wbY©q Ki‡eb,
†mB msL¨v`ywUi gv‡S †QvU msL¨vwU w`‡q eo
msL¨vwU‡K fvM ïiæ Ki‡eb| me‡k‡l †h fvRKwU
w`‡q wbt‡k‡l wefvR¨ n‡e, †mwU n‡e M.mv.¸|
48. 12 I 30 Gi M.mv.¸ wbY©q Kiæb|
(12 †QvU msL¨v, ZvB 12 w`‡q 30 †K fvM Ki‡Z n‡e|)
12) 30 ( 2
24
6 ) 12 ( 2
12
0
 12 w`‡q 30 †K fvM Kivi ci 6 fvM‡kl _vKj|
Zvici 6 w`‡q cÖ_g fvRK 12 †K fvM Kiv nj|
GeviI hw` fvM‡kl _vKZ, †mB fvM‡kl w`‡q Zvi
c~‡e©i fvRK 6 †K fvM KiZvg|
49. 28, 48 Ges 72 Gi M.mv.¸ wbY©q Kiæb|
(Gevi `ywU msL¨vi cwie‡Z© 3wU msL¨v †`qv Av‡Q|
cÖ_‡g 28 I 48 w`‡q ïiæ Kiv hvK|)
28) 48 ( 1
28
20) 28 ( 1
20
8) 20 ( 2
16
4) 8 ( 2
8
0
(Gevi 28 I 48 †_‡K cÖvß M.mv.¸ 4 w`‡q 72 †K
fvM Ki‡Z n‡e|)
4) 72 ( 18 (†kl fvRK 4-B msL¨v 3wU M.mv.¸)
72
0
 28, 48 Ges 72 Gi M.mv.¸ 4|
(L) mswÿß c×wZ‡Z M.mv.¸ wbY©qt GLb †h c×wZ wb‡q
Av‡jvPbv Kie, GwU LyeB ¸iæZ¡c~Y© c×wZ| G
c×wZwU A‡bKUv j.mv.¸ wbY©‡qi c×wZi KvQvKvwQ|
50. 144, 240, 612 Gi M.mv.¸ wbY©q Kiæb|
2 144, 240, 612
2 72, 120, 306
3 36, 60, 153
12, 20, 51
(jÿ¨ Kiæb- fvRK 2, 2 I 3 Øviv cÖ‡Z¨K‡K fvM Kiv
†M‡jI GLb Avi Ggb †Kvb fvRK cvIqv hv‡”Q bv, hv‡K
w`‡q 12, 20 I 51 msL¨v wZbwU‡K GKmv‡_ fvM Kiv
hvq| GLv‡b meKqwU‡K fvM Ki‡Z cviv fvRK 2, 2, 3
Gi ¸YdjB n‡”Q M.mv.¸ |
 M.mv.¸ = 223 = 12|
 G c×wZi myweav n‡”Q, j.mv.¸ I M.mv.¸ GKmv‡_
†ei Kiv hvq| Pjyb j.mv.¸ †ei Kiv hvK-
2 144, 240, 612
2 72, 120, 306
3 36, 60, 153
4 12, 20, 51
3 3, 5, 51
1, 5, 17
 j.mv.¸ = 22343517
= 12,240
51. †Kvb e„nËg msL¨v Øviv 57, 93, 183 †K fvM Ki‡j
†Kvb fvM‡kl _vK‡e bv? wb¤œ gva¨.MwYZ(6ô†kÖwY)Abykxjbx1.3
(cyivZb)
3 57, 93, 183
19, 31, 61
M.mv.¸ = 3|
 wb‡Y©q e„nËg msL¨v = 3|
(M) M.mv.¸ KLb 1 nq?: †h me msL¨vi M.mv.¸ wbY©q
Kie, Zv‡`i gv‡S hw` 1 e¨ZxZ Avi †Kvb fvRK/
Drcv`K Kgb ev mv`„k¨ bv _v‡K| †hgb- 5 I 7 Gi
M.mv.¸ 1| KviY 5 I 7 Gi gv‡S 1 Qvov Avi †Kvb
Kgb Drcv`K †bB|
52. 2x I 3x Gi M.mv.¸ KZ?
2x I 3x Gi M.mv.¸ = x |
53. `ywU msLvi AbycvZ 11 : 17 n‡j msL¨v `ywUi M.mv.¸
KZ?
awi, msL¨v `ywU n‡”Q 11x I 17x
 M.mv.¸ = x
54. `ywU msL¨vi M.mv.¸ 3 n‡j msL¨v `ywU Kx Kx?
msL¨v `ywU n‡e 3x I 3y | KviY 3x, 3y aivi
Kvi‡Y GLv‡b 3 mv`„k¨ ev wgj Av‡Q, ZvB 3 M.mv.¸
mgvavb
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14. 2  Math Tutor
n‡e|
 g‡b ivLyb, Dc‡ii 05, 06 I 07 bs cÖ‡kœi
mgvavb¸‡jv fv‡jv K‡i eySzb, Avgiv cieZx©‡Z GB
AvBwWqv¸‡jv Kv‡R jvMve Bbkvjøvn&|
 04.08 M.mv.¸Õi †ewmK mgm¨vewj
g‡b ivLyb 29
 mgvbfv‡e fvM K‡i †`qvi A_© n‡”Q M.mv.¸‡K wfwË a‡i fvM K‡i †`qv| ZvB †Kvb mgm¨vq mgvbfv‡e fvM K‡i
†`qv eySv‡j g‡b Ki‡eb M.mv.¸ Ki‡Z ejv n‡q‡Q| 01 I 02 bs cÖkœ †`Lyb|
 ÔcÖ_g I wØZxq msL¨vi ¸Ydj Ges wØZxq I Z…Zxq msL¨vi ¸YdjÕ Gi gv‡S wØZxq msL¨vwU `ywU ¸Yd‡jB Av‡Q e‡j
wØZxq msL¨vwU n‡”Q M.mv.¸| 03 bs cÖkœ †`Lyb|
55. 125 wU Kjg I 145 wU †cwÝj KZR‡bi g‡a¨
mgvbfv‡M fvM Kiv hv‡e? wbev©Pb Kwgkb mwPevj‡qi
†Rjv wbev©Pb Awdmvi I mnKvix mwPe-04
10 15
5 20 DËi: M
5 125, 145
25 , 29
 5 evj‡Ki gv‡S Kjg I †cwÝj¸‡jv mgvbfv‡e
fvM Kiv hv‡e|
 125) 145 ( 1
125
20) 125 ( 6
120
5) 20 ( 4
20
0
 5 evj‡Ki gv‡S Kjg I †cwÝj¸‡jv mgvbfv‡e
fvM Kiv hv‡e|
56. KZRb evjK‡K 125 wU Kgjv‡jey Ges 145 wU Kjv
mgvbfv‡e fvM K‡i †`qv hvq? A_© gš¿Yvj‡qi Awdm
mnKvix -11
10 Rb‡K 05 Rb‡K
15 Rb‡K 25 Rb‡K DËi: L
57. cª_g I wØZxq msL¨vi ¸Ydj 35 Ges wØZxq I
Z…Zxq msL¨vi ¸Ydj 63| wØZxq msL¨vwU KZ?
cwiKíbv gš¿Yvj‡qi WvUv cÖ‡mwms Acv‡iUi: 02
5 6
7 8 DËi: M
7 35, 63
5, 9
 wØZxq msL¨vwU 7|
 1g 2q = 5  7
2q 3q = 7  9
 wØZxq msL¨vwU 7|
58. GKwU †jvnvi cvZ I GKwU Zvgvi cv‡Zi •`N©¨
h_vµ‡g 672 †m. wg. I 960 †m. wg.| cvZ `yBwU
†_‡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
Ki| wb¤œ gva¨wgKMwYZ(6ô†kÖwY)Abykxjbx1.3
cvZ `yBwU †_‡K †K‡U †bIqv GKB gv‡ci me‡P‡q
eo UzKivi •`N©¨ nj 672 I 960 Gi gv‡S me‡P‡q
eo Kgb ¸YbxqK ev M.mv.¸|
3 672, 960
8 224, 320 (Avcbviv †QvU †QvU msL¨v
4 28, 40 w`‡qI fvM Ki‡Z cv‡ib)
7, 10
 me‡P‡q eo UzKivi •`N©¨ (M.mv.¸) = 384
= 96
†jvnvi cv‡Zi UzKivi msL¨v = 672  96 = 7wU
Zvgvi cv‡Zi UzKivi msL¨v = 960  96 = 10wU|
59. `yBwU Wªv‡g h_vµ‡g 868 wjUvi I 980 wjUvi `ya
Av‡Q| me‡P‡q eo gv‡ci cvÎ Øviv `yBwU Wªv‡gi
`ya c~Y©msL¨K ev‡i gvcv hv‡e? wb¤œ gva¨wgKMwYZ(6ô†kÖwY)
Abykxjbx1.3(cyivZb)
868 I 980 Gi M.mv.¸ 28 n‡”Q DËi|
60. `ywU AvqZvKvi ¸`vg N‡ii •`N©¨ h_vµ‡g 28 I 20
wgUvi Ges cÖ¯’ h_vµ‡g 12, 14 wgUvi| me‡P‡q eo
†Kvb AvqZ‡bi cv_i w`‡q †Nii †g‡S cyivcywi †X‡K
†djv hv‡e (Ges †Kvb cv_i AcPq n‡e bv) ? beg-
`kg †kÖwY MwYZ, (1983 ms¯‹iY) Abykxjbx 1.1 Gi 46 bs cÖkœ
cÖ_‡g ¸`vg Ni `ywUi †ÿÎdj †ei K‡i wb‡Z n‡e
Ges Zv‡`i M.mv.¸ †ei Ki‡Z n‡e|
1g N‡ii †ÿÎdj = 2812 = 336 eM©wgUvi
2q N‡ii †ÿÎdj = 20  14 = 280 eM©wgUvi
mgvavb
mgvavb
mgvavb
mgvavb
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mgvavb
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15. 2  Math Tutor
336 I 280 Gi M.mv.¸ wbY©q Ki‡Z n‡e|
4 336, 280
2 84, 70
7 42, 35
6, 5
M.mv.¸ = 427 = 56|
myZivs, wb‡Y©q cv_‡ii AvqZb = 56 eM©wgUvi|
61. GKwU AvqZvKvi N‡ii •`N©¨ 30 wgUvi, cÖ¯’ 12 wgUvi
Av‡iKwU AvqZvKvi nj N‡ii •`N©¨ 20 wgUvi I cÖ¯’
15 wgUvi| me‡P‡q eo †Kvb AvqZ‡bi Kv‡Vi UyKiv
w`‡q Dfq N‡ii †g‡S cyivcywi †X‡K †djv hv‡e? beg-
`kg †kÖwY MwYZ, (1983 ms¯‹iY) Abykxjbx 1.1 Gi 46 bs cÖkœ
1g N‡ii †ÿÎdj = 3012 = 360 eM©wgUvi
2q N‡ii †ÿÎdj = 2015 = 300 eM©wgUvi
15 360, 300
4 24, 20
6, 5
(eo eo msL¨v w`‡q fvM Ki‡j mgq Kg jvM‡e, Z‡e
Avcwb `ye©j n‡j †QvU †QvU msL¨v w`‡q †Póv Kiæb)
M.mv.¸ = 154 = 60|
myZivs, wb‡Y©q eo Kv‡Vi UzKivi AvqZb = 60
eM©wgUvi|
 04.09 cÖ‡Z¨Kevi 1 wU msL¨v Aewkó _vK‡j I †mwU D‡jøL _vK‡j
†R‡b wbb-30
 cÖ‡Z¨Kevi 1wU msL¨v Aewkó _vK‡j †mwU cÖ‡kœ cÖ`Ë cÖwZwU ¸wYZK †_‡K fvM‡kl/Aewkó we‡qvM Ki‡Z n‡e|
 Zvici we‡qvMK…Z ¸wYZK¸‡jvi M.mv.¸ Ki‡Z n‡e|
62. †Kvb e„nËg msL¨v w`‡q 102 I 186 †K fvM Ki‡j
cÖ‡Z¨Kevi 6 Aewkó _vK‡e| WvKI†Uwj‡hvMv‡hvMwnmveiÿK
Kg©KZv© :03
12 15
16 22 DËi: K
†h e„nËg msL¨v Øviv 102 I 186 †K fvM
Ki‡j cÖ‡Z¨Kevi fvM‡kl _vK‡e †mwU nj (102
- 6) = 96 I (186 - 6) = 180|
96 ) 180 ( 1
96
84) 96 ( 1
84
12) 84 ( 7
84
0
 96 I 180 Gi M.mv.¸ = 12|
 wb‡Y©q e„nËg msL¨v = 12|
 Drcv`‡K we‡kølY cÖ‡qvM K‡i M.mv.¸ wbY©q:
2 96, 180
2 48, 90
3 24, 45
8, 15
 M.mv.¸ = 223 = 12|
63. †Kvb e„nËg msL¨v Øviv 100 I 184 †K fvM Ki‡j
cÖ‡Z¨Kevi fvM‡kl 4 _vK‡e? wb¤œ gva¨wgKMwYZ(6ô†kÖwY)
Abykxjbx1.3Gi 5bscÖkœ DËi: 12
 04. 10 cÖwZwU ¸wYZ‡Ki Rb¨ c„_K c„_K Aewkó/fvM‡kl _vK‡j
†R‡b wbb-31
 cÖwZwU ¸wYZ‡Ki Rb¨ c„_K c„_K fvM‡kl _vK‡j cÖ`Ë ¸YwZKmg~n †_‡K c„_K c„_Kfv‡e fvM‡kl we‡qvM Ki‡Z
n‡e|
 Zvici we‡qvMK…Z ¸wYZK ¸‡jvi M.mv.¸ wbY©q Ki‡Z n‡e|
64. †Kvb e„nËg msL¨v Øviv 27, 40 I 65 †K fvM Ki‡j
h_vµ‡g 3, 4, 5 fvM‡kl _vK‡e? ewnivMgb I cvm‡cvU©
Awa`߇ii mnKvix cwiPvjK-11; ¯^ivóª gš¿Yvj‡qi Kviv
Z¡Ë¡veavqK -10; wb¤œ gva¨wgK MwYZ (6ô †kÖwY), 1.3 Gi 6 bs
16 14
12 10 DËi: M
27 - 3 = 24
40 - 4 = 36
65 - 5 = 60
mgvavb
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mgvavb

16. 2  Math Tutor
24, 36 I 60 Gi M.mv.¸ wbY©q Ki‡Z n‡e|
12 24, 36, 60 (fvM c×wZ‡ZI Ki‡Z cv‡ib)
2, 3, 5
 24, 36 I 60 Gi M.mv.¸ = 12|
 wb‡Y©q e„nËg msL¨v 12|
65. 728 Ges 900 †K mev©‡cÿv eo †Kvb msL¨v Øviv
fvM Ki‡j h_vµ‡g 8 Ges 4 Aewkó _vK‡e?
cÖwZiÿv gš¿Yvj‡qi Aaxb GWwgwb‡÷ªwUf Awdmvi I cv‡mv©bvj
Awdmvi-06
12 13
14 16 DËi:
728 - 8 = 720
900 - 4 = 896
720) 896 ( 1
720
176) 720 ( 4
704
16) 176 ( 11
176
0
 720 I 896 Gi M.mv.¸ = 16|
 4 720, 896
4 180, 224 (wefvR¨Zvi bxwZ e¨envi Ki‡j
45, 56 Lye `ªæZ mgvavb Kiv hvq)
 720 I 895 Gi M.mv.¸ = 44 = 16|
66. 159 wU Avg, 227 wU Rvg I 401 wU wjPz me‡P‡q
†ewk KZRb evj‡Ki g‡a¨ mgvbfv‡e fvM K‡i w`‡j
3 wU Avg, 6wU Rvg I 11wU wjPz Szwo‡Z _vK‡e?
159 - 3 = 156 wU Avg
227 - 6 = 221 wU Rvg
401 - 11 = 390 wU wjPz
156, 221 I 390 Gi M.mv.¸ wbY©q Ki‡Z n‡e|
13 156, 221, 390
12, 17, 30
 156, 221 I 390 Gi M.mv.¸ = 13|
myZivs, wb‡Y©q me‡P‡q †ewk evj‡Ki msL¨v 13 Rb|
 g‡b ivLyb: `ªæZ wn‡me Kivi Rb¨ Avcbv‡K Aek¨B
bvgZv I wefvR¨Zvi bxwZi Dci `Lj _vK‡Z n‡e|
 04.11 cÖ‡Z¨Kevi 1 wU msL¨v Aewkó _vK‡j I †mwU D‡jøL bv _vK‡j
†R‡b ivLyb-32
 cÖwZ †ÿ‡Î GKB Aewkó n‡j I †mwU D‡jøL bv _vK‡j cÖ`Ë msL¨v a, b, c Gi Rb¨ (b - a) (c - b) (c - a) m~Î
cÖ‡qvM K‡i M.mv.¸ Kivi msL¨v¸‡jv Lyu‡R †ei Ki‡Z n‡e|
67. †Kvb& e„nËg msL¨v Øviv 1305, 4665 I 6905-†K
fvM Ki‡j cÖwZ‡ÿ‡Î GKB Aewkó _v‡K? beg-
`kg †kÖwY MwYZ, (1983 ms¯‹iY) Abykxjbx 1.1 Gi 46 bs cÖk
4665 - 1305 = 3360 (b - a)
6905 - 4665 = 2240 (c - b)
6905 - 1305 = 5600 (c - a)
3360, 2240 I 5600 Gi M.mv.¸ wbY©q Ki‡Z
n‡e|
2240) 3360 ( 1
2240
1120) 2240 ( 2
2240
0
1120 ) 5600 ( 5
5600
0
3360, 2240 I 5600 Gi M.mv.¸ = 1120
wb‡Y©q e„nËg msL¨v 1120|
68. e„nËg msL¨v N Øviv 1305, 4665 I 6905 †K fvM
fvM Ki‡j cÖwZ‡ÿ‡Î GKB Aewkó _v‡K| N Gi
A¼¸‡jvi mgwó KZ? WvPevsjve¨vsKwj.(cÖ‡ekbvwiAwdmvi)
2012
4 5
6 8 DËi: K
67 bs Gi gZ mgvavb Kivi ci e„nËg msL¨vwU
(N) wbY©q Kiæb| Gici cÖkœvbyhvqx, D³ e„nËg
msL¨vwUi A¼¸‡jvi †hvM Kiæb, Zvn‡jB DËi cvIqv
hv‡e, †mwUi †hvMdjB n‡e DËi|
e„nËg msL¨vwU = 1120
AZGe, e„nËg msL¨vwUi A¼¸‡jvi mgwó
= 1 + 1 + 2 + 0 = 4|
69. †Kvb& e„nËg msL¨v Øviv 4003, 4126 I 4249-‡K
fvM Ki‡j cÖwZ‡ÿ‡Î GKB Aewkó _v‡K? cjøxmÂq
e¨vsK(K¨vk)2018
43 41
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mgvavb
mgvavb
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17. 2  Math Tutor
45 50 DËi: L
4126 - 4003 = 123
4249 - 4126 = 123
4249 - 4003 = 246
123, 123 I 246 Gi M.mv.¸ wbY©q Ki‡Z n‡e|
123) 123 ( 1
123
0
123 ) 246 ( 2
246
0
cÖ‡kœ †QvU GKUv RwUjZv Av‡Q| mivmwi DËi †bB|
Ack‡b 123 Gi ¸wYZK _vK‡jI DËi nZ, wKš‘
†mwUI †bB| ZvB 123 †K Drcv`‡K we‡kølY K‡i
†`L‡Z n‡e, 123 Gi †Kvb msL¨vi ¸wYZK|
3 123
41
41 Gi ¸wYZK n‡”Q 123, ZvB 123 Øviv fvM Ki‡j
†hgb cÖwZ‡ÿ‡Î GKB Aewkó _vK‡e GKBfv‡e 41
Øviv fvM Ki‡jI GKB Aewkó _vK‡e|
 wb‡Y©q e„nËg msL¨v = 41|
 04.12 fMœvs‡ki j.mv.¸ I M.mv.¸
†R‡b wbb-33
 fMœvskmg~‡ni j.mv.¸ : fMœvs‡ki j.mv.¸ wbY©q Kivi wbqg n‡”Q fMœvskmg~‡ni je¸‡jvi j.mv.¸ †ei Ki‡Z n‡e
Ges ni¸‡jvi M.mv.¸ †ei Ki‡Z n‡e|
fMœvs‡ki j.mv.¸ =
M.mv.¸
jvi
ni¸‡
j.mv.¸
jvi
je¸‡
 fMœvskmg~‡ni M.mv.¸ : fMœvs‡ki M.mv.¸ wbY©q Kivi wbqg n‡”Q fMœvskmg~‡ni je¸‡jvi M.mv¸ †ei Ki‡Z n‡e Ges
ni¸‡jvi j.mv.¸ †ei Ki‡Z n‡e|
fMœvs‡ki M.mv.¸ =
j.mv.¸
jvi
ni¸‡
M.mv.¸
jvi
je¸‡
 g‡b ivLyb: fMœvs‡ki j.mv.¸ PvB‡j j‡e j.mv¸ Ges fMœvs‡ki M.mv.¸ PvB‡j  j‡e M.mv.¸ |
70.
3
1
,
6
5
,
9
2
Ges
27
4
Gi j.mv.¸ KZ?
54
1
27
10
3
20
†Kv‡bvwUB bq DËi: M
1, 5, 2 Ges 4 Gi j.mv.¸ = 20
3, 6, 9 Ges 27 Gi M.mv.¸ = 3
 wb‡Y©q j.mv.¸ =
3
20
|
71.
3
2
,
9
8
,
81
64
Ges
27
10
Gi M.mv.¸ KZ?
3
2
81
2
3
160
81
160
DËi: L
2, 8, 64 Ges 10 Gi M.mv.¸ = 2
3, 9, 81 Ges 27 Gi j.mv.¸ = 81
 wb‡Y©q M.mv.¸ =
81
2
|
72.
3
2
,
5
3
,
7
4
Ges
13
9
Gi j.mv.¸ KZ?
36
36
1
1365
1
455
12
DËi: K
73.
10
9
,
25
12
,
35
18
Ges
40
21
Gi M.mv.¸ KZ?
5
3
5
252
1400
3
700
63
DËi: M
N
M
L
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N
M
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mgvavb
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mgvavb
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mgvavb
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18. 2  Math Tutor
74. wZbwU N›Uv GK‡Î †e‡R 1
2
1
wgwbU, 2
2
1
wgwbU,
3
2
1
wgwbU AšÍi evR‡Z jvMj| b~¨bZg KZÿY ci
NÈv¸‡jv cybivq GK‡Î evR‡e? cwiKíbvgš¿YvjqGescÖevmx
Kj¨vYI•e‡`wkKKg©ms¯’vbgš¿Yvj‡qimnKvixcwiPvjK2006
12
2
1
wgwbU 32
2
1
wgwbU
52
2
1
wgwbU 72
2
1
wgwbU DËi: M
(wgkÖ fMœvsk¸‡jv‡K AcÖK…Z fMœvs‡k iƒcvšÍi K‡i wbb)
1
2
1
wgwbU ev
2
3
wgwbU, 2
2
1
wgwbU ev
2
5
wgwbU,
3
2
1
wgwbU ev
2
7
wgwbU Gi j.mv.¸-B n‡e DËi|
3, 5 Ges 7 Gi j.mv.¸ = 105 Ges 2, 2 Ges 2
Gi M.mv.¸ = 2
 wb‡Y©q j.mv.¸ =
2
105
= 52
2
1
|
 04.13 j.mv.¸ I M.mv.¸I m¤úK©
(K) msL¨v `ywUi ¸Ydj = j. mv. ¸  M. mv. ¸
75. `yBwU msL¨vi M. mv.¸ I j. mv.¸ -Gi ¸Ydj msL¨v
`ywUi - mve-†iwR÷vit92
†hvMd‡ji mgvb ¸Yd‡ji mgvb
we‡qvMd‡ji mgvb fvMd‡ji mgvb DËi: L
76. `ywU msL¨vi ¸Ydj 1376| msL¨v `ywUi j. mv. ¸ 86
n‡j M. mv. ¸ KZ? Z_¨gš¿Yvj‡qiAax‡bmnKvixcwiPvjK,
†MÖW-2t03
16 18
24 22 DËi: K
msL¨v `ywUi ¸Ydj = j. mv. ¸  M. mv. ¸
ev, 1376 = 86  M. mv. ¸
 M.mv.¸ =
86
1376
= 16|
77. `yBwU msL¨vi ¸Ydj 4235 Ges Zv‡`i j. mv. ¸
385| msL¨v `yBwUi M. mv. ¸ KZ? KvwiMwiwkÿv
Awa`߇iiAax‡bPxdBÝUªv±i:03
17 15
11 13 DËi: M
78. `ywU msL¨vi M. mv.¸ 16 Ges j. mv. ¸ 192| GKwU
msL¨v 48 n‡j, Aci msL¨vwU KZ? cÖv_wgKwe`¨vjq
mnKvixwkÿK:01
60 62
64 68 DËi: M
GKwU msL¨v  Aci msL¨v = j. mv. ¸  M. mv. ¸
ev, 48  Aci msL¨v = 192  16
 Aci msL¨v =
48
16
192 
= 64|
79. `ywU msL¨vi j.mv.¸ 48 Ges M. mv. ¸ 4 | GKwU
msL¨v 16 n‡j Aci msL¨vwU KZ? ¯’vbxqmiKvigš¿Yvj‡qi
Aax‡bGjwRBwW‡ZmnKvixcÖ‡KŠkjx:05
12 22
24 32 DËi: K
Aci msL¨v =
16
4
48 
= 12|
†R‡b wbb -34
 GKwU m~Î w`‡q wZbwU m~Î AvqË¡ Kiv hvK-
 msL¨v¸‡jvi j.mv.¸ = msL¨v¸‡jvi Abycv‡Zi ¸Ydj  msL¨v¸‡jvi M. mv. ¸
 msL¨v¸‡jvi M.mv.¸ =
¸Ydj
Zi
Abycv‡
jvi
msL¨v¸‡
j.mv.¸
jvi
msL¨v¸‡
 AbycvZ¸wji ¸Ydj =
M.mv.¸
jvi
msL¨v¸‡
j.mv.¸
jvi
msL¨v¸‡
  bs m~Î †_‡K evKx m~θ‡jv G‡m‡Q| LvZvq wj‡L evievi cÖ¨vKwUm Kiæb, mn‡RB Avq‡Ë¡ P‡j Avm‡e|
mgvavb
N
M
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mgvavb
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mgvavb
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mgvavb
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19. 2  Math Tutor
(L) `ywU msL¨vi AbycvZ I M.mv.¸ †_‡K j.mv.¸ wbY©q
80. `ywU msL¨vi AbycvZ 3 : 4 Ges Zv‡`i M.mv.¸ 4
n‡j, Zv‡`i j.mv.¸ KZ?
12 16
24 48 DËi: N
g‡bKwi, msL¨v `ywU = 3x I 4x
Ges Zv‡`i Mmv¸ = x
cÖkœg‡Z, x = 4
∴ msL¨v `ywU : 3 x = 3  4 = 12
I 4x = 4  4 = 16|
myZivs, 2 12, 16
2 6, 8
3, 4
12 I 16 Gi j.mv.¸- 2  2  3  4 = 48|
 kU©KvUt msL¨v¸‡jvi j.mv.¸ = msL¨v¸‡jvi
Abycv‡Zi ¸Ydj  M. mv. ¸ = 3  4  4= 48|
 kU©KvU †UKwb‡Ki e¨vL¨vt- msL¨v `ywUi AbycvZ‡K
¸.mv.¸ Øviv ¸Y Ki‡j msL¨v `ywU cvIqv hvq| †hgb-
3 : 4 AbycvZ‡K M.mv.¸ 4 Øviv ¸Y K‡i cÖvß msL¨v
`ywU n‡”Q- 34 = 12 Ges 44 = 16|
∴ 12 I 16 Gi j.mv.¸ n‡”Q = 4 12, 16
3, 4
= 4 34 = 48|
(M.mv.¸ 4 I AbycvZ 3 : 4)
81. `ywU msL¨vi AbycvZ 5 : 6 Ges Zv‡`i M. mv. ¸ 4
n‡j, msL¨v `ywUi j. mv. ¸ KZ? _vbvmn wkÿvAwdmvi:99
100 120
150 180
msL¨v¸‡jvi j.mv.¸ = 564 = 120
82. `yBwU msL¨vi AbycvZ 5 : 6 Ges Zv‡`i M.mv.¸ 6
n‡j, msL¨v `yBwUi j.mv.¸ KZ? `ybx©wZ`gbey¨‡ivcwi`k©K:04
210 180
150 120 DËi: L
83. `yBwU msL¨vi AbycvZ 5 : 6 Ges Zv‡`i M.mv.¸ 8
n‡j Zv‡`i j.mv.¸ KZ? RbcÖkvmbgš¿Yvj‡qicÖkvmwbK
Kg©KZ©v:16
200 224
240 248 DËi: M
84. `yBwU msL¨vi 5 : 7 Ges Zv‡`i M.mv.¸ 6 n‡j, msL¨v
`yBwUi j.mv.¸ KZ? cÖvK-cÖv_wgKmnKvixwkÿK(†WjUv):14
210 180
150 120 DËi: K
 (M) `ywU msL¨vi AbycvZ I j.mv.¸ †_‡K M.mv.¸ wbY©q
85. `ywU msL¨vi AbycvZ 4 : 5 Ges msL¨v `ywUi j.mv.¸
60 n‡j, M.mv.¸ KZ?
5 1
2 3 DËi: N
g‡bKwi, M.mv.¸ = x Ges `ywU msL¨v = 4x I 5x
(AbycvZ‡K ¸.mv.¸ Øviv ¸Y Ki‡j msL¨v `ywU cvIqv hvq)
∴ `ywU msL¨vi j.mv.¸ = x 4x, 5x
4, 5
= x 45 = 20x
cÖkœg‡Z, 20x = 60
ev, x =
20
60
= 3
myZivs, wb‡Y©q M.mv.¸ = 3|
 kU©KvUt msL¨v؇qi M.mv.¸ =
¸Ydj
Zi
Abycv‡
j.mv.¸
= 3
2
60
5
4
60


 o
86. `yBwU msL¨vi AbycvZ 7 : 5 Ges Zv‡`i j.mv.¸ 140
n‡j msL¨v `yBwUi M.mv.¸ KZ? 39Zg wewmGm (we‡kl)
wcÖwjwgbvwi
12 6
9 4 DËi: N
msL¨v؇qi M.mv.¸ =
¸Ydj
Zi
Abycv‡
j.mv.¸
=
5
7
140

= 4|
87. `ywU msL¨vi AbycvZ 7 : 8 Ges Zv‡`i j.mv.¸ 280
n‡j msL¨v `yBwUi M.mv.¸ KZ? 15Zg †emiKvwi cÖfvlK
wbeÜb (K‡jR/ch©vq) : 18
4 5
6 7 DËi: L
N
M
L
K
mgvavb
N
M
L
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mgvavb
N
M
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N
M
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M
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mgvavb
N
M
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mgvavb
N
M
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20. 2  Math Tutor
msL¨v؇qi M.mv.¸ =
8
7
280

= 5|
88. `ywU msL¨vi AbycvZ 5 : 7 Ges Zv‡`i j.mv.¸ 350|
msL¨v `ywUi M.mv.¸- ¯^v¯’¨ cÖ‡KŠkj Awa`߇ii mn. cÖ‡KŠkjx
(wmwfj) : 17
50 70
35 10 DËi: N
89. `ywU msL¨vi AbycvZ 5 : 6 Ges Zv‡`i j.mv.¸ 120;
msL¨v `ywUi M.mv.¸ KZ? evsjv‡`k †ijI‡qi Dcmn.
cÖ‡KŠkjx (wmwfj) : 16
3 4
5 6 DËi: L
90. wZbwU msL¨vi AbycvZ 3 : 4 : 5 Ges Zv‡`i j.mv.¸
660 n‡j, msL¨v wZbwUi M.mv.¸ KZ?
5 4
3 11 DËi: N
g‡bKwi, msL¨v wZbwUi M.mv.¸ = x
Ges msL¨v wZbwU = 3x, 4x I 5x
∴ msL¨v·qi j.mv.¸ = x 3x, 4x, 5x
3, 4, 5
= x345 = 60x
kZ©g‡Z, 60x = 660
ev, x =
60
660
= 11
myZivs, msL¨v wZbwUi M.mv.¸ = 11|
91. wZbwU msL¨vi AbycvZ 3 : 4 : 5 Ges Zv‡`i j.mv.¸
240| G‡`i M.mv.¸ KZ?
8 7
6 4 DËi: N
kU©KvUt msL¨v¸‡jvi M.mv.¸ =
5
4
3
240


4
6
240


o
|
 (N) `ywU msL¨vi AbycvZ I M.mv.¸/j.mv.¸ †_‡K msL¨v `ywU wbY©q, ÿz`ªZg ev e„nËg msL¨v wbY©q
†R‡b wbb - 35
 msL¨v `ywUi AbycvZ‡K ¸.mv.¸ Øviv ¸Y Ki‡j msL¨v `ywU cvIqv hvq|
92. `ywU msL¨vi AbycvZ 7 : 9 Ges msL¨v `ywUi M.mv.¸ 5
n‡j, ÿz`ªZg msL¨vwU KZ?
21 14
7 35 DËi: N
g‡b Kwi, ÿz`ªZg msL¨vwU = 7x I e„nËg msL¨vwU
= 9x Ges msL¨v `ywUi M.mv.¸ = x
cÖkœg‡Z, x = 5
(M.mv.¸ x Gi mv‡_ `v‡Mi M.mv.¸ 5 Gi Zzjbv Kiv n‡q‡Q)
myZivs, ÿz`ªZg msL¨vwU = 7x = 7  5 = 35|
 kU©KvU t `ywU msL¨vi AbycvZ‡K M.mv.¸ Øviv ¸Y
Ki‡j msL¨v `ywU cvIqv hvq| myZivs, msL¨v `ywU-
7 : 9
75 95
= 35 = 45
(ÿz`ªZg) (e„nËg)
mivmwi Abycv‡Zi ÿz`ªZg ivwkwU‡K M.mv.¸ 5 Øviv
¸Y Kiæb, 75 = 35 (ÿz`ªZg msL¨v)
93. wZbwU msL¨vi M.mv.¸ 12 Ges Zv‡`i AbycvZ 1 : 2 :
3 n‡j, msL¨v wZbwU nj-
6, 12, 18 10, 20, 30
12, 24, 36 24, 48, 72 DËi: M
g‡bKwi, msL¨v 3wU = x : 2x : 3x
Ges Zv‡`i M.mv.¸ = x |
cÖkœg‡Z, x = 12
myZivs, msL¨v 3 wU n‡”Q- x = 12, 2x = 212 =
24 I 3x = 312 = 36 |
 kU©KvUt 1 : 2 : 3
M.mv.¸ 12 Øviv ¸Y K‡i- 12 24 36
94. `ywU msL¨vi AbycvZ 5 : 6 Ges Zv‡`i j.mv.¸ 360
n‡j msL¨v `yÕwU wK wK ? AvbmviI wfwWwcAwa`߇iimv‡K©j
A¨vWRy‡U›U:05
45, 54 50, 60
60, 72 75, 90 DËi: M
g‡bKwi, †QvU msL¨vwU = 5x I eo msL¨vwU = 6x
(GLv‡b x n‡”Q msL¨v `ywUi M.mv.¸)
msL¨v `ywUi j.mv.¸ = x 5x, 6x
5, 6
= x56 = 30x
cÖkœg‡Z, 30x = 360
mgvavb
N
M
L
K
mgvavb
N
M
L
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mgvavb
N
M
L
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mgvavb
N
M
L
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mgvavb
N
M
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N
M
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N
M
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mgvavb

21. 2  Math Tutor
ev, x = 12
30
360

myZivs, msL¨v `ywU = 5x = 512 = 60
Ges 6x = 6 12 = 72|
 kU©KvUt j.mv.¸ †K AbycvZ `ywUi ¸Ydj Øviv fvM
Ki‡j M.mv.¸ cvIqv hvq|
A_©vr, M.mv.¸ =
30
360
6
5
360


12
 Ges M.mv.¸
Øviv Abycv‡Zi ivwk `ywU‡K ¸Y Ki‡j msL¨v `ywU
cvIqv hvq| msL¨v `ywU n‡”Q-
5 : 6
512 612
= 60 = 72
 msL¨v `ywU = 60 I 72|
95. `ywU msL¨vi AbycvZ 3 : 4 Ges msL¨v `ywUi j.mv.¸
84 n‡j, eo msL¨vwU KZ?
21 24
28 84 DËi: M
msL¨v `ywUi M.mv.¸ = 7
4
3
84


Ges msL¨v
`ywU n‡”Q- 3 : 4
37 47
= 21 = 28
(†QvU) (eo)
 eo msL¨vwU = 28|
 Abycv‡Zi eo ivwkwU‡K M.mv.¸ 7 Øviv ¸Y Ki‡jB
mivmwi eo msL¨vwU cvIqv hv‡e|
 eo msL¨v = 47 = 28|
96. wZbwU msL¨vi AbycvZ 3 : 4 : 5 Ges Zv‡`i j.mv.¸
1200 n‡j, msL¨v wZbwU Kx Kx?
60 80
100 me KqwU DËi: N
msL¨v wZbwUi M.mv.¸ = 20
5
4
3
1200



Ges
msL¨v wZbwU n‡”Q- 3 : 4 : 5
M.mv.¸ 20 w`‡q ¸Y K‡i- 60 80 100
97. `yÕwU msL¨vi AbycvZ 5 : 8 Ges Zv‡`i j.mv.¸ 120
n‡j, msL¨v `ywU KZ? Rb¯^v¯’¨ cÖ‡KŠkj Awa`߇ii Dc-
mnKvix cÖ‡KŠkjx (wmwfj) : 15
25, 40 20, 32
15, 24 10, 16 DËi: M
 (O) cvuPwgkvwj
98. `ywU msL¨vi AbycvZ 2 : 3 Ges Zv‡`i M.mv.¸ I
j.mv.¸-Gi ¸Ydj 33750 n‡j, msL¨v `ywUi
†hvMdj KZ?
250 425
325 375 DËi: N
g‡bKwi, msL¨v `ywU = 2x I 3x
∴ msL¨v `ywUi ¸Ydj = 2x3x = 6x2
(msL¨v`ywUi ¸Ydj = j. mv. ¸  M. mv. ¸ m~Îvbymv‡i
kZ©g‡Z wjL‡Z cvwi|)
kZ©g‡Z, 6x2
= 33750
ev, x2
=
6
33750
ev, x2
= 5625
∴ x = 5625 = 75
 msL¨v `ywUi †hvMdj = 2x + 3x
= 5x = 5  75 = 375
99. `ywU msL¨vi M.mv.¸ 2 Ges Zv‡`i j.mv.¸ 70 n‡j,
msL¨v `ywU wK wK?
14, 10 2, 35
6, 70 4, 70 DËi: K
g‡bKwi, msL¨v `ywU = 2x I 2y
(msL¨v؇qi gv‡S 2 n‡”Q M.mv.¸ 2)
 j. mv. ¸ = 2 2x, 2y = 2xy = 2xy
x , y
cÖkœg‡Z, 2xy = 70
ev, xy = 35
2
70

xy = 35 †_‡K x I y Gi gvb n‡Z cv‡i-
1  35 = 35  1 I 35 A_ev
5  7 = 35  5 I 7
GLb, x I y Gi gvb 1 I 35 n‡j, msL¨v `ywU n‡e,
2 1 = 2 I 2 35 = 70
A_ev, x I y Gi gvb 5 I 7 n‡j, msL¨v `ywU n‡e,
2 5 = 10 I 27 = 14
†h‡nZz Ack‡b 14 I 10 Av‡Q †m‡nZz msL¨v `ywU
mgvavb
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M
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mgvavb
N
M
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N
M
L
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mgvavb
N
M
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mgvavb
N
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22. 2  Math Tutor
n‡e 14, 10| (DËi)
 Option Test:
msL¨v`ywUi j.mv.¸  M.mv.¸ = msL¨v؇qi ¸Ydj
ev, 70  2 = 140
(cÖgvY Ki‡Z n‡e Ack‡bi msL¨v؇qi ¸Ydj = 140)
14, 10  14  10 = 140 (mwVK DËi)
2, 35  235 = 70 (mwVK bq) ...
100. `ywU msL¨vi mgwó 1000 Ges Zv‡`i j.mv.¸ 8919
n‡j msL¨v `ywU wK wK?
993, 7 989, 11
987, 13 991, 9 DËi: N
g‡bKwi, msL¨v `ywU = x I y, mgwó = x + y Ges
j.mv.¸ = xy (msL¨vØq mn‡gŠwjK n‡j Zv‡`i
¸YdjB j.mv.¸)
GLv‡b, x + y = 1000 Ges xy = 8919
me KqwU Ack‡bi msL¨v؇qi †hvMdj 1000, ZvB
†h Ack‡bi msL¨v؇qi ¸Ydj 8919 †mwUB DËi
n‡e| ïay Ackb Gi msL¨v؇qi ¸Ydj wg‡j
hv‡”Q- 9919 = 8919|
101. `ywU msL¨vi M.mv.¸ , mgwó I j.mv.¸ h_vµ‡g 36,
252 I 432| msL¨v `ywU wbY©q Ki| ? beg-
`kg †kÖwY MwYZ, (1983 ms¯‹iY) Abykxjbx 1.1 Gi 21 bs cÖk
g‡b Kwi, msL¨v `ywU 36x I 36y (x I y mn‡gŠwjK)
kZ©vbymv‡i, 36x + 36y = 252
ev, 36 (x + y) = 252
 x + y =
36
252
= 7
Avevi, 36x I 36y Gi j.mv.¸ = 36xy
kZv©bymv‡i, 36xy = 432
 xy =
36
432
= 12
GLv‡b, xy =12 †_‡K x I y Gi †h †h gvb n‡Z cv‡i-
xy = 1  12, 2 6 , 34 |
†h‡nZz, x + y = 7, †m‡nZz x = 3 I y = 4 n‡e|
 msL¨vØq = 36x = 363 = 108 I
36y = 364 = 144
 g‡b ivLyb: Dˇii w`‡K jÿ¨ K‡i †`Lyb, msL¨v`ywU
M.mv.¸ 36 Gi ¸wYZK| me mgq g‡b ivL‡eb, cÖ‡kœ
M.mv.¸ _vK‡j msL¨vØq memgq M.mv.¸Õi ¸wYZK
n‡e|
102. `ywU msL¨vi M.mv.¸ 15 Ges j.mv.¸ 180| msL¨v
`ywUi mgwó 105 n‡j, msL¨v `ywU wK wK?
30, 75 35, 70
45, 60 40, 65 DËi: M
msL¨v`ywU Aek¨B M.mv.¸ 15 Gi ¸wYZK n‡e, GB
`„wó‡KvY †_‡K Ackb I ev` hv‡e| evKx I
Ack‡bi gv‡S Gi msL¨v`ywUi j.mv.¸ 180, hv
cÖ‡kœ D‡jøwLZ msL¨v؇qi j.mv.¸ 180 Gi mv‡_ wg‡j
hvq|
103. `ywU msL¨vi M.mv.¸ I j.mv.¸ h_vµ‡g 12 I 72|
msL¨v `ywUi mgwó 60 n‡j, Zv‡`i g‡a¨ GKwU nj-
12 60
24 72 DËi: M
meKqwU Ackb M.mv.¸ 12 Gi ¸wYZK| ZvB
evKx kZ©¸‡jv hvPvB K‡i †`L‡Z n‡e| msL¨v `ywUi
mgwó 60 Abymv‡i, `ywU msL¨vi GKwUB hw` 60 ev 72
nq, Znv‡j Ackb I ev` c‡i hv‡”Q|
 Ackb K Abymv‡i, GKwU msL¨v 12 n‡j, AciwU
60 -12 = 48, hv‡`i j.mv.¸ 48 nIqvq ev`|
 Ackb M Abymv‡i, GKwU msL¨v 24 n‡j, AciwU
60 - 24 = 36, hv‡`i j.mv.¸ 72 nIqvq GwUB
mwVK DËi|
104. `ywU msL¨vi j.mv.¸ I M.mv.¸ Gi ¸Ydj 32| msL¨v
`ywUi AšÍi 14 n‡j, eo msL¨vwU KZ?
4 2
18 16 DËi: N
g‡bKwi, `ywU msL¨v x I y, G‡`i ¸Ydj = xy
Ges AšÍi = x - y (†hLv‡b x > y)
Avgiv Rvwb, `ywU msL¨vi ¸Ydj = M.mv.¸ I
j.mv.¸Õi ¸Ydj, Zvn‡j xy = 32 ej‡Z cvwi|
xy = 32 †_‡K 321, 162 I 84 msL¨v
hyMj cvIqv hvq| cÖkœvbymv‡i, †h‡nZz msL¨v `ywUi
AšÍi 14 n‡e, †m‡nZz 162 msL¨vhyMjwU mwVK|
 eo msL¨vwU = 16|
105. `ywU msL¨vi M.mv.¸, AšÍi I j.mv.¸ h_vµ‡g 12, 60
I 2448| msL¨v `ywU wbY©q Ki| beg-`kg †kÖwY MwYZ,
(1983 ms¯‹iY) Abykxjbx 1.1 Gi 22 bs cÖk , 17Zg wewmGm;
15Zg wkÿK wbeÜb (¯‹zj ch©vq-2)
104, 204 104, 144
104, 244 144, 204 DËi: N
g‡b Kwi, msL¨v `ywU 12x I 12y (†hLv‡b x > y
mgvavb
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M
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mgvavb
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M
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L
mgvavb
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M
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mgvavb
N
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mgvavb
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mgvavb
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L
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23. Math Tutor  3
Ges x, y mn‡gŠwjK )
kZ©vbymv‡i, 12x - 12y = 60
ev, 12 (x - y) = 60
 x - y =
12
60
= 5 |
Avevi, 12x I 12y Gi j.mv.¸ = 12xy
kZv©bymv‡i, 12xy = 2448
 xy =
12
2448
= 204
GLv‡b, xy = 204 I x - y = 5 Abymv‡i x I y Gi
gvb wbY©q Kiv hvK-
2 204
2 102
3 51
17
x y = 1712 A_©vr, x = 17 Ges y = 12 msL¨v
`ywU wbY©xZ nj|
 msL¨vØq = 12x = 1217 = 204 I
12y = 1212 = 144
 kU©KvUt msL¨v`ywU cÖ`Ë M.mv.¸ 12 Gi ¸wYZK n‡Z
n‡e| wKš‘ Ackb , I Gi 104, 12 Gi
¸wYZK bq, ZvB GB Ackb wZbwU ev` hv‡e| Ackb
Gi 144 I 204, 12 Gi ¸wYZK, ZvB GwUB
mwVK DËi|
 msL¨v؇qi AšÍidj 60 wKbv GB kZ©wU Ackb †_‡K
hvPvB K‡i DËi †ei Kiv hvq| Ackb¸‡jvi gv‡S
ïay Gi msL¨v؇qi AšÍi 60 n‡e|
106. `ywU msL¨vi M.mv.¸, AšÍi I j.mv.¸ h_vµ‡g 12,
60 I 2448| msL¨v `ywU wbY©q Kiæb| 33 Zg wewmGm
wjwLZ
105 bs cÖ‡kœi mgvavb †`Lyb|
107. `ywU msL¨vi M.mv.¸ 21 Ges j.mv.¸ 4641| GKwU
msL¨v 200 I 300 Gi ga¨eZx©; AciwU KZ? beg-
`kg †kÖwY MwYZ, (1983 ms¯‹iY) Abykxjbx 1.1 Gi 39 bs cÖkœ ;
34 Zg wewmGm wjwLZ; 20Zg wewmGm wjwLZ
g‡b Kwi, msL¨v `ywU 21x I 21y
(GLv‡b x I y ci¯úi mn‡gŠwjK)
21x I 21y Gi j.mv.¸ = 21xy
cÖkœg‡Z, 21xy = 4641
 xy =
21
4641
= 221
xy = 221 †_‡K x I y Gi msL¨vhyMj¸‡jv †ei Kiv
hvK-
13 221
17
xy = 1  221
xy = 13  17
x I y mn‡gŠwjK nIqvq x = 1, y = 221 I x =
13, y = 17 Dfq gvb wVK Av‡Q| wKš‘ ÔmsL¨vwU
200 I 300 Gi ga¨eZx© n‡Z n‡eÕ GB k‡Z© x =
13, y = 17 gvb mwVK|
AZGe, wb‡Y©q msL¨v؇qi GKwU = 21x = 2113
= 273 I AciwU = 21y = 2117 = 357|
DËi: Aci msL¨vwU 357|
108. Qq A‡¼i ÿz`ªZg msL¨vi mv‡_ †Kvb ÿz`ªZg msL¨v
†hvM Ki‡j mgwó 2, 4, 6, 8, 10 I 12 Øviv wefvR¨
n‡e? cywjk mv‡R©›U wb‡qvM cixÿv 1997
43 bs cÖ‡kœi mgvavb †`Lyb|
109. cuvP A‡¼i e„nËg msL¨vi mv‡_ †Kvb ÿz`ªZg msL¨v
†hvM Ki‡j †hvMdj 5, 8, 12 I 14 Øviv wefvR¨?
ewnivMgb I cvm‡cvU© Awa`߇ii Awdm mnKvix Kvg
Kw¤úDUvi gy`ªvÿwiK 2013
5, 8, 12 I 14-Gi j. mv. ¸ wbY©q Kwi|
2 5, 8, 12, 14
2 5, 4, 6, 7
5, 2, 3, 7
 j.mv.¸ = 225237 = 840
cvuP A‡¼i e„nËg msL¨v = 99999
840) 99999 ( 119
840
1599
840
7599
7560
39
mgvavb
mgvavb
mgvavb
mgvavb
N
N
M
L
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ïay wjwLZ Av‡jvPbv
 

24. 2  Math Tutor
AZGe, wb‡Y©q ÿz`ªZg msL¨v = 840 - 39 = 801|
(DËi)
110. cvuP A‡¼i e„nËg msL¨vi m‡½ †Kvb ÿz`ªZg msL¨v
†hvM Ki‡j †hvMdj 6, 8, 10 I 14 Øviv wefvR¨
n‡e? 10g wewmGm wjwLZ
109 bs mgvav‡bi Abyiƒc | wWwR‡Ui cv_©K¨ †`Lv
†M‡jI j.mv.¸ I DËi GKB n‡e A_©vr DËi 801|
111. GKwU ¯‹z‡j wWªj Kivi mgq 8, 10 ev 12 wU jvBb Kiv
hvq| H ¯‹z‡j AšÍZ c‡ÿ KZ Rb QvÎQvÎx wQj? RvZxq
wbivcËv †Mv‡q›`v (NSI) 2019
8, 10 Ges 12 Gi j.mv.¸-B n‡e b~¨bZg QvÎQvÎx
msL¨v|
2 8, 10, 12
2 4, 5, 6
2, 5, 3
 8, 10 Ges 12 Gi j.mv.¸ = 22253
= 120
AZGe, H ¯‹z‡j QvÎ-QvÎx msL¨v 120 Rb|
112. †Kvb •mb¨`‡ji •mb¨‡K 8, 10 ev 12 mvwi‡Z Ges
eM©vKv‡iI mvRv‡bv hvq| †mB •mb¨`‡ji ÿz`ªZg msL¨vwU
wbY©q Kiæb hv Pvi A¼wewkó| 15 Zg wewmGm wjwLZ
2 8, 10, 12
2 4, 5, 6
2, 5, 3
∴ j.mv.¸ = 222 5 3 = 120
Drcv`K Abymv‡i 120 Rb •mb¨‡K eM©vKv‡i mvRv‡bv
hv‡e bv, ZvQvov msL¨vwU Pvi A¼wewkóI bv| GRb¨
120 †K c~Y©eM© Kivi Rb¨ (253) Øviv ¸Y
Ki‡Z n‡e|
c~Y©eM© msL¨vwU = 1202 5 3 = 3600|
Gevi cÖvß c~Y©eM© msL¨vwU‡K eM©vKv‡i mvRv‡bv hv‡e
Ges GwU GKwU Pvi A¼wewkó msL¨v|
myZivs, wb‡Y©q •mb¨ msL¨v 3600| (DËi)
113. 400 I 500-Gi ga¨eZx© †Kvb †Kvb msL¨v‡K 12, 15 I
20 Øviv fvM w`‡j cÖwZ †ÿ‡Î 10 Aewkó _v‡K? beg-`kg
†kÖwY MwYZ, (1983 ms¯‹iY) Abykxjbx 1.1 Gi 28 bs cÖkœ
12, 15 I 20 Gi j.mv.¸ wbY©q Ki‡Z n‡e|
2 12, 15, 20
2 6, 15, 10
3 3, 15, 5
5 1, 5, 5
1, 1, 1
j.mv.¸ = 2235 = 60|
myZivs wb‡Y©q msL¨v 60 Gi ¸wYZK + 10 n‡e Ges
Dnv 400 I 500 Gi ga¨eZx© n‡e|
601 + 10 = 70, 400 I 500 Gi ga¨eZx© bq
602 + 10 = 130, 400 I 500 Gi ga¨eZx© bq
603 + 10 = 190, 400 I 500 Gi ga¨eZx© bq
604 + 10 = 250, 400 I 500 Gi ga¨eZx© bq
605 + 10 = 310, 400 I 500 Gi ga¨eZx© bq
606 + 10 = 370, 400 I 500 Gi ga¨eZx© bq
607 + 10 = 430, 400 I 500 Gi ga¨eZx©
608 + 10 = 490, 400 I 500 Gi ga¨eZx©
AZGe, msL¨v `ywU 430 I 490|
114. †Kvb ÿz`ªZg msL¨v‡K 3, 4, 5, 6 I 7 Øviv fvM
Ki‡j cÖwZ‡ÿ‡Î 1 Aewkó _v‡K wKš‘ 11 Øviv fvM
w`‡j †Kvb Aewkó _v‡K bv? beg-`kg †kÖwY MwYZ, (1983
ms¯‹iY) Abykxjbx 1.1 Gi 35 bs cÖkœ
2 3, 4, 5, 6, 7
3 3, 2, 5, 3, 7
1, 2, 5, 1, 7
 j.mv.¸ = 23257 = 420|
myZivs wb‡Y©q ÿz`ªZg msL¨v 420 Gi ¸wYZK + 1
n‡e Ges Dnv 11 Øviv wefvR¨ n‡e|
420  1 + 1 = 421, 11 Øviv wefvR¨ bq
420  2 + 1 = 841, 11 Øviv wefvR¨ bq
420  3 + 1 = 1261, 11 Øviv wefvR¨ bq
420  4 + 1 = 1681, 11 Øviv wefvR¨ bq
420  5 + 1 = 2101, 11 Øviv wefvR¨|
AZGe, wb‡Y©q msL¨v 2101 |
115. KZK¸‡jv PvivMvQ cÖwZ mvwi‡Z 3, 5, 6, 8, 10 I
12 wU K‡i jvMv‡Z wM‡q †`Lv †Mj †h cÖwZev‡i 2 wU
Pviv evwK _v‡K wKš‘ cÖwZ mvwi‡Z 19wU K‡i jvMv‡j
GKwU PvivI Aewkó _v‡K bv| b~¨bZg KZK¸‡jv
PvivMvQ wQj? beg-`kg †kÖwY MwYZ, (1983 ms¯‹iY)
Abykxjbx 1.1 Gi 28 bs cÖkœ
114 bs Gi Abyiƒc |
j.mv.¸ = 120
----------------
----------------
1206 + 2 = 722, 19 Øviv wefvR¨ n‡e|
 wb‡Y©q PvivMv‡Qi msL¨v 722 wU| (DËi)
116. †gvUvgywU GK nvRvi wjPz _vKvi K_v, Ggb GK Szwo
wjPz 80 Rb evj‡Ki g‡a¨ fvM Ki‡Z wM‡q †`Lv †Mj
†h 30wU wjPz DØ„Ë _v‡K; wKš‘ evj‡Ki msL¨v 90 n‡j
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25. Math Tutor  3
wjPz¸‡jv mgvb fv‡M fvM Kiv †hZ| SzwowU‡Z
cÖK…Zc‡ÿ KZwU wjPz wQj? 11Zg wewmGm wjwLZ
wjPzi msL¨v n‡e 80 Gi ¸wYZK + 30 Ges Dnv
90 Øviv wefvR¨ n‡e|
80  1 + 30 = 110 , 90 Øviv wefvR¨ bq
80  2 + 30 = 190, 90 Øviv wefvR¨ bq
80  3 + 30 = 270, 90 Øviv wefvR¨ bq
80  4 + 30 = 350, 90 Øviv wefvR¨ bq
80  5 + 30 = 430, 90 Øviv wefvR¨ bq
80  6 + 30 = 510, 90 Øviv wefvR¨ bq
80  7 + 30 = 590, 90 Øviv wefvR¨ bq
80  8 + 30 = 670, 90 Øviv wefvR¨ bq
80  9 + 30 = 750, 90 Øviv wefvR¨ bq
80  10 + 30 = 830, 90 Øviv wefvR¨ bq
80  11 + 30 = 910, 90 Øviv wefvR¨ bq
80  12 + 30 = 990, 90 Øviv wefvR¨|
 wb‡Y©q wjPzi msL¨v = 990 wU|
117. ÿz`ªZg msL¨vwU wbY©q Kiæb hvnv 13 Øviv wefvR¨ wKš‘
4, 5, 6 I 9 Øviv fvM Ki‡j cÖwZ‡ÿ‡Î 1 Aewkó
_v‡K| 29 Zg wewmGm wjwLZ
114 bs Gi Abyiƒc| DËi: 1261
118. 13 Øviv wefvR¨ ÿz`ªZg †Kvb msL¨v‡K 3, 4, 5, 6 I
7 Øviv fvM Ki‡j h_vµ‡g 1, 2, 3, 4 I 5 Aewkó
_v‡K? beg-`kg †kÖwY MwYZ, (1983 ms¯‹iY) Abykxjbx 1.1
Gi 36 bs cÖkœ
cÖ`Ë fvRK I fvM‡k‡li ga¨Kvi e¨eavb-
3 - 1 = 2
4 - 2 = 2
5 - 3 = 2
6 - 4 = 2
7 - 5 = 2
3, 4, 5, 6 I 7 Gi j.mv.¸ wbY©q Kiv hvK-
2 3, 4, 5, 6, 7
3 3, 2, 5, 3, 7
1, 2, 5, 1, 7
 j.mv.¸ = 23257 = 420|
myZivs, wb‡Y©q ÿz`ªZg msL¨vwU n‡e 420 Gi ¸wYZK
 2 Ges Dnv 13 Øviv wefvR¨|
4201  2 = 418, 13 Øviv wefvR¨ bq
4202  2 = 838, 13 Øviv wefvR¨ bq
4203  2 = 1258, 13 Øviv wefvR¨ bq
420 4  2 = 1678, 13 Øviv wefvR¨ bq
420 5  2 = 2098, 13 Øviv wefvR¨ bq
420 6  2 = 2518, 13 Øviv wefvR¨ bq
4207  2 = 2938, 13 Øviv wefvR¨ |
AZGe, wb‡Y©q msL¨v 2938|
119. Qq A‡¼i ÿz`ªZg msL¨v wbY©q Kiæb hv‡K 5, 7, 12,
15 Øviv fvM Ki‡j Aewkó h_vµ‡g 2, 4, 9, 12
_vK‡e| 25Zg wewmGm wjwLZ
cÖ`Ë fvRK I fvM‡k‡li ga¨Kvi e¨eavb-
5 - 2 = 3
7 - 4 = 3
12 - 9 = 3
15 - 12 = 3
5, 7, 12 I 15 Gi j.mv.¸ wbY©q Kiv hvK-
5 5, 7, 12, 15
3 1, 7, 12, 3
1, 7, 4, 1
 j.mv.¸ = 5374 = 420|
Qq A‡¼i ÿz`ªZg msL¨v = 100000
420) 100000 ( 238
840
1600
1260
3400
3360
40
fvRK I fvM‡k‡li cv_©K¨ = 420 - 40 = 380
 wb‡Y©q msL¨v = 100000 + 380 - 3
= 100377 (DËi)
 wba©vwiZ A‡¼i ÿz`ªZg msL¨v‡K wbt‡k‡l wefvR¨
Ki‡Z ÔfvRK I fvM‡kl Gi cv_©K¨Õ †hvM Ki‡Z
nq|
 fvRKmg~n I Zv‡`i cÖ‡Z¨‡Ki fvM‡kl _vK‡j
wbt‡k‡l wefvR¨ msL¨v †_‡K †mwU we‡qvM Ki‡Z nq|
 cÖ`Ë A‡¼ cÖ_‡g Qq A‡¼i ÿz`ªZg msL¨v‡K wbt‡k‡l
wefvR¨ Kiv n‡q‡Q Ges c‡i 3 we‡qvM K‡i cÖwZwU
fvR‡Ki h_vµ‡g Aewkó ivLvi kZ© c~iY Kiv n‡q‡Q|
120. GKwU AvqZvKvi N‡ii •`N©¨ 30 wgUvi, cÖ¯’ 12
wgUvi| Av‡iKwU AvqZvKvi nj N‡ii •`N©¨ 20 wgUvi
I cÖ¯’ 15 wgUvi| me‡P‡q eo †Kvb AvqZ‡bi UzKiv
w`‡q Dfq N‡ii †g‡S cy‡ivcywi †X‡K †djv hv‡e?
beg-`kg †kÖwY MwYZ, (1983 ms¯‹iY) Abykxjbx 1.1 Gi 36 bs
cÖkœ ; 11Zg wewmGm
1g N‡ii †ÿÎdj = 30 wgUvi  12 wgUvi
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26. 4  Math Tutor
= 360 eM©wgUvi
2q N‡ii †ÿÎdj = 20 wgUvi  15 wgUvi
= 300 eM©wgUvi
360 I 300 Gi M.mv.¸ wbY©q Ki‡Z n‡e| wb‡Y©q
M.mv.¸ n‡e me‡P‡q eo AvqZ‡bi Kv‡Vi UzKiv|
300) 360 ( 1
300
60) 300 ( 5
300
0
M.mv.¸ = 60 |
AZGe, wb‡Y©q Kv‡Vi UzKivi AvqZb = 60 eM©wgUvi
1g N‡i Kv‡Vi msL¨v = 360  60 = 6 wU
2q N‡i Kv‡Vi msL¨v = 300  60 = 5 wU
†gvU Kv‡Vi msL¨v = 5 + 6 = 11 wU|
121. `ywU msL¨vi AbycvZ 3 : 4 Ges Zv‡`i j.mv.¸ 180
n‡j msL¨v `ywU wbY©q Kiæb| ewnivMgb I cvm‡cvU©
Awa`߇ii Awdm mnKvix Kvg Kw¤úDUvi gy`ªvÿwiK 2013
g‡b Kwi, msL¨v `ywU = 3x I 4x
Ges G‡`i j.mv.¸ = 3x  4x = 12x
kZ©g‡Z, 12x = 180
 x =
12
180
= 15|
myZivs, msL¨v `ywU h_vµ‡g 3x = 3  15 = 45
Ges 4x = 4 15 = 60| (DËi)
122. `ywU msL¨vi AbycvZ 5 : 7 Ges Zv‡`i M.mv.¸ 4 n‡j
msL¨v `ywUi j.mv.¸ KZ? weÁvb I cÖhyw³ gš¿Yvj‡qi
mvuU gy`ªvÿwiK Kvg-Kw¤úDUvi Acv‡iUi/Awdm mnKvix
Kvg-Kw¤úDUvi gy`ªvÿwiK 2017
g‡b Kwi, msL¨v `ywU = 5x I 7x Ges G‡`i
M.mv.¸ = x
kZ©g‡Z, x = 4
 msL¨v `ywU h_vµ‡g 5x = 54 = 20
Ges 7x = 74 = 28|
GLb, 20 I 28 Gi j.mv.¸ wbY©q Ki‡Z n‡e|
2 20, 28
2 10, 14
5, 7
wb‡Y©q j.mv.¸ = 2257 = 140| (DËi)
123. `ywU msL¨vi ¸Ydj 3380, Gi M.mv.¸ 13, msL¨v
`ywUi j.mv.¸ KZ? evsjv‡`k †ijI‡qi mnKvix
†÷kb gv÷vi 2016
Avgiv Rvwb,
msL¨v `ywUi ¸Ydj = msL¨v `ywUi j.mv.¸  M.mv.¸
ev, 3380 = msL¨v `ywUi j.mv.¸  13
 msL¨v `ywUi j.mv.¸ =
13
3380
= 260| (DËi)
 wet `ªt avivevwnK †cv÷¸‡jv‡Z †cR bv¤^vi wVK †bB
Ges cÖwZwU Aa¨v‡qi web¨vm g~j eB‡q wKQzUv cwieZ©b
I cwiea©b n‡q hy³ n‡e|
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