05. average full chapter (math tutor by kabial noor [www.itmona.com]

Math Tutor  1
CHAPTER- 05
ga¨g msL¨v, Mo I mgwó
05.01 µwgK †e‡Rvo, µwgK †Rvo I µwgK msL¨vi †hvMdj
†R‡b wbb - 35
 µwgK †e‡Rvo msL¨vi †hvMdj †ei Kivi wbqg n‡”Q, †h KqwU µwgK †e‡Rvo msL¨vi †hvMdj †ei Ki‡Z n‡e, H
msLvi mv‡_ H msL¨v‡KB ¸Y K‡i ev H msL¨v‡K eM© Ki‡jB †hvMdj cvIqv hvq| †hgb- cÖ_g 4wU µwgK †e‡Rvo
msL¨vi †hvMdj KZ? mnR DËi n‡”Q- 4 4 = 42
= 16| Giƒc nIqvi KviY wb‡Pi D`vniY¸‡jv †`‡L wbb-
 4 wU µwgK †e‡Rvo msL¨vi †hvMdj = 1 + 3 + 5 + 7 = 16 , hv 4 Gi eM©|
 5 wU µgwK †e‡Rvo msL¨vi †hvMdj = 1 + 3 + 5 + 7 + 9 = 25, hv 5 Gi eM© |
GKBfv‡e, n msL¨K µwgK ¯^vfvweK †e‡Rvo msL¨vi †hvMdj n‡e n × n ev, n Gi eM© = n2
.
 µwgK †Rvo msL¨vi †hvMdj †ei Kivi wbqg n‡”Q, †h KqwU µwgK †Rvo msL¨vi †hvMdj †ei Ki‡Z n‡e, H
msL¨vi mv‡_ Zvi c‡ii msL¨v ¸Y Ki‡Z n‡e| †hgb- 5 wU µwgK †Rvo msL¨vi †hvMdj KZ? mnR DËi n‡”Q,
5  6 = 30| Giƒc nIqvi KviY wb‡Pi D`vnviY¸‡jv †`‡L wbb-
 5 wU µgwK †Rvo msL¨vi †hvMdj = 2 + 4 + 6 + 8 + 10 = 30, hv 5 × 6 = 30
 6 wU µwgK †Rvo msL¨vi †hvMdj = 2 + 4 + 6 + 8 + 10 + 12 = 42, hv 6 × 7 = 42
GKBfv‡e, n msL¨K µwgK †Rvo msL¨vi †hvMdj n‡e- n Gi mv‡_ Zvi c‡ii msL¨v n + 1 Gi ¸Y A_©vr, n 
(n + 1) = n (n + 1)
 µwgK †Rvo msL¨vi †hvMdj n (n + 1) †K 2 Øviv fvM Ki‡j µwgK msL¨vi †hvMdj †ei nq| †hgb- 7 wU µwgK
†Rvo msL¨vi †hvMdj = 78 = 56| Gevi 56 †K 2 Øviv fvM Ki‡j 7 wU µwgK msL¨vi †hvMdj †ei n‡e|
A_©vr, 7 wU µwgK msL¨vi †hvMdj = 56  2 = 28| Zvn‡j µwgK msL¨vi †hvMdj †ei Kivi m~ÎwU `vov‡”Q-
2
)1n(n 
|
 †h †Kvb msL¨K µwgK msL¨vi ¸Yd‡ji mv‡_ 1 †hvM Ki‡j c~Y©eM© msL¨v cvIqv hvq| †hgb- 4 wU µwgK msL¨vi
¸Yd‡ji mv‡_ 1 †hvM Kiv hvK- 1 × 2 × 3 × 4 = 24 + 1 = 25| cÖvß 25 GKwU c~Y© eM© msL¨v| (4bs cÖkœ †`Lyb)
 m~Î¸‡jv gyL¯Í bv K‡i ev¯Í‡e cÖ‡qvM K‡i †`Lyb Zvn‡j mn‡RB g‡b _vK‡e|
01. x msL¨K µwgK ¯^vfvweK †e‡Rvo msL¨vi †hvMdj
KZ? mnKvixcwievicwiKíbvKg©KZ©v:16
x 2x
𝑥2
2x + 1 DËi: M
02. cÖ_g `kwU we‡Rvo msL¨vi †hvMdj KZ? RbcÖkvmb
gš¿Yvj‡qiAax‡bPSC-GimnKvixcwiPvjK:16
100 81
1000 109 DËi: K
cÖ_g 10wU we‡Rvo msL¨vi †hvMdj= 102
= 100|
03. cÖ_g 6wU ¯^vfvweK msL¨vi mgwó KZ? mvaviYexgv
K‡c©v‡ikbmnKvixe¨e¯’vcKc‡`wb‡qvMcixÿ:16
15 18
21 24 DËi: M
6 Gi mv‡_ Zvi c‡ii msL¨v 7 ¸Y Kiæb Gfv‡e-
6 × 7 = 42 Ges 42 †K 2 Øviv fvM Kiæb |
A_©vr, 42 ÷ 2 = 21|
04. a, b, c, d PviwU µwgK ¯^vfvweK msL¨v n‡j wb‡Pi
†KvbwU c~Y©eM© msL¨v? ICB CapitalManagement
Ltd.-2019
abcd ab + cd
abcd + 1 abcd - 1 DËi: MNM
LK
mgvavb
NM
LK
mgvavb
NM
LK
NM
LK
†ewmK, GgwmwKD I wjwLZ Av‡jvPbv 

2  Math Tutor
05.02 ga¨g msL¨v wbY©q
†R‡b wbb -36
 ga¨g msL¨v (middle number) : aiæb, 1, 2, 3 GLv‡b 3wU µwgK msL¨v (consecutive number) Av‡Q,
G‡ì gv‡S 2 n‡”Q ga¨g msL¨v (middle number)| A_©vr, we‡Rvo msL¨K µwgK msL¨vi Mo ev gv‡SiwU n‡”Q
ga¨g msL¨v (middle number)|
 †e‡Rvo msL¨K µwgK msL¨vi †hvMdj †_‡K ga¨g msL¨v wbY©q Kivi wbqg: aiæb, 1, 2, 3 µwgK msL¨v wZbwUi
†hvMdj (1+2+3) n‡”Q 6| Avcbv‡K ejv nj, 3wU µwgK msL¨vi †hvMdj 6 n‡j gaëZx©/ga¨g msL¨v KZ?
G‡ÿ‡Î Lye mn‡RB Avcwb ga¨g msL¨vwU †ei Ki‡Z cv‡ib| wKfv‡e? Rv÷, †h KqwU msL¨vi †hvMdj ejv n‡q‡Q,
†mwU w`‡q †hvMdj‡K fvM Kiæb| A_©vr, 3 wU µwgK msL¨vi †hvMdj 6 n‡j,
ga¨g msL¨v =
c`msL¨v
hvMdj/mgwó†
=
3
6
= 2|
AviI †`Lyb-
 3 wU µwgK msL¨vi †hvMdj 12 n‡j, Zv‡ì gaëZx©/ga¨g msL¨vwU KZ? DËi: 12 ÷ 3 = 4 (ga¨g msL¨v)
 5 wU µwgK msL¨vi †hvMdj 25 n‡j, Zv‡ì gaëZx©/ga¨g msL¨vwU KZ? DËi: 25 ÷ 5 = 5(ga¨g msL¨v)
 †e‡Rvo msL¨K µwgK †Rvo I µwgK †e‡Rvo msL¨vi †hvMdj †_‡K ga¨g msL¨v wbY©q: †e‡Rvo msL¨K µwgK
msL¨vi ga¨g msL¨v †ei Kivi †h wbqg, †e‡Rvo msL¨K µwgK †Rvo msL¨v I µwgK †e‡Rvo msL¨vi ga¨g msL¨v
†ei KiviI GKB wbqg|
 wZbwU µwgK †e‡Rvo msL¨vi †hvMdj 57| ga¨g msL¨vwU KZ?
DËi: 57 ÷ 3 = 19 (ga¨g msL¨v)|  18, 19 , 20
 cvuPwU µwgK †Rvo msL¨vi †hvMdj 40| ga¨g msL¨vwU KZ?
DËi: 40 ÷ 5 = 8 (ga¨g msL¨v)  4, 6, 8 , 10, 12
 ga¨g msL¨v †_‡K evKx msL¨v¸‡jv wbY©q Kivi wbqg:
 †e‡Rvo msL¨K µwgK msL¨vi †ÿ‡Î : ïiæ‡ZB a‡i wbw”Q 3wU µwgK msL¨vi ga¨g msL¨v 5| GLv‡b 5 Gi
ev‡g GKwU I Wv‡b GKwU msL¨v Av‡Q| ev‡giwU †ei Ki‡Z 5 †_‡K 1 we‡qvM Kie Ges Wv‡biwU †ei Ki‡Z 5
Gi mv‡_ 1 †hvM Kie A_©vr, 4  5  6
 5wU µwgK msL¨vi ga¨g msL¨v 3 n‡j, evKx msL¨v¸‡jv Kx Kx?
 ga¨g msL¨v †_‡K ev‡g †M‡j 1 K‡i n«vm Ki‡Z _vKe Ges Wv‡b †M‡j 1 K‡i e„w× Ki‡Z _vKe|
12345
 †e‡Rvo msL¨K µwgK †Rvo I we‡Rvo msL¨vi †ÿ‡Î: †Rvo I †e‡Rv‡oi †ÿ‡Î ev‡g †M‡j 2 K‡i n«vm I Wv‡b
†M‡j 2 K‡i e„w× †c‡Z _vK‡e|
 5wU µwgK †Rvo msL¨vi ga¨g msL¨v 6 n‡j, evKx msL¨v¸‡jv wK wK?
 6 †_‡K ev‡g 2 K‡i n«vm I Wv‡b 2 K‡i e„w× Kiv n‡q‡Q- 246810
 7 wU µwgK we‡Rvo msL¨vi ga¨g msL¨v 21 n‡j, evKx msL¨v¸‡jv wK wK?
 15171921232527
05. wZbwU we‡Rvo µwgK msL¨vi †hvMdj 57 n‡j
gaëZ©x msL¨v KZ? evsjv‡`ke¨vsKmnKvixcwiPvjK:10
19 21
17 16 DËi: K
57 †K 3 Øviv fvM Ki‡jB ga¨g msL¨vwU cvIqv
hv‡e|  gaëZx© msL¨v = 57  3 = 19|
06. wZbwU µwgK msL¨vi †hvMdj 30, eowU I †QvUwUi
we‡qvMdj 2 n‡j †QvU msL¨vwU- †m‡KÛvixGWz‡Kkb†m±i
Bb‡f÷‡g›U†cÖvMÖvg,mn._vbvgva¨wgKwkÿvKg©KZ©vc‡`wbe©vPbxcixÿv:15
7 9
10 11 DËi: L
ga¨g msL¨v : 30 ÷ 3 = 10|  msL¨v¸‡jv =
91011| cÖ‡kœ ejv n‡q‡Q, eo msL¨v I †QvU
msL¨vi we‡qvMdj 2| A_©vr, 11 - 9 = 2|
mgvavb
NM
LK
mgvavb
NM
LK

Math Tutor  3
myZivs, †QvU msL¨vwU n‡”Q- 9|
07. wZbwU µwgK msL¨vi †hvMdj 240 n‡j, eo `ywUi
mgwó KZ? evsjv‡`ke¨vsKmnKvixcwiPvjK:09
79 159
169 161 DËi: N DËi: N
ga¨g msL¨v : 240  3 = 80|
 msL¨v¸‡jv = 798081
 eo `ywU msL¨vi †hvMdj = 80 + 81 = 161.
08. cvuPwU µwgK c~Y©msL¨vi †hvMdj n‡”Q 105| cÖ_g
`ywU msL¨vi †hvMdj n‡”Q- BangladeshCommercial
BankLtd.JuniorOfficer:08
39 21
19 41 DËi: K DËi: K
ga¨g msL¨v = 105 ÷ 5 = 21|
msL¨v¸‡jv = 1920212223
 cÖ_g `ywU msL¨vi †hvMdj = 19 + 20 = 39|
09. wZbwU µwgK msL¨vi †hvMdj 123| ÿz`ªZg msL¨v
`yBwUi ¸Ydj KZ? 12ZgwkÿKwbeÜb(¯‹zj/mgch©vq2)-15
625 1640
1600 900 DËi: L DËi: L
ga¨g msL¨v = 123  3 = 41|
msL¨v¸‡jv = 404142
 ÿ`ªZg msL¨v `ywUi ¸Ydj = 4041 = 1640
10. wZbwU µwgK msL¨vi †hvMdj 123| †QvU msL¨vwU
KZ? †m‡KÛvwiGWz‡Kkb†m±iBb‡f÷‡g›U†cÖvMÖvgDc‡Rjv,_vbv
GKv‡WwgKmycvifvBRvi15
30 45
40 49 DËi: M
11. 5wU µwgK msL¨vi †hvMdj 100 n‡j, cÖ_g msL¨v I
†kl msL¨vi ¸Ydj KZ? WvKI†Uwj‡hvMv‡hvMwefv‡MiWvK
Awa`ß‡iiwewìs Ifviwkqvi2018
246 242
396 484 DËi: M
12. 17wU µwgK †Rvo msL¨vi Mo 42| avivwU wb¤œZi
†_‡K D”PZi w`‡K mvRv‡bv n‡j Z…Zxq msL¨vwU KZ?
evsjv‡`ke¨vsKmnKvix cwiPvjK:11
28 29
30 34 DËi: M
avivwUi ga¨g msL¨v ev Mo n‡”Q 42|
msL¨v¸‡jv = 262830323436
38404244464850525
45658| wb¤œZi 26 †_‡K Z…Zxq †Rvo
msL¨vwU n‡”Q 30|
 kU©KvU: Mo 42 Gi Av‡M 8wU msL¨v Av‡Q, cÖwZwU 2
K‡i n«vm cv‡i| Zvigv‡b me‡P‡q ev‡gi msL¨v
msL¨vwU 82 = 16 Kg n‡e- 42 - 16 = 26|
Gevi 26 †_‡K Z…Zxq msL¨vwU n‡”Q 30|
13. cuvPwU avivevwnK we‡Rvo msL¨vi Mo 61| me‡P‡q
eo I †QvU msL¨vØ‡qi cv_©K¨ KZ?evsjv‡`ke¨vsKmnKvix
cwiPvjK(†Rbv‡ijmvBW):16
2 8
5 †Kv‡bvwUB bq DËi: L
ga¨g msL¨v ev Mo = 61|
msL¨vMy‡jv = 5759616365
 me‡P‡q eo I †QvU msL¨vØ‡qi cv_©K¨ =
65 - 57 = 8|
05.03 cici wKQzmsL¨K msL¨vi cÖ_g K‡qKwUi †hvMdj w`‡q c‡ii K‡qKwUi †hvMdj PvB‡j
†R‡b wbb - 37
 cici x msL¨K msL¨vi cÖ_g K‡qKwUi †hvMdj w`‡q c‡ii x msL¨K msL¨vi †hvMdj PvB‡j Avgiv GB kU©KvUwU
e¨envi Ki‡Z cvwi- †kl x msL¨K msL¨vi †hvMdj = cÖ_g x msL¨K msL¨vi †hvMdj + †klmsL¨K Gi eM©|
Pjyb †UKwbKwU †ewmK †_‡K ey‡S †bqv hvK- cici 6wU msL¨vi cÖ_g wZbwU msL¨vi †hvMdj 6 n‡j, †kl wZbwU
msL¨vi †hvMdj KZ? aiæY 6 wU msL¨v n‡”Q- 1, 2, 3, 4, 5, 6
1 + 2 + 3 + 4 + 5 + 6
= 6 (3wU) + 15(3wU)
 6 + 32
= 15 (1g wZbwU †hvMdj 6 Gi mv‡_ †kl 3 msL¨K Gi eM© †hvM Ki‡j 15 nq)
14. cici `kwU msL¨vi cÖ_g 5wUi †hvMdj 560 n‡j,
†kl 5wUi †hvMdj KZ? 18ZgwewmGm
540 565
570 585 DËi: N
g‡b Kwi, cici 10 wU msL¨v = x, x + 1, x +
2, x + 3, x + 4, x + 5, x + 6, x + 7, x + 8
, x + 9|
cÖ_g 5wU msL¨vi †hvMdj = x + (x + 1) + (x +
2) + ( x + 3) + (x + 4) = 5x + 10
†kl 5wU msL¨vi †hvMdj = (x + 5) + (x + 6) +mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
NM
LK
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK

4  Math Tutor
( x + 7) + (x + 8) + (x + 9) = 5x + 35
kZ©g‡Z, 5x + 10 = 560
ev, 5x = 560 - 10  x =
5
550
= 110|
myZivs, †kl 5wU msL¨vi †hvMdj = 5x + 35
= (5110) + 35 = 550 + 35 = 585
 kU©KvUt †kl msL¨v¸‡jvi †hvMdj = cÖ_g
msL¨v¸‡jvi †hvMdj + †klmsL¨K Gi eM©
= 560 + 52
= 560 + 25 = 585
15. QqwU cici c~Y©msL¨v †`qv Av‡Q| cÖ_g 3wUi
†hvMdj 27 n‡j †kl 3wUi †hvMdj KZ? mve†iwR÷vi:
01/13Zg†emiKvix wkÿKwbeÜbIcÖZ¨qbcixÿv(¯‹zj/mgch©vq):16
36 33
32 30 DËi: K
†kl 3wU †hvMdj = 27 + 32
= 27 + 9 = 36
16. QqwU µwgK c~Y© msL¨vi cÖ_g wZbwUi Mo 8 n‡j,
†kl wZbwUi †hvMdj- ¯^v¯’ÏcwieviKj¨vYgš¿Yvj‡qi
wmwbqi÷vdbvm© (evwZjK…Z)2017
29 31
33 35 DËi: M
1g wZbwUi †gvU = 8 3= 24|
†kl wZbwUi †gvU = 24 + 32
= 33|
05.04 Mo wb‡q †ewmK Av‡jvPbv
†R‡b wbb - 38
 cvuP wfÿyK K, L, M, N Ges O mvivw`b wfÿv K‡i h_vµ‡g 81, 87, 90, 95 Ges 82 UvKv Avq K‡i, Zvn‡j
Zv‡ì Mo UvKv KZ n‡e?
 ïiæ‡Z GKwU NUbvi Dci wbf©i Kiv hvK| aiv hvK, GB cvuP Rb wfÿzK GKwU Pzw³ Kij Giƒc- Zviv mvivw`b
cÖ‡Z¨‡K hv Avq Ki‡e, w`b †k‡l me UvKv GKÎ K‡i mevB mgvbfv‡e fvM K‡i wb‡e| GKw`b K, L, M, N Ges
O h_vµ‡g 81, 87, 90, 95 Ges 82 UvKv Avq Kij, Zv‡ì Pzw³ †gvZv‡eK cÖ‡Z¨‡Ki UvKv GKÎ K‡i mgvb K‡i
fvM K‡i wb‡Z GKwU cv‡K© G‡m emj| Zv‡ì `j‡bZv me UvKv GKÎ K‡i †djj| †`Lv †Mj me wg‡j †gvU 81 + 87
+ 90 + 95 + 82 = 435 UvKv n‡q‡Q| a‡i †bqv hvK, wfÿzKiv me GK UvKvi K‡qb wfÿv †c‡qwQj| hv‡nvK,
Gevi `j‡bZv Kzievwbi gvs‡mi gZ K‡i 5 Uv _‡ji cÖwZwU‡Z cÖwZevi 1 wU K‡i K‡qb ivL‡Z jvMj| fvM‡k‡l †`Lv
†Mj cÖ‡Z¨‡K 87 UvKv K‡i fv‡M †c‡q‡Q| g~j K_v- GKvwaK Dr‡mi wfbœ wfbœ cwigvY †hvM K‡i c~Yivq cÖwZwU
Drm‡K mgvb K‡i fvM K‡i †`qvB n‡”Q Mo| wfÿyK‡ì GB m¤ú~Y© wn‡mewU MvwYwZK dgy©jvq wnmve Kiv hvK-
Mo 87
K L M N O
 M‡oi AmvaviY †UKwbKt Avevi M‡í †diZ hvIqv hvK| Gevi wfÿyK‡ì `j‡bZv c~‡e©i b¨vq GKUv K‡i K‡qb
fvM bv GKUz wfbœ Dcvq Aej¤^b Ki‡jb| wZwb †`L‡jb cÖ‡Z¨‡K †h UvKv AvR‡K Avq K‡i‡Q, Zv‡ì g‡a¨
me‡P‡q Kg K Avq K‡i‡Q 81 UvKv| wfÿzK `j‡bZv Gevi mevB‡K ejj- cÖ‡Z¨‡K 80 UvKv K‡i †Zvgv‡ì
Kv‡Q †i‡L AwZwi³ LyuPiv UvKv¸‡jv Rgv Ki| cÖ‡Z¨‡K 80 UvKv K‡i †i‡L AwZwi³ LyPiv UvKv¸‡jv mevB Rgv
Kij = K 1 UvKv + L 7 UvKv + M 10 UvKv + N 15 UvKv + O 2 UvKv = 35 UvKv| Gevi wfÿzK `j‡bZv GB
35 UvKv‡K 5 R‡bi g‡a¨ fvM K‡i wbj, Gevi cÖ‡Z¨‡K AviI 7 UvKv K‡i ‡cj| c~‡e© †c‡qwQj 80 UvKv Ges c‡i
†cj 7 UvKv K‡i A_©vr cÖ‡Z¨‡K †cj 80 + 7 = 87 UvKv K‡i| GB 87 UvKvB n‡”Q Mo| GB AvBwWqv eënvi
K‡i Lye `ªæZ M‡oi mgm¨v mgvavb Kiv hvq|
mgvavb
NM
LK
mgvavb
NM
LK
81
87
90
95
82
 avc t-
†gvU wbY©q: 81 + 87 + 90 + 95 + 82 = 435
 avc t-
Mo =
msL¨vivwki
mgwóivwkiKwZcqGKRvZxq
=
5
435
= 87|
 g‡b ivLyb: wfÿzK‡ì †gvU UvKv n‡”Q ÔGKRvZxq
KwZcq ivwki mgwóÕ Ges wfÿzK‡ì msL¨v n‡”Q
Ôivwki msL¨vÕ|

Math Tutor  5
 we¯ÍvwiZ wbq‡g M‡oi A¼ mgvavb Kivi Rb¨ memgq Mo‡K †gvU K‡i mgvavb Kiæb|
17. 73, 75, 77, 79 I 81 Gi Mo KZ? cÖv_wgKwe`¨vjq
mnKvixwkÿK1992
75 76
77 78 DËi: M
ivwk¸‡jvi mgwó = 73 + 75 + 77 + 79 + 81
= 385  Mo =
5
385
= 77 |
 kU©KvU:  Mo 70 (hv me msL¨vi gv‡S Av‡Q) aiv
nj  70 e¨ZxZ evKx As‡ki mgwó- 3 + 5 + 7
+ 9 + 11 = 35 | GB mgwó 35 †K Gevi 5 w`‡q
fvM K‡i Mo Kiv nj - 35  5 = 7|
 Mo = 70 + 7 = 77|
 g‡b ivLyb: 1g av‡ci Mo I 2q av‡ci Mo †hvM K‡i
Kvw•LZ ÔMoÕ cvIqv hvq|
18. 11, 12, 13, 14 Ges 15 GB msL¨v¸‡jvi Mo n‡e-
cwievicwiKíbvAwa`ß‡Iwewfbœ c‡`wb‡qvM2014
24 13
15 †Kv‡bvwUB bq DËi: L
ivwk¸‡jvi mgwó = 11 + 12 + 13 + 14 + 15
= 65|  Mo =
5
65
= 13 |
 kU©KvU:  cÖ_g Mo = 10|  evKx As‡ki mgwó
= 1 + 2 + 3 + 4 + 5 = 15  Mo =
5
15
= 3|
 Mo = 10 + 3 = 13|
19. 13, 16, 18, 24, 34 Gi Mo KZ? ¯^v¯’¨wkÿvIcwievi
Kj¨vYwefv‡MiAwdmmnvqK2019
16 18
19 21 DËi: N
 cÖ_g Mo = 10|  evKx As‡ki mgwó = 3
+ 6 + 8 + 14 + 24 = 55 Ges Mo =
5
55
= 11
 Mo = 10 + 11 = 21|
 g‡b ivLyb: M‡o ivwk¸‡jvi cv_©K¨ Kg ev ivwk¸‡jv
mn‡R wn‡me Kiv †M‡j Ôgy‡L gy‡L ey‡S mgvavbÕ Kiv
fv‡jv | wKš‘ cv_©K¨ †ewk ev ivwk¸‡jv wn‡me Ki‡Z
RwUj g‡b n‡j mvaviY wbq‡g mgvavb KivB DËg|
20. 6, 8 I 10 Gi Mo KZ? †m‡KÛvwiGWz‡Kkb†m±i
†W‡fjc‡g›U†cÖvMÖvgM‡elYvKg©KZ©v 2015
6 8
10 12 DËi: L
21. 5, 11, 13, 7, 8 Ges 10 msL¨v¸‡jvi Mo KZ?
13ZgwkÿKwbeÜb(¯‹zj-2) 2016
6 7
8 9 DËi: N
05.05 mgwó wbY©q
†R‡b wbb- 39
 †hvM K‡i mgwó wbY©q Kiv: cÖ‡kœ wfbœ cwigvY †`qv _vK‡j †m¸‡jv †hvM K‡i mgwó †ei Ki‡Z n‡e| †hgb-
22. †`‡jvqvi †Kvb 5 wel‡q 75, 81, 84, 90 Ges 80 b¤î †cj †m †gvU KZ b¤î †cj?
†gvU b¤î = 75 + 81 + 84 + 90 + 80 = 410|
 ¸Y K‡i mgwó wbY©q Kiv : cÖ‡kœ Mo †`qv _vK‡j ¸Y K‡i mgwó †ei Ki‡Z nq|
23. 3 wU msL¨vi Mo 20 n‡j, msL¨v 3wUi mgwó KZ?
3 wU msL¨vi mgwó = 3 × 20 = 60| A_©vr, †h KqwU Dcv`v‡bi msL¨v †`qv _vK‡e Zvi mv‡_ Zv‡ì Mo‡K
¸Y Ki‡jB mgwó cvIqv hvq| myZivs, mgwó = Mo × Dcv`v‡bi msL¨v|
 g‡b ivLyb: we¯ÍvwiZ wbq‡g M‡oi A¼ Ki‡Z n‡j Avcbv‡K evi evi mgwó †ei Ki‡Z n‡e, ZvB Dc‡iv³ mgwó
wbY©‡qi wbqg `ywU me mgq g‡b ivLyb|
24. RyjvB gv‡mi •`wbK e„wócv‡Zi Mo 0.65 †m.wg.
wQj| H gv‡mi †gvU e„wócv‡Zi cwigvY KZ? cÖwZiÿv
gš¿Yvj‡qi Aax‡b¸ßms‡KZcwi`ß‡iimvBeviAwdmvi05
20.15 †m.wg. 20.20 †m.wg.
20.25 †m.wg. 65 †m.wg. DËi: K
RyjvB gvm 31 w`‡b nq|
 RyjvB gv‡mi †gvU e„wócv‡Zi cwigvY
= (0.65  31) †m.wg. = 20.15 †m.wg. |
25. 2000 mv‡j †deªæqvwi gv‡mi •`wbK e„wócv‡Zi Mo
wQj 0.55 †m.wg.| H gv‡mi †gvU e„wócv‡Zi cwigvY
KZ? mv‡K©j A¨vWRy‡U›U, Dc‡Rjv Avbmvi wfwWwc
Kg©KZ©v 2015
mgvavb
NM
LK
mgvavb
mgvavb
NM
LK
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK

6  Math Tutor
15.5 †m.wg. 15.4 †m.wg.
15.95 †m.wg. 15.55 †m.wg. DËi: M
2000 mvj Awael©| Awae‡l© †d«eyqvwi gvm nq 29
w`‡b|  †d«eyqvwi gv‡mi †gvU e„wócvZ
= (0.55  29) †m.wg. = 15.95 †m.wg.|
26. 3wU msL¨vi Mo 33| `yBwU msL¨v 24 Ges 42 n‡j
Aci msL¨vwU KZ? Lv`Äwa`ß‡iimn.DcLv`¨cwi`k©K12
12 22
32 33 DËi: N
3 wU msL¨vi mgwó = 333 = 99 Ges 2 wU
msL¨vi mgwó = 24 + 42 = 66
 Aci msL¨vwU = 99 - 66 = 33|
05.06 †hvM K‡i mgwó I ¸Y K‡i mgwó †_‡K GKwU msL¨vi gvb wbY©q
†R‡b wbb-40
 Mo A‡¼i we¯ÍvwiZ wbqg eyS‡Z Avgv‡ì Kv‡Q KwVb g‡b nq bv, Z‡e we¯ÍvwiZ wbq‡g mgvav‡b mgq †ewk jv‡M
weavq ÔkU©KvU wbqgÕ d‡jv Kiv A‡bK mgq Riæwi n‡q c‡o| Avgvi GB As‡k Avev‡iv wfÿzK`‡ji M‡í
wd‡i hve| hw` MíwU Avcwb c‡o bv _v‡Kb Zvn‡j Ô‡R‡b wbb-00Õ c‡o Zvici GLvb †_‡K ïiæ Kiæb-
wfÿK`‡ji c~‡e©i cÖkœwU Gevi GKUz Nywi‡q Kiv hvK- wfÿzK K, L, M Ges N mvivw`‡b wfÿv K‡i h_vµ‡g
81, 87, 90 Ges 95 UvKv, Zvn‡j wfÿyK O Avi KZ UvKv wfÿv Ki‡j Zv‡ì Mo wfÿv n‡e 87 UvKv?
GB mgvavb GKvwaK wbq‡g Ki‡Z cvwi| Avwg GLv‡b `ywU wbqg wb‡q K_v eje, `ywUB †ewmK|
 wbqg : `ywU †gvU †ei Kivi gva¨‡g- K, L, M Ges N Gi †gvU †ei Kie (†hvM K‡i) Ges K, L, M, N I O
Gi †gvU (¸Y K‡i)| ( †R‡b wbb -00 †`‡L wbb)
K, L, M Ges N (4 R‡bi) Gi †gvU wfÿv = 81 + 87 + 90 + 95 = 353
K, L, M, N Ges O (5R‡bi) Gi †gvU wfÿv = 87  5 = 435
Gevi 5 R‡bi †gvU †_‡K 4 R‡bi †gvU we‡qvM Kiæb, Zvn‡j 1 Rb (O) Gi wfÿv †ei n‡q Avm‡e|
 O Gi wfÿv = 435 - 353 = 82|
 wbqg : GB wbqgwU †ewmK I kU©‡UKwbK DfqB Kvfvi Ki‡e| Pjyb wfÿzK`‡ji cÖ_g MÖvdwU‡Z wd‡i hvB-
Mo 87
K L M N O
 avc 02: Avcwb B‡Zvg‡a¨ †`L‡Z †c‡q‡Qb A¨vk Kvjv‡ii Kjvg¸‡jv eø¨vK Kvjv‡ii ÔMo ev mgvb ev mgZjÕ
†j‡f‡ji †P‡q Kg ev †ewk| GLb Avcbv‡K hw` GKUv KvR Ki‡Z ejv nq †h, Avcwb A¨vk Kvjv‡ii Ge‡ov-
†_e‡ov Kjvg¸‡jv KvUQvuU K‡i eø¨vK Kvjv‡ii gZ mgvb Ki‡eb| Zvn‡j Avcwb wK Ki‡eb? Avcwb wbðq
†hLv‡b †ewk Av‡Q †mLvb †_‡K †K‡U wb‡q †hLv‡b Kg Av‡Q †mLv‡b †Rvov jvMv‡eb| A‡bKUv Ge‡ov †_e‡ov
gvwU‡K Avgiv †Kv`vj w`‡q mgZj Kwi †hfv‡e wVK †miKg|
 avc 03: Gevi Avgiv Avevi MÖv‡di msL¨vi w`‡K g‡bvwb‡ek Kie| msL¨v¸‡jv‡K Avgiv KvUQvuU (Kg/†ewk)
K‡i cÖwZwU‡K 87 Gi mgvb Kie| Pjyb ïiæ Kiv hvK-
K Gi mv‡_ 6 †hvM Ki‡Z n‡e KviY 6 NvUwZ Av‡Q = 81 + 6 = 87 (Aci c„ôvq)
mgvavb
mgvavb
NM
LK
mgvavb
NM
LK
81
87
90
95
x
avç‡jv co–b
 avc 01: Avmyb Avgiv GZÿY wfÿvi UvKvi †h wPšÍv
K‡iwQ, †mUv fz‡j wM‡q Rv÷ MÖv‡di Ge‡ov-†_e‡ov
Kjvg¸‡jvi w`‡K bRi †`B Ges fv‡jvfv‡e jÿ¨
Ki‡j †`L‡eb- A¨vk Kvjvi Gi Kjvg¸‡jv Ge‡ov-
†_e‡ov Ges eø¨vK Kvjv‡ii Kjvg mgvb ev mgZj
(2/3 evi LuywU‡q LuywU‡q †`Lyb)| me‡P‡q †ewk bRi
w`b eø¨vK Kvjv‡ii mgvb ev mgZ‡ji w`‡K, hv Zxi
wPý Øviv eySv‡bv n‡q‡Q|

Math Tutor  7
L Gi mv‡_ †hvM ev we‡qvM Ki‡Z n‡e bv, KviY NvUwZ ev †ewk †bB = 87 + 0 = 87
M Gi mv‡_ 3 we‡qvM Ki‡Z n‡e, KviY 3 †ewk Av‡Q = 90 - 3 = 87
N Gi mv‡_ 8 we‡qvM Ki‡Z n‡e, KviY 8 †ewk Av‡Q = 95 - 8 = 87
K, L, M Ges N GB 4 wU †K mgvb Ki‡Z wM‡q †ewk Av‡Q = 6 + 0 - 3 - 8 = - 5| (gy‡L gy‡L wn‡me K‡i
GB jvBbwU mivmwi cÖkœc‡Î wj‡L †dj‡eb)| cÖvd Abyhvqx, †h‡nZz O Gi wfÿv 87 †_‡K Kg ZvB O Gi wfÿvi
mv‡_ 5 †hvM K‡i w`‡j O Gi Mo wfÿv 87 cvIqv hv‡e|
aiv hvK, O Gi wfÿv x
x + 5 = 87 ev, x = 87 - 5 = 82|
 mycvi kU©KvU t mycvi kU©KvU n‡”Q- mivmwi Mo 87 Gi †P‡q †Kvb msL¨v eo n‡j Ô†ewk AskUzKzÕ we‡qvM Ges †QvU
n‡j ÔKg AskUzKzÕ †hvM Ki‡Z n‡e| Pjyb welqwU GKwU KvíwbK NUbvi mv‡_ eY©bv Kiv hvK|
(NUbvi eY©bvi mgq Avcbvi KvR n‡e- Avwg hv hv eje Zvi me wb‡P ewY©Z ÔO Gi wfÿvÕ jvBbwUi mv‡_ cÖwZevi
wgjv‡Z _vK‡eb|)
 cÖ_‡g mvwe©K Mo 87 wjL‡jb| wjLv gvÎ 87 msL¨vwU Avcbvi mv‡_ K_v ej‡Z ïiæ Kij! Avcwb wb‡Ri Kvb‡K
wek¦vm Ki‡Z cvi‡Qb bv, wK n‡”Q Gme! 87 Avcbv‡K wRÁvmv Kij, ÔZzwg wK ej‡Z cv‡iv Avgvi cv‡k wKQz msL¨v
†Kb †hvM (+) I we‡qvM (-) K‡iwQ? Avcwb bv m~PK gv_v SvwK‡q ej‡jb- Ôbv ej‡Z cvwi bv| Avcwb KviY Rvb‡Z
PvB‡jb, Ô †Kb wKQz msL¨v †hvM-we‡qvM K‡iQ?Õ 87 Avevi ej‡Z ïiæ Kij, ÔPj, †Zvgv‡K Avmj NUbvUv Ly‡j
ewj|Õ Avcwb g‡bvgy»Ki n‡q 87 Gi K_v ïbwQ‡jb| 87 ejj, Ô K, L, M, N G‡ì mevi msL¨v Avgvi gZ 87
nIqvi K_v wQj| wKš‘ †`L, K Gi Av‡Q 81, ej‡Z cvi‡e K-Gi evKx 6 †Mj †Kv_vq? ïb, H 6 Avwg Avgvi
c‡K‡U cy‡i †i‡LwQ! Õ Avcwb †`L‡jb ZvB‡Zv! 87 Gi cv‡k H 6 †K †hvM Ae¯’vq †`L‡Z cvIqv hv‡”Q! 87 Avevi
ejj, Ô L Gi msL¨vwU †`‡LQ? †m wKš‘ Avgvi gZB †`L‡Z, †ePviv Lye PvjvK, ZvB Zvi KvQ †_‡K wK”Qz evwM‡q
wb‡Z cvwiwb, GRb¨ Avgvi cv‡k †`L k~b¨ (0) wj‡L †i‡LwQ|Õ Avcwb †`L‡jb K_v mZ¨| 87 Avevi ej‡Z ïiæ
Kij, Ô M Gi msL¨v n‡”Q 90, Zvi Avgvi gZ 87 nIqvi K_v wQj, evKx 3 †m †Kv_vq †cj? ejwQ, M GKUv Av¯Í
e`gvBk| †m Avgvi KvQ †_‡K 3 †Rvi K‡i wQwb‡q wb‡q †M‡Q| †`LQ bv, GRb¨ Avwg Avgvi cv‡k 3 we‡qvM
K‡iwQ!Õ 87 Gevi GKUy `g wbj| ÔH †h N Gi 95 msL¨vwU‡Z †h AwZwi³ 8 Av‡Q IUvI Avgvi KvQ †_‡K wQwb‡q
wb‡q‡Q, ZvB GeviI 8 we‡qvM Ki‡Z n‡q‡Q| ey‡SQ Avmj NUbvwU wK?Õ Avcwb 87 Gi †kl cÖ‡kœ nVvr †Nvi KvU‡ZB
†`L‡Z †c‡jb-
 O Gi wfÿv = 87 + 6 + 0 - 3 - 8 = 87 - 5 = 82
27. GKRb e¨vUmg¨vb cÖ_g wZbwU T-20 ‡Ljvq 82, 85
I 92 ivb K‡ib| PZz_© †Ljvq KZ ivb Ki‡j Zvi
Mo ivb 87 n‡e? cÖv_wgK mnKvix wkÿK -2018
86 87
88 89 DËi: N
(3wU †Ljvi †gvU I 4 wU †Ljvi †gvU †ei K‡i
we‡qvM Ki‡Z n‡e)
cÖ_g 3wU T-20 †Ljvi iv‡bi mgwó = 82 + 85 +
92 = 259
4wU †Ljvi Mo ivb = 87
∴ Ó Ó †gvU Ó = 87  4 = 348
myZivs, 4_© †Ljvq ivb Ki‡Z n‡e = 348 - 259
= 89|
 kU©KvUt PZz_© †Ljvi ivb = 87 + 5 + 2 - 5
= 87 + 2 = 89|
28. cixÿvq ÔKÕ Gi cÖvß b¤î h_vµ‡g 70, 85 I 75|
PZz_© cixÿvq Zv‡K KZ b¤î †c‡Z n‡e †hb Zvi Mo
cÖvß b¤î 80 nq? cÖv_wgK mnKvix wkÿK - 2018;
cÖavb wkÿK (bvMwj½g)-12
82 88
90 78 DËi: M
3 cixÿvq cÖvß bv¤^v‡ii mgwó = 70 + 85 + 75mgvavb
NM
LK
mgvavb
NM
LK

8  Math Tutor
= 230
4wU cixÿvq cÖvß bv¤^v‡ii mgwó = 80  4 = 320
 PZz_© cixÿvq bv¤^vi †c‡Z n‡e = 320 - 230 = 90|
 kU©KvUt PZz_© cixÿvq cÖvß b¤î: 80 + 10 - 5 + 5
= 80 + 10 = 90|
29. cixÿvq ÔKÕ Gi cÖvß b¤î h_vµ‡g 82, 85 I 92|
cÖvß b¤î 87 nq? cÖavb wkÿK 2012 (evMvbwejvm)
K. 89 L. 88
M. 86 N. 91 DËi: K DËi: K
28 bs Gi Abyiƒc mgvavb|
 kU©KvUt 87 + 5 + 2 - 5 = 89|
30. cixÿvq K Gi cÖvß b¤î h_vµ‡g 75, 85 I 80|
cªvß b¤î 82 nq? cÖavb wkÿK (wµmvbw_gvg)-12
90 89
92 88 DËi: N
82 + 7 - 3 + 2 = 88|
31. 0.6 n‡jv 0.2, 0.8, 1 Ges x Gi Mo gvb | x Gi
gvb KZ? cÖwZiÿvgš¿Yvj‡qiAax‡bmvBdviAwdmvit99
0.2 0.4
.67 2.4 DËi: L
x e¨ZxZ 3 wU msL¨vi mgwó = 0.2 + 0.8 + 1 = 2
x mn 4wU msL¨vi mgwó = 0.6  4 = 2.4
 x Gi gvb = 2.4 - 2 = .4
 kU©KvU: x = 0.6 + 0.4 - 0.2 - 0.4 = 0.4
32. 4, 6, 7 Ges x Gi Mo gvb 5.5 n‡j x-Gi gvb
KZ? cÖvK-cÖv_wgKmnKvixwkÿK-15;cÖvK-cÖv_wgKmnKvix
wkÿK(Mvgv)2014;
5.0 7.5
6.8 6.5 DËi: K
x = 5.5 + 1.5 -.5 -1.5 = 5
05.07 ¸Y K‡i `ywU mgwó wbY©q I †mLvb †_‡K GKwU msL¨vi gvb wbY©q
33. 11wU msL¨vi Mo 30| cÖ_g cvuPwU msL¨vi Mo 25 I
†kl cvuPwU msL¨v Mo 28| lô msL¨vwU KZ? mnKvix
wkÿK (Rev)-11; mnKvix wkÿK (hgybv)-08
55 58
65 67 DËi: M
11wU msL¨vi mgwó = 3011 = 330
cÖ_g 5wU msL¨vi mgwó = 255 = 125
†kl 5wU msL¨vi mgwó = 285 = 140
∴ 10wU msL¨vi mgwó = 125 + 140 = 265
∴ 6ô msL¨v = 330 - 265 = 65|
 kU©KvUt lô msL¨v = 30 + (55) + (52)
= 30 + 25 + 10 = 65|
34. †Kv‡bv GK ¯’v‡b mßv‡ni Mo ZvcgvÎv 300
†mj-
wmqvm| cÖ_g 3 w`‡bi Mo ZvcgvÎv 280
†mjwmqvm
I †kl 3 w`‡bi Mo ZvcgvÎv 290
†mjwmqvm n‡j
PZz_© w`‡bi ZvcgvÎv KZ? mnKvix wkÿK (nvmbv‡nbv)-
11, mnKvix wkÿK (Lyjbv)-06
330
†m. 360
†m.
390
†m. 430
†m. DËi: M
7 w`‡bi ZvcgvÎvi mgwó = 300
× 7 = 2100
cÖ_g 3 w`‡bi ZvcgvÎvi mgwó = 280
× 3 = 840
†kl 3 w`‡bi ZvcgvÎvi mgwó = 290 × 3 = 870
6 w`‡bi ZvcgvÎvi mgwó = 840
+ 870
= 1710
myZivs, PZz_© w`‡bi ZvcgvÎv = 2100
- 1710
= 390
 kU©KvUt PZz_© w`‡bi ZvcgvÎv = 300
+ (20
3)
+ (10
3) = 390
|
35. 10wU msL¨vi †hvMdj 462| cÖ_g PviwUi Mo 52,
†kl 5wUi Mo 38| cÂg msL¨vwU KZ? cÖvK-cÖv_wgK
mnKvix wkÿK - 2015; cÖvK-cÖv_wgK mnKvix wkÿK
(ivBb) 2015
50 62
64 60 DËi: M
10wU msL¨vi Mo = 46.2 , cÖ_g PviwU‡Z Kgv‡Z
n‡e 5.8  5 = 23.2 Ges †kl 5wU‡Z e„w× Ki‡Z
n‡e 8.2  5 = 41|
A_©vr, cÂg msL¨vwU = 46.2 - 23.2 + 41 = 64|
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb

Math Tutor  9
05.08 mvwe©K Mo wbY©q
†R‡b wbb - 41
 `yB ev Z‡ZvwaK Mo w`‡q Zv‡ì mvwe©K Mo †ei Ki‡Z ejv n‡j, Avgiv cÖ_‡g Mo¸‡jv‡K †gvU-G iƒcvšÍi Kie
Ges cÖvß c„_K c„_K †gvU †hvM K‡i mvwe©K †gvU †ei Kie| Gevi †gvU ivwki msL¨v Øviv mvwe©K †gvU‡K fvM K‡i
mvwe©K Mo †ei Kie|
(K) c„_K c„_K Mo †_‡K mvwe©K Mo †ei Kiv
36. †Kv‡bv †kÖwYi 12 Rb Qv‡Îi †Kv‡bv cixÿvq djv-
d‡ji Mo 70 | Aci 18 R‡bi djvd‡ji Mo 80|
Z‡e 30 Rb Qv‡Îi djvd‡ji mvwe©K Mo KZ?
wnmveiÿY Kg©KZv© : 96; cÖv_wgKwkÿvAwa`ß‡iimnKvix
jvB‡eªwiqvbKvgK¨vUvjMvi2018
73.75 75.25
76 77.125 DËi: M
12 Rb Qv‡Îi djvd‡ji Mo 70 n‡j, Zv‡ì
†gvU djvdj = 7012 = 840
18 Rb Qv‡Îi djvd‡ji Mo 80 n‡j, Zv‡ì
†gvU djvdj = 80 18 = 1440
†gvU QvÎ = 12 + 18 = 30 Rb Ges Zv‡ì
djvd‡ji †hvMdj = 840 + 1440 = 2280
 30 Rb Qv‡Îi djvd‡ji mvwe©K Mo =
30
2280
= 76
 ey‡S ey‡S kU©KvU:
12 Rb 18 Rb
 wP‡Î jÿ¨ Kiæb, 12 Rb Qv‡Îi M‡oi †P‡q 18 Rb
Qv‡Îi Mo 10 K‡i †ewk| Avgiv hw` 12 + 18 =
30 Rb Qv‡Îi Mo •Zwi Ki‡Z PvB, Zvn‡j GB 18
R‡bi AwZwi³ Mo 10 K‡i †gvU 180 (1018),
30 R‡bi gv‡S mgvb K‡i fvM K‡i w`‡Z n‡e|
 Mo 70 †h‡nZz 12 I 18 Df‡qi gv‡S _vKvi A_©
30 R‡bi gv‡S _vKv| GRb¨ cÖ_‡g 30 R‡bi Mo
70 a‡i AwZwi³ 180 Mo K‡i †hvM K‡i w`‡Z
n‡e|
30 R‡bi Mo = 70 +
30
1810 
= 70 + 6 = 76
 g‡b ivLyb:
 hZ R‡bi M‡oi gv‡S ÔAwZwi³ MoÕ
_vK‡e ZZRb Øviv ÔAwZwi³ MoÕ ¸Y K‡i †gvU
AwZwi³ †ei Ki‡Z n‡e| †hgb: cÖ`Ë A‡¼ 18
R‡bi gv‡S ÔAwZwi³ MoÕ 10 wQj, ZvB 18 Øviv ¸Y
K‡i †gvU AwZwi³ 180 †ei Kiv n‡q‡Q|
 Zvici †gvU AwZwi³‡K mvwe©K msL¨v Øviv fvM
Ki‡Z n‡e| †hgb- †gvU AwZwi³ 180 †K mvwe©K
msL¨v 30 Øviv fvM Kiv n‡q‡Q|
37. GKRb †evjvi M‡o 20 ivb w`‡q 12 DB‡KU cvb|
cieZ©x †Ljvq M‡o 4 ivb w`‡q 4wU DB‡KU cvb|
wZwb M‡o DB‡KU cÖwZ KZ ivb w`‡q‡Qb? cÖvK-cÖv_wgK
mnKvix wkÿK(‡nvqvs‡nv)2013
14 16
18 19 DËi: L
12 DB‡KU †c‡Z ivb †`q = 20 12 = 240
4 DB‡KU †c‡Z ivb †`q = 4  4 = 16
me©‡gvU ivb = 240 + 16 = 256
me©‡gvU cÖvß DB‡KU = 12 + 4 = 16
 DB‡KU cÖwZ Mo ivb †`q =
16
256
= 16|
12 DB‡K‡U Mo ivb 4 + AwZwi³ Mo 16 ivb
4 DB‡K‡U Mo ivb 4
12 DB‡K‡U †gvU AwZwi³ 1612
 Mo = 4 +
16
1216 
= 4 + 12 = 16|
38. GKRb †evjvi M‡o 22 ivb w`‡q 6wU DB‡KU cvb|
wZwb M‡o DB‡KU cÖwZ KZ ivb w`‡q‡Qb ? cÖvK-
cÖv_wgK mnKvix wkÿK (wSjvg) 2013
14.0 16.0
18. 0 18.8 DËi: N
ey‡S ey‡S kU©KvU:mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
80
70
10

10  Math Tutor
6 DB‡K‡U Mo ivb 14 + AwZwi³ Mo 8 ivb
6 DB‡K‡U †gvU AwZwi³ 86
 Mo = 14 +
10
68 
= 14 + 4.8 = 18.8
39. GKRb †evjvi M‡o 18 ivb w`‡q 10 DB‡KU cvb|
cieZ©x Bwbs‡m M‡o 4 ivb w`‡q 4 DB‡KU cvb| wZwb
DB‡KU cÖwZ M‡o KZ ivb w`‡q‡Qb? cÖvK-cÖv_wgK
mnKvix wkÿK (wgwmwmwc) 2013
12 13
14 16 DËi: M
DB‡KU cÖwZ Mo ivb = 4 +
14
1410
= 4 + 10= 14|
cieZ©x Bwbs‡m M‡o 8 ivb w`‡q 3wU DB‡KU cvb|
wZwb DB‡KU cÖwZ KZ ivb w`‡q‡Qb? cÖvK-cÖv_wgK
mnKvix wkÿK (`vRjv) 2013
12 14.3
15.5 16 DËi: L
ey‡S ey‡S kU©KvU:
 Mo = 8 +
10
79
= 8 + 6.3 = 14.3
GLb Zvi DB‡KU cÖwZ Mo ivb KZ? cÖavb wkÿK
(cÙ)-09
9 10
12 14 DËi: M
ey‡S ey‡S kU©‡UKwbK:
Mo = 6 +
16
128
= 6 + 6 = 12
42. M msL¨K msL¨vi Mo A Ges N msL¨K Mo B
n‡j me¸‡jv msL¨vi Mo KZ? 23Zg wewmGm
2
BA 
2
BNAM 
NM
BNAM


BA
BNAM


DËi: M
M msL¨K msL¨vi Mo A n‡j,
M msL¨K msL¨vi †hvMdj = MA
N msL¨K msL¨vi Mo B n‡j,
N msL¨K msL¨vi †hvMdj = NB
†gvU msL¨v = M+N
Ges msL¨v¸‡jvi †hvMdj = AM+BN
∴ Zv‡ì Mo =
NM
BNAM


43. p msL¨K msL¨vi Mo m Ges q msL¨K Mo n|
me¸‡jv msL¨vi Mo KZ? cÖavb wkÿK (cÙ)-09
p+q
2
m+n
2
pm +qn
p+q
pm +qn
m+n
DËi: M
44. x msL¨K †Q‡ji eq‡mi Mo y eQi Ges a msL¨K
†Q‡ji eq‡mi Mo b eQi| me †Q‡ji eq‡mi Mo
KZ? cÖavb wkÿK (wkDjx)-09
x+2
2
y+b
2
xy +ab
y+b
xy+ab
x+a
DËi: N
(L) mvwe©K M‡oi Zzjbvq cÖ`Ë GKvs‡ki Mo eo n‡j
45. 100 Rb wkÿv_x©i cwimsL¨v‡b Mo b¤î 70| G‡ì
g‡a¨ 60 Rb QvÎxi Mo b¤î 75 n‡j, QvÎ‡ì Mo
b¤î KZ? 35ZgwewmGmwcÖwj.
55.5 65.5
60.5 62.5 DËi: N
100 Rb wkÿv_x©i †gvU b¤î = 70  100
= 7000
60 Rb QvÎxi †gvU b¤î = 75  60
= 4500
(hv‡ì Mo †ei Ki‡eb, Zv‡ì †gvU Av‡M †ei Kiæb)
(100 - 60 ) ev 40 Rb Qv‡Îi †gvU b¤î
= 7000 - 4500 = 2500
 40 Rb Qv‡Îi Mo b¤î =
40
2500
= 62.5
mgvavb
NM
LK
NM
LK
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK

Math Tutor  11
75
70
60 Rb 100 Rb 40 Rb
 100 ev 60 + 40 R‡bi Mo n‡”Q 70| GLv‡b 60
ev 40 Gi †h‡Kvb GKwU Mo e„w× †c‡j Ab¨ GKwU
Mo n«vm cv‡e|
 cÖkœvbyhvqx, 60 Gi Mo 70 A‡cÿv e„w× †c‡q‡Q,
Zvn‡j wbwðZfv‡e 40 Gi Mo 70 A‡cÿv n«vm cv‡e
 60 R‡b hZ e„w× †c‡q‡Q, 40 R‡b ZZ n«vm †c‡q‡Q|
KviY †gvU cwigv‡Yi †Kvb cwieZ©b nq bv, ïay GK
RvqMv †_‡K Av‡iK RvqMvq ¯’vbvšÍi nq|
 60 R‡bi Mo 5 K‡i e„w× cvIqvq †gvU e„w× †c‡q‡Q
560 = 300, GB 300, 40 R‡bi cÖ‡Z¨‡K
mgvb K‡i †Q‡o w`‡q‡Q| A_©vr, Zviv cÖ‡Z¨‡K †Q‡o
w`‡q‡Q -
40
300
= 7.5 K‡i|
 7.5 K‡i †Q‡o †`qvi ci 40 R‡bi Mo
n‡e = 70 - 7.5 = 62.5 |
 g‡b ivLyb: mvwe©K Mo †_‡K cÖ`Ë MowU eo n‡j †h
MowU PvIqv n‡e †mB MowU Aek¨B n‡e †QvU| ZvB
mvwe©K Mo †_‡K we‡qvM K‡i †QvU MowU †ei Ki‡Z
n‡e| †hgb: mvwe©K Mo 70 †_‡K cÖ`Ë Mo 75 eo
wQj, ZvB we‡qvM K‡i mgvavb Kiv n‡q‡Q| Avevi
mvwe©K Mo †_‡K cÖ`Ë Mo †QvU n‡j †hvM K‡i
mgvavb Ki‡Z n‡e| †hgb: 50 bs cÖkœ †`Lyb|
46. 100 Rb wkÿv_©xi Mo b¤î 90, hvi g‡a¨ 75 Rb
wkÿv_x©i Mo b¤î 95| Aewkó wkÿv_x©‡ì Mo b¤î
KZ? gv`K`ªe¨ wbqš¿Y Awa`ß‡ii Dc-cwi`k©K-18
75 70
80 65 DËi: K
75 Rb wkÿv_x©i Mo b¤î 5 K‡i e„w× cvIqvq
†gvU 5  75 = 375 e„w× †c‡q‡Q, hv 25 Rb
†_‡K n«vm †c‡q‡Q| 25 Rb †_‡K Mo AvKv‡i n«vm
†c‡q‡Q-
25
375
= 15 K‡i|
 25 R‡bi b¤^‡ii Mo = 90 - 15 = 75|
47. 15 Rb Qv‡Îi Mo eqm 29 eQi| Zv‡ì Avevi `yRb
Qv‡Îi eq‡mi Mo 55 eQi| Zvn‡j evwK 13 Rb
Qv‡Îi eq‡mi Mo KZ n‡e? †iwR. †emiKvwi mnKvix
wkÿK (UMi)-11; mnKvix wkÿK (Ki‡Zvqv)-10; cÖavb
wkÿK (K¨vwgwjqv)-12; cÖavb wkÿK (ewikvj wefvM)-08
25 eQi 26 eQi
27 eQi 28 eQi DËi: K
13 Rb Qv‡Îi Mo = 29 -
13
226 
= 29 - 4 = 25|
48. 9 Rb Qv‡Îi Mo eqm 15 ermi| 3 Rb Qv‡Îi
eq‡mi Mo 17 ermi n‡j evwK 6 Rb Qv‡Îi eq‡mi
Mo KZ? †iwR. cÖv_wgK we`¨v. mn. wkÿK(kvcjv)-11
14 ermi 15 ermi
16 ermi 17 ermi DËi: K
6 Rb Qv‡Îi eq‡mi Mo = 15 -
6
32 
= 15 - 1 = 14|
49. 20 Rb evjK I 15 Rb evwjKvi Mo eqm 15 eQi|
evjK‡ì Mo eqm 15.5 eQi n‡j evwjKv‡ì Mo
eqm KZ? cÖvK-cÖv_wgK mnKvix wkÿK (`vwbqye) -13
14 eQi 14 eQi 4 gvm
14 eQi 6 gvm 14 eQi 8 gvm DËi: L
15 Rb evwjKvi Mo = 15 -
15
10
= 15 -
3
2
= 14
3
1
eQi = 14 eQi
3
1
 12 gvm
= 14 eQi 4 gvm|
(M) mvwe©K M‡oi Zzjbvq cÖ`Ë GKvs‡ki Mo †QvU n‡j
50. 9 wU msL¨vi Mo 12| Gi g‡a¨ cÖ_g 7 wU msL¨vi
Mo 10| evKx msL¨v `ywUi Mo KZ? BADC (‡÷vi
wKcvi) 17
17 18
19 20 DËi: M
9 wU msL¨vi mgwó = 12  9 = 108
7wU msL¨vi mgwó = 10  7 = 70
evKx (9 - 7) ev 2 wU msL¨vi †gvU = 108 - 70
= 38
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK

12  Math Tutor
 evKx msL¨vi Mo =
2
38
= 19|
12
10
7wU msL¨v 9wU msL¨v 2wU msL¨v
 mvwe©K M‡oi Zzjbvq 7wU msL¨vi Mo †QvU nIqvq
Aci 2wU msL¨vi Mo eo n‡e|
 7 wU msL¨vi Mo 2 K‡i n«vm cvIqvq †gvU 27 =
14 n«vm †c‡q‡Q, GB 14, 2 wU msL¨v‡Z wM‡q hy³
n‡e Ges Zv Mo AvKv‡i A_©vr,
2
14
= 7 K‡i|
 2wU msL¨vi Mo = 12 + 7 = 19|
 mivmwi wjL‡eb:
2 wU msL¨vi Mo = 12 +
2
14
= 12 + 7 = 19|
51. GK †`vKvb`vi 12 w`‡b 504 UvKv Avq Ki‡jb|
cÖ_g 4 w`‡b Mo Avq 40 UvKv n‡j evwK w`b¸‡jvi
Mo Avq KZ UvKv n‡e? mnKvix wkÿK (BQvgwZ)-10;
40 UvKv 42 UvKv
43 UvKv 47 UvKv DËi: M
†`qv Av‡Q, 12 w`‡bi cÖvß Avq 504 UvKv
cÖ_g 4 w`‡bi †gvU Avq = 40  4 = 160 UvKv
evKx (12 - 4) ev 8 w`‡bi †gvU Avq = 504 - 160
= 344 UvKv
 evKx w`b¸‡jvi Mo Avq =
8
344
UvKv
= 43 UvKv|
 ey‡S ey‡S kU©KvU: cÖ`Ë cÖ‡kœ mvwe©K MowU †`qv †bB,
ZvB Avgiv †mwU Av‡M †ei K‡i wbe-
12
504
= 42|
 8 w`‡bi Mo = 42 +
8
8
= 42 + 1 = 43|
(N) Dc‡iv³ (L) I (M) Gi kU©KvU wbqg `ywU GKB A‡¼ cÖ‡qvM
52| 6 Rb cyiæl, 8 Rb ¯¿x‡jvK Ges 1 Rb evj‡Ki
eq‡mi Mo 35 eQi| cyiæ‡li eq‡mi Mo 40 Ges
¯¿x‡jvK‡`i eq‡mi Mo 34 eQi| evj‡Ki eqm KZ?
cÖvK-cÖv_wgK mnKvix wkÿK-(Avjdv)-14; mnKvix
wkÿK-(cÙ)-12
14 eQi 13 eQi
16 eQi 15 eQi DËi: L
6 Rb cyiæl + 8 Rb ¯¿x †jvK + 1 Rb evjK =
15 R‡bi †gvU eqm = 35  15 = 525 |
6 Rb cyiæ‡li †gvU eqm = 40  6 = 240
8 Rb ¯¿x‡jv‡Ki †gvU eqm = 34  8 = 272
6 Rb cyiæl + 8 Rb ¯¿x‡jv‡Ki †gvU eqm = 240 +
272 = 512
 1 Rb evj‡Ki eqm = 525 - 512 = 13|
1 evj‡Ki eqm = 35 -
1
65 
+
1
81
= 35 - 30 + 8 = 13 eQi|
05.09 `ywU mgwói cv_©K¨ †_‡K 1wU msL¨v wbY©q
†R‡b wbb- 42
 `ywU Mo _vK‡e, GKwU Mo †_‡K Ab¨ GKwU M‡o 1wU msL¨v †ewk _vK‡e| †hgb- 2 wU msL¨vi Mo I 3wU msL¨vi
Mo , 10wU msL¨vi Mo I 11 wU msL¨vi Mo BZ¨vw` |
 `ywU M‡oi mgwói cv_©K¨ †_‡K 1wU msL¨v wbY©q Kiv nq| †hgb- 3wU msL¨vi mgwó †_‡K 2 wUi mgwó, 5wU msL¨vi
mgwó †_‡K 4 wUi mgwó , 7wU msL¨vi mgwó †_‡K 6 wUi mgwó we‡qvM K‡i 1 wU msL¨v wbY©q Kiv nq|
 Avgiv mvg‡b GKwU‡K Kg msL¨vi Mo Ges Ab¨ GKwU‡K †ewk msL¨vi Mo wn‡m‡e wPwýZ Kie|
mgvavb
NM
LK
mgvavb
NM
LK

Math Tutor  13
(K) †ewki msL¨vi Mo hw` Kg msL¨vi Mo †_‡K gv‡b eo nq
†R‡b wbb- 43
 evwn¨K `„wó‡Z A¼¸‡jvi `vM wfbœ wfbœ g‡b n‡jI †gBb ÷ªvKPvi ûeû GKB| ZvB cÖwZwU A¼ wb‡q fv‡jvfv‡e wPšÍv
Kiæb, †`L‡eb G‡ì gv‡S PgrKvi †hvMm~Î †c‡q †M‡Qb| Avwg `xN© mgq wb‡q A¼¸‡jv G¸‡jv web¨¯Í K‡iwQ,
Avcbv‡ì KvR n‡”Q evievi PP©v g~j †d«gUv‡K AvqË¡ Kiv| GB e‡qi cÖwZwU P¨vÞv‡ii †ÿ‡Î GKB K_v cÖ‡hvR¨|
 g‡b ivLyb: kU© †UKwb‡K †h msL¨vi gvb †ei Ki‡eb, †mB msL¨vwU †h M‡oi gv‡S Av‡Q, Zv‡K wfwË a‡i mgvavb
Ki‡Z n‡e| †hgb 53 bs cÖ‡kœ ÔZ…Zxq msL¨vwUi gvb †ei Ki‡Z n‡e, †hwU 30 M‡oi gv‡S Av‡Q, ZvB kU©‡UKwb‡K
30 †K wfwË a‡i mgvavb Kiv n‡q‡Q| g~j Av‡jvPbvq †ewmKmn Rvb‡Z cvi‡eb|
 GKwU gRvi Z_¨: kU© †UKwb‡K mgvav‡bi †ÿ‡Î eqm msµvšÍ A‡¼ GKUv gRvi welq n‡”Q, wcZv ev gvZv I
mšÍvb‡ì Mo eqm †`qv _vK‡j Aek¨B wcZv ev gvZvi eqm H Mo A‡cÿv †ewk n‡e Ges mšÍv‡bi eqm H Mo
A‡cÿv Aek¨B Kg n‡e| GRb¨ cÖ‡kœ wcZv ev gvZvi eqm PvIqv n‡j H M‡oi mv‡_ †hvM K‡i mgvavb †ei Ki‡Z
n‡e| Ges cÖ‡kœ mšÍv‡bi eqm PvIqv n‡j H Mo †_‡K we‡qvM K‡i mgvavb Ki‡Z n‡e|
53. cÖ_g I wØZxq msL¨vi Mo 25| cÖ_g, wØZxq I Z…Zxq
msL¨vi Mo 30 n‡j, Z…Zxq msL¨vwU KZ? b¨vkbvj
GwMÖKvjPvivj†UK‡bvjwR†cÖvMÖvg2019
25 40
90 50 DËi: L
1g I 2 q msL¨vi mgwó = 252 = 50
1g, 2q I 3q msL¨vi mgwó = 30  3 = 90
3q msL¨vwU = 90 - 50 = 40|
 ey‡S ey‡S kU©KvUt
30
25
1g 2q 1g 2q 3q
 ïiæ‡Z 1g I 2q msL¨vi Mo wQj 25| c‡i 3q
msL¨vwU hy³ nIqvq Zv‡ì Mo e„w× †c‡q nq 30|
G‡ÿ‡Î 1g I 2q msL¨vi Mo 5 K‡i e„w× cvIqvq
†gvU 52 = 10 e„w× cvq| GB 10 G‡m‡Q Z…Zxq
msL¨vwU †_‡K (wP‡Îi A¨vk Kvjvi jÿ¨ Kiæb)|
Zvn‡j Z…Zxq msL¨vwU n‡e- Z…Zxq msL¨v Mo + 1g I
2q msL¨vi e„w×K…Z Ask|
 Z…Zxq msL¨v = 30 + (52) = 30 + 10 = 40|
 g‡b ivLyb: †h msL¨vwUi gvb †ei Ki‡Z n‡e †mwUi
Mo Gi mv‡_ evKx msL¨v¸‡jvi e„w×K…Z Ask †hvM
Ki‡Z n‡e| †h msL¨vwU cÖ‡kœ PvIqv nq †mwU
mvaviYZ †ewk msL¨vi M‡o cvIqv hvq|
54. wcZv, gvZv I cy‡Îi eq‡mi Mo 37| Avevi wcZv I
cy‡Îi eq‡mi Mo 35 eQi| gvZvi eqm KZ?
evsjv‡`kcjøxDbœqb †ev‡W©iDc‡RjvcjøxDbœqbKg©KZv2015
38 eQi 41 eQi
wcZv, gvZv I cy‡Îi †gvU eqm = 37  3
= 111
wcZv I cy‡Îi †gvU eqm = 35  2 = 70
 gvZvi eqm = 111 - 70 = 41|
 ey‡S ey‡S kU©KvUt
gvZvi eqm = 37 + (22) = 37 + 4 = 41
55. wcZv I 2 cy‡Îi eq‡mi Mo 30 eQi| `yB cy‡Îi
eq‡mi Mo 20 eQi| wcZvi eqm KZ? cÖvK-cÖv_wgK
mnKvixwkÿK (Ki‡Zvqv)2013
20 eQi 40 eQi
50 eQi 60 eQi DËi: M
ey‡S ey‡S kU©KvUt
wcZvi eqm = 30 + (102) = 30 + 20 = 50|
56. wcZv I `yB mšÍv‡bi eq‡mi Mo 27 eQi| `yB
mšÍv‡bi eq‡mi Mo 20 eQi n‡j wcZvi eqm KZ?
WvK I †Uwj‡hvMv‡hvM wefv‡Mi Gw÷‡gUi 2018
47 eQi 41 eQi
wcZvi eqm = 27 + (72) = 27 + 14 = 41
57. `yB mšÍv‡bi eq‡mi Mo 10 ermi I gvZvmn Zv‡ì
eq‡mi Mo 17 ermi n‡j, gvZvi eqm KZ? cÖvK-
cÖv_wgK mnKvix wkÿK (wSjvg) 2013
28 ermi 30 ermi
31 ermi 32 ermi DËi: M
gvZvi eqm = 17 + (7  2) = 17 + 14 = 31
58. wZb fvB‡qi eq‡mi Mo 16 eQi| wcZvmn 3 fvB‡qi
eq‡mi Mo 25 eQi n‡j, wcZvi eqm KZ?
cÖvK-cÖv_wgK mnKvix wkÿK - 2016 (gyw³‡hv×v)
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK

14  Math Tutor
45 42
52 41 DËi: M
wcZvi eqm = 25 + (93) = 25 + 27 = 52|
59. wZb mšÍv‡bi eq‡mi Mo 6 ermi I wcZvmn Zv‡ì
eq‡mi Mo 13 ermi n‡j wcZvi eqm KZ? cÖvK-
cÖv_wgK mnKvix wkÿK (‡nvqvs‡nv) 2013
32 ermi 33 ermi
wcZvi eqm = 13 + (73) = 13 + 21 = 34
60. wZb cy‡Îi eq‡mi Mo 16 eQi| wcZvmn cyÎ‡ì
eq‡mi Mo 25 eQi| wcZvi eqm KZ? mnKvix wkÿK
(BQvgwZ)-10;mnKvix wkÿK (XvKv wefvM)-06
45 eQi 48 eQi
50 eQi 52 eQi DËi: N
61. cvuP mšÍv‡bi Mo 7 eQi Ges wcZvmn Zv‡ì eq‡mi
Mo 13 eQi| wcZvi eqm KZ? cÖvK-cÖv_wgKmnKvix
wkÿK(weUv) 2014
43 eQi 33 eQi
53 eQi 63 eQi DËi: K
wcZvi eqm = 13 + (65) 13 + 30 = 43
62. wcZv I 2 cy‡Îi eq‡mi Mo 30 eQi| `yB cy‡Îi
eq‡mi Mo 20 eQi| wcZvi eqm KZ? cÖvK-cÖv_wgK
mnKvix wkÿK-13 (Ki‡Zvqv); mnKvix wkÿK
(gyw³‡hv×v)-10; mnKvix wkÿK (Ki‡Zvqv)-10
20 eQi 40 eQi
63. †Kv‡bv †kÖwY‡Z 20 Rb Qv‡Îi eq‡mi Mo 10 eQi|
wkÿKmn Zv‡ì eq‡mi Mo 12 eQi n‡j, wkÿ‡Ki
eqm KZ? cÖvK-cÖv_wgK mnKvix wkÿK (ivBb)-13
32 eQi 42 eQi
64. †Kv‡bv †kÖwYi 24 Rb Qv‡Îi Mo eqm 14 eQi| hw`
GKRb †kÖwY wkÿ‡Ki eqm Zv‡ì eq‡mi mv‡_ †hvM
Kiv nq Z‡e eq‡mi Mo GK ermi e„w× cvq|
wkÿ‡Ki eqm KZ? evsjv‡`k †ijI‡qi Dc-
mnKvix cÖ‡KŠkjx 2013
40 ermi 39 ermi
45 ermi 35 ermi DËi: L
24 Rb Qv‡Îi eq‡mi mgwó = 14 24
= 336 eQi
Zv‡ì mv‡_ GKRb wkÿK †hvM n‡j Zv‡ì msL¨v
n‡e 25 Rb Ges Zv‡ì Mo eqm 1 ermi e„w×
cvIqvq bZzb Mo n‡e 15 eQi|
wkÿKmn 25 R‡bi eq‡mi mgwó = 15  25
= 375 eQi
 wkÿ‡Ki eqm = 375 - 336 = 39 eQi|
 wkÿ‡Ki eqm = 15 + (241) = 15 + 24 = 39
 g‡b ivLyb: Dc‡iv³ mgm¨v¸‡jvi mv‡_ GB mgm¨vi
†Kvb cv_©K¨ †bB| GLv‡b Z_¨¸‡jv c~‡e©i b¨vq
mivmwi bv w`‡q c‡ivÿfv‡e w`‡q‡Q| GB †QvÆ
welqwU eyS‡Z cvi‡j MwYZ Avcbvi Rb¨ mnR n‡q
DV‡e|
65. 10 Rb Qv‡Îi Mo eqm 15 eQi| bZzb GKRb QvÎ
Avmvq Mo eqm 16 eQi n‡j bZzb Qv‡Îi eqm KZ
eQi? gnv-wnmve wbixÿK I wbqš¿‡Ki Kvh©vj‡qi Aaxb
AwWUi 2015
20 24
26 †Kv‡bvwUB bq DËi: M
10 Rb Qv‡Îi eq‡mi mgwó = 1510 = 150 eQi
bZzb QvÎ mn 11 Qv‡Îi eq‡mi mgwó = 1611
= 176 eQi
 bZzb Qv‡Îi eqm = 176 - 150 = 26 eQi|
 ey‡S ey‡S kU©KvUt 16 + (110) = 16 + 10 = 26
66. GKRb wµ‡KUv‡ii 10 Bwbs‡mi iv‡bi Mo 45.5|
11 Zg Bwbs‡m KZ ivb K‡i AvDU n‡j me Bwbsm
wgwj‡q Zvi iv‡bi Mo 50 n‡e? Dc‡RjvmnKvwiK…wl
Kg©KZv© 2011
55 ivb 45 ivb
130 ivb 95 ivb DËi: N
10 Bwbs‡mi iv‡bi mgwó = 45.5  10 = 455
11 Bwbs‡mi iv‡bi mgwó = 5011 = 550
 11 Zg Bwbs‡mi ivb = 550 - 455 = 95 ivb|
 ey‡S ey‡S kU©KvU: 50 + (4.510) = 50 + 45
= 95 ivb|
67. Kwi‡gi †bZ…Z¡vaxb 6 m`‡mï GKwU `‡ji m`m¨‡ì
eq‡mi Mo 8.5| Kwig‡K ev` w`‡j Aewkó‡ì
eq‡mi Mo K‡g ùvovq 7.2| Kwi‡gi eqm KZ?
we‡Kwe Awdmvi: 07
7.8 10.8
12.6 15 DËi: N
6 m`m¨ `‡ji m`m¨‡ì eq‡mi mgwó = 8.56mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
mgvavb
NM
LK
NM
LK
NM
LK
mgvavb
NM
LK
NM
LK
mgvavb
NM
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Math Tutor  15
Ô = 51 eQi
Kwig e¨ZxZ evKx m`m¨‡ì eq‡mi mgwó = 7.25
= 36 eQi
 Kwi‡gi eqm = 51 - 36 = 15 eQi|
 Kwi‡gi eqm = 8.5 + (1.35) = 8.5 + 6.5
= 15
68. 7 msL¨vi Mo 12 | GKwU b¤î evwZj Ki‡j Mo nq
11| evwZjK…Z msL¨vwU KZ? †mvbvjx, RbZv Ges
AMÖYx e¨vsK Awdmvi: 08
7 10
12 18 DËi: N
evwZjK…Z msL¨vwU = 12 + (1 6) = 18 |
69. 11 Rb †jv‡Ki Mo IRb 70 †KwR| 90 †KwR
IR‡bi GKRb †jvK P‡j †M‡j evKx‡ì Mo IRb
KZ nq? cÖvK-cÖv_wgK mnKvix wkÿK (†WjUv) 2014
68 †KwR 72 †KwR
80 †KwR 62 †KwR DËi: K
11 Rb †jv‡Ki IR‡bi mgwó = 7011 = 770
1 Rb †jvK P‡j †M‡j evKx 10 Rb †jv‡Ki IR‡bi
mgwó = 770 - 90 = 680 ivb
 evKx‡ì (10 Rb) Mo IRb =
10
680
= 68|
 †h P‡j wM‡q‡Q Zvi Mo IRb 70 †KwR wQj, wKš‘ †m
wb‡q †M‡Q 90 †KwR| A_©vr, †ewk wb‡q †M‡Q 20
†KwR, hv evKx 10 R‡bi cÖ‡Z¨‡Ki KvQ †_‡K 2
†KwR K‡i wb‡q †M‡Q| GRb¨ evKx 10 R‡bi eqm
Mo IRb 70 †KwR †_‡K 2 †KwR K‡i n«vm cv‡e|
 evKx‡ì Mo IRb = 70 - 2 = 68 †KwR|
70. 11 Rb evj‡Ki Mo IRb 50 †KwR| 40 †KwR
IR‡bi GKRb evjK P‡j †M‡j evwK‡ì Mo IRb
KZ n‡e? †c‡Uªvevsjv Gi D”Pgvb mnKvix 2017
50 †KwR 49 †KwR
51 †KwR 60 †KwR DËi: M
whwb P‡j †M‡jb wZwb 50 Gi cwie‡Z© 40 eQi
wb‡q †M‡Qb, †i‡L †M‡Qb 10 eQi| GB 10 eQi
evwK 10 R‡bi gv‡S Mo K‡i w`‡j 1 K‡i e„w× cv‡e|
 evwK‡ì Mo IRb = 50 + 1 = 51 †KwR|
71. iweevi †_‡K kwbevi ch©šÍ †Kv‡bv ¯’v‡bi Mo e„wócvZ
3Ó| iweevi †_‡K ïµevi ch©šÍ Mo e„wócvZ 2”| H
mßv‡ni kwbev‡i e„wócv‡Zi cwigvY KZ? mnKvix
wkÿK -07 (XvKv wefvM)
1” 5”
7” 9” DËi: N
iweevi †_‡K kwbevi 7 w`‡bi e„wócv‡Zi mgwó
= 3Ó  7 = 21Ó|
iweevi †_‡K ïµevi 6 w`‡bi e„wócv‡Zi mgwó
= 2Ó 6 = 12Ó|
 kwbev‡ii e„wócv‡Zi cwigvY = 21Ó - 12Ó
= 9Ó|
 ey‡S ey‡S kU©KvUt kwbev‡i e„wócv‡Zi cwigvY
= 3 + (16) = 3 + 6 = 9Ó|
72. 3wU msL¨vi Mo 6 Ges H 3wU msL¨vmn †gvU 4wU
msL¨vi Mo 8 n‡j PZz_© msL¨vwUi A‡a©‡Ki gvb KZ?
cÖv_wgK we`¨vjq mnKvix wkÿK 2019
7 8
5 6 DËi: K
PZz_© msL¨vwU = 8 + (23) = 8 + 6 = 14
 msL¨vwUi A‡a©K =
2
14
= 7|
(L) †ewk msL¨vi Mo hw` Kg msL¨vi Mo †_‡K gv‡b †QvU nq
73. wcZv I gvZvi eq‡mi Mo 45 eQi| Avevi wcZv,
gvZv I GK cy‡Îi eq‡mi Mo 36 eQi| cy‡Îi eqm
KZ? cÖv_wgKmnKvixwkÿKwb‡qvMcixÿv2015
20 eQi 18 eQi
wcZv I gvZvi eq‡mi mgwó = 45  2
= 90 eQi|
wcZv, gvZv I GK cy‡Îi eq‡mi mgwó = 36  3
= 108 eQi|
 cy‡Îi eqm = 108 - 90 = 18 eQi|
 ey‡S ey‡S kU©‡UKwbK:
18 eQi
wcZv gvZv wcZv gvZv cyÎ
 ïiæ‡Z wcZv I gvZvi eq‡mi Mo wQj 45| c‡i
cyÎ hy³ nIqvq eq‡mi Mo n«vm †c‡q nq 36|
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16  Math Tutor
G‡ÿ‡Î wcZv I gvZvi eq‡mi Mo (45-36) ev 9
K‡i †gvU 18 n«vm cvq| GB n«vmK…Z eqm wM‡q
cy‡Îi eq‡mi mv‡_ hy³ n‡q cy‡Îi Mo eqm n‡q‡Q
36| wKš‘ cÖK…Zc‡ÿ cy‡Îi eqm 36 †_‡K Av‡iv
Kg| cy‡Îi eqm †ei Ki‡Z n‡j wcZv I gvZvi
n«vmK…Z eqm cy‡Îi eqm †_‡K ev` w`‡Z n‡e|
cy‡Îi eqm : 36 - (92) = 36 - 18 = 18 eQi|
74. wcZv I gvZvi eq‡mi Mo 20 ermi| wcZv, gvZv I
cy‡Îi eq‡mi Mo 16 ermi n‡j cy‡Îi eqm KZ?
cÖvK-cÖv_wgK mnKvixwkÿK(`vRjv,†nvqvs‡nv,wgwmwmwc)13
8 ermi 15 ermi
16 ermi 16
2
1
ermi DËi: K
ey‡S ey‡S kU©KvU: cy‡Îi eqm = 16 - (42)
= 8 ermi|
cÖvK-cÖv_wgK mnKvix wkÿK (‡nvqvs‡nv) 2013
2 ermi 4 ermi
ey‡S ey‡S kU©KvU: cy‡Îi eqm = 18 - (72)
= 18 - 14 = 4 ermi|
cy‡Îi Mo eqm 24 ermi n‡j, cy‡Îi eqm KZ?
cÖvK-cÖv_wgK mnKvix wkÿK (wgwmwmwc) 2013
8 ermi 10 ermi
11 ermi 12 ermi DËi: N
cy‡Îi eqm = 24 - (62) = 24 - 12
= 12 eQi|
cÖvK-cÖv_wgK mnKvix wkÿK (`vRjv) 2013
8 ermi 15 ermi
16 ermi 16.5 ermi DËi: K
cy‡Îi eqm = 16 - 8 = 8 eQi|
78. wcZv I gvZvi eq‡mi Mo 42 eQi| Avevi wcZv,
gvZv I GK cy‡Îi Mo eqm 32 eQi| cy‡Îi eqm
KZ? cÖvK-cÖv_wgK mnKvix wkÿK-13 (eywoM½v)
8 eQi 10 eQi
cy‡Îi eqm = 32 - 20 = 12 eQi|
mnKvix wkÿK (nvmbv‡nbv)-11
12 ermi 14 ermi
cy‡Îi eqm = 32 - 16 = 16 ermi|
80. wcZv I gvZvi Mo eqm 36 eQi| wcZv, gvZv I
cy‡Îi Mo eqm 28 eQi n‡j, cy‡Îi eqm KZ?
mnKvix wkÿK (gyw³‡hv×v)-10
9 eQi 11 eQi
cy‡Îi Mo eqm 27 eQi n‡j cy‡Îi eqm KZ?
mnKvix wkÿK (gyw³‡hv×v)-10
9 eQi 11 eQi
cy‡Îi Mo eqm 32 eQi n‡j, cy‡Îi eqm KZ? cÖavb
wkÿK (XvKv wefvM)-07
14 eQi 16 eQi
83. 12 Rb Qv‡Îi eq‡mi Mo 20 eQi | hw` bZzb
GKRb Qv‡Îi eqm AšÍf~©³ Kiv nq, Z‡e eq‡mi Mo
1 eQi K‡g hvq| bZzb Qv‡Îi eqm KZ? IFICBank
probationaryofficer:09
5 7
9 11 DËi: L
12 Rb Qv‡Îi eq‡mi mgwó = 2012 = 240 eQi
GKRb Qv‡Îi eqm AšÍfz©³ Kivq 13 Rb Qv‡Îi
eq‡mi Mo 1 n«vm †c‡q 19 nq|
13 Rb Qv‡Îi eq‡mi mgwó = 19  13 = 247 eQi
 bZzb Qv‡Îi eqm = 247 - 240 = 7 eQi&
bZzb Qv‡Îi eqm = 19 - (12 1) = 19 - 12 = 7
84. †Kv‡bv †kÖwY‡Z 20 Rb QvÎxi eq‡mi Mo 12 eQi| 4
Rb bZzb QvÎx fwZ© nIqv‡Z eq‡mi Mo 4 gvm K‡g
†Mj| bZzb 4 Rb QvÎxi eq‡mi Mo KZ? cÖvK-
cÖv_wgK mnKvix wkÿK 2013 (hgybv); mnKvix
wkÿK 2012 (cÙ); mnKvix wkÿK 2008 (†Mvjvc)
11 eQi 9 eQiLK
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Math Tutor  17
20 Rb QvÎxi eq‡mi mgwó = 1220 eQi
= 240 eQi
4 Rb bZzb QvÎx fwZ© nIqvq 24 QvÎxi eq‡mi Mo
nq (12 eQi - 4 gvm) ev 11 eQi 8 gvm|
24 Rb QvÎxi eq‡mi mgwó = 11 eQi 8 gvm24
= (11 +
3
2
)24 eQi
= 11  24 eQi +
3
2
24 eQi
= 264 eQi + 16 eQi = 280 eQi|
bZzb 4 Rb QvÎxi eqm = 280 - 240 = 40 eQi|
 bZzb 4 Rb QvÎxi eq‡mi Mo =
4
40
eQi
= 10 eQi|
 g‡b ivLyb: GLv‡b 8 gvm‡K eQ‡i iƒcvšÍi Kiv n‡q‡Q
12
8
ev
3
2
eQi| A‡¼ gvm I eQi GKmv‡_ _vK‡j
Ges Avcwb MYbvq `ÿ bv n‡j we¯ÍvwiZ wbq‡gB
mgvavb Ki‡eb)
4 Rb bZzb QvÎxi eq‡mi Mo
= 11 eQi 8 gvm - (
4
204
gvm)
= 11 eQi 8 gvm - 20 gvm
= 11 eQi 8 gvm - 1 eQi 8 gvm
= 10 eQi|
(GLv‡b 4 Rb QvÎxi Mo nIqvq (420) †K 4 Øviv
fvM Kiv n‡q‡Q| MYbvq `ÿ _vK‡j K‡qK †m‡K‡Û
mgvavb Kiv m¤¢e)
(M) GB av‡c (K) I (L) wbqg AbymiY Kiv n‡q‡Q wKš‘ `v‡M GKUz wfbœZv Av‡Q
85. wZb eQi Av‡M iwng I Kwi‡gi iq‡mi Mo wQj 18
eQi| Avjg Zv‡ì mv‡_ †hvM`vb Kivq Zv‡ì
eq‡mi Mo ‡e‡o 22 eQi nq| Avj‡gi eqm KZ?
mnKvixwkÿK(K‡cvZvÿ)-10;†iwR÷vW© cÖv_wgKwe`¨vjqmnKvix
wkÿK(wkDwj)2011
30 eQi 28 eQi
(cÖ‡kœ wZb eQi Av‡M iwng I Kwi‡gi eq‡mi Mo
†`qv Av‡Q, †mwU‡K eZ©gvb eq‡mi M‡o iƒcvšÍi Ki‡Z n‡e)
wZb eQi Av‡M iwng I Kwi‡gi eq‡mi Mo wQj 18
eZ©gv‡b iwng I Kwi‡gi eq‡mi Mo = 18+3 = 21
eZ©gv‡b Zv‡ì eq‡mi mgwó = 212 = 42 eQi
iwng, Kwig I Avj‡gi eq‡mi Mo = 22 eQi
Zv‡ì eq‡mi mgwó = 223 = 66 eQi
Avj‡gi eqm = 66 - 42 = 24 eQi|
 ey‡S ey‡S kU©KvU: (cÖ_‡g eZ©gvb eqm †ei K‡i wb‡Z
n‡e, Zvici c~‡e© wbq‡g mgvavb Ki‡Z n‡e)
iwng I Kwi‡gi eZ©gvb eq‡mi Mo = 18+3 = 21
iwng, Kwig I Avj‡gi eq‡mi Mo = 22
 Avj‡gi eqm = 22 + (1  2) = 24 eQi|
86. wcZv I `yB cy‡Îi Mo eqm 20 eQi| `yB eQi c~‡e©
`yB cy‡Îi Mo eqm wQj 12 eQi| wcZvi eZ©gvb
eqm KZ? cÖvK-cÖv_wgKmnKvixwkÿK(myigv,kxZjÿ¨v)2013
26 eQi 28 eQi
`yB cy‡Îi eZ©gvb eq‡mi Mo = 12 + 2 = 14
 wcZvi eZ©gvb eqm = 20 + (62)
= 20 + 12 = 32 eQi|
87. wZb eQi Av‡M `yB †ev‡bi eq‡mi Mo wQj 24 eQi|
eZ©gv‡b `yB †evb I Zv‡ì GK fvB‡qi eq‡mi Mo
25 eQi| fvB‡qi eZ©gvb eqm KZ eQi? evsjv‡`k
wkwcs K‡cv©‡ik‡b wmwbqi A¨vwm÷¨v›U 2018
27 24
21 18 DËi: M
`yB †ev‡bi eZ©gvb eq‡mi Mo = 24 + 3 = 27
fvB‡qi eZ©gvb eqm = 25 - (22) = 21
88. wcZv I `yB cy‡Îi eZ©gvb Mo eqm 20 ermi| 2
ermi ci `yB cy‡Îi Mo eqm 12 ermi n‡j wcZvi
eZ©gvb eqm KZ? cÖvK-cÖv_wgKmnKvixwkÿK(`vRjv,
wgwmwmwc)2013
40 ermi 42 ermi
43 ermi 44 ermi DËi: K
(cÖ‡kœ `yB ermi ci `yB cy‡Îi eq‡mi Mo †`qv Av‡Q, †mwU‡K
eZ©gvb eq‡mi M‡o iƒcvšÍi Ki‡Z n‡e)
`yB cy‡Îi eZ©gvb eq‡mi Mo = 12 - 2 = 10 ermi
Zv‡ì eZ©gvb eq‡mi mgwó = 102 = 20 ermi
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18  Math Tutor
wcZv I `yB cy‡Îi eq‡mi Mo = 20 ermi
Zv‡ì eq‡mi mgwó = 20  3 = 60 ermi
 wcZvi eZ©gvb eqm = 60 - 20 = 40 ermi|
`yB cy‡Îi eZ©gvb eq‡mi Mo = 12 - 2 = 10
wcZv I `yB cy‡Îi eq‡mi Mo = 20 eQi
 wcZvi eqm = 20 + (102) = 40 eQi|
89. wcZv I `yB cy‡Îi eZ©gvb Mo eqm 22 ermi| 3
ermi ci `yB cy‡Îi Mo eqm 13 ermi n‡j wcZvi
eZ©gvb eqm KZ? cjøx we`y¨Zvqb †ev‡W©i mnKvix
Gb‡dvm©‡g›U †Kv-AwW©‡bUi 2017
40 ermi 42 ermi
`yB cy‡Îi eZ©gvb eq‡mi Mo = 13 - 3 = 10
wcZvi eZ©gvb eqm = 22 + (102) = 42|
(N) GK R‡bi cwie‡Z© bZzb GKRb †hvM †`qvi d‡j Mo e„w× ev n«vm nIqv
90. 8 R‡bi GKwU `‡j 65 †KwR IR‡bi GKR‡bi
cwie‡Z© bZzb GKRb †hvM †`qvq Zv‡ì Mo IRb
2.5 †KwR †e‡o hvq| bZzb e¨w³i IRb KZ †KwR?
evsjv‡`kK…wlDbœqbK‡cv©‡ik‡bimnKvixcÖmvkwbKKg©KZ©v2017
45 76
80 85 DËi: N
bZzb e¨w³ †hvM`vb Kivq IRb e„w× cvq
= 2.5  8 †KwR = 20 †KwR|
 bZzb e¨w³i IRb = 65 + 20 = 85 †KwR|
8 R‡bi `‡j 65 †KwR IR‡bi GKRb P‡j hvq Ges
Zvi cwie‡Z© whwb Av‡mb Zvui IRbI hw` 65 †KwR
nZ, Zvn‡j Zv‡ì Mo IR‡bi †Kvb cwieZ©b nZ
bv| wKš‘ H e¨w³ `‡j †hvM`vb Kivi ci Zv‡ì
cÖ‡Z¨‡Ki Mo IRb 2.5 †KwR K‡i †e‡o †M‡Q| Zvi
gv‡b, bZzb †hvM`vbK…Z e¨w³i eqm Ô65 + AviI
†ewkÕ n‡e| GLb cÖkœ n‡”Q 65 Gi †P‡q KZUzKz
†ewk n‡e? mnR DËi n‡”Q- 8 R‡bi Mo hZ e„w×
†c‡q‡Q Zvi mgwói mgvb|
bZzb e¨w³i IRb = 65 + (82.5)
= 65 + 20 = 85 eQi|
 g‡b ivLyb: e„w× bv †c‡q hw` n«vm †cZ Zvn‡j †hvM
bv n‡q we‡qvM nZ| 92 bs cÖkœ †`Lyb
91. 8 Rb †jv‡Ki Mo eqm 2 e„w× cvq hLb 24 eQi
eqmx †Kvb †jv‡Ki cwie‡Z© GKRb gwnjv †hvM
†`q| gwnjvwUi eqm KZ? RbZv e¨vsK (EO)-17
35 eQi 28 eQi
gwnjvi eqm = 24 + (28) = 24 + 16 = 40|
92. 7 Rb †jv‡Ki Mo IRb 3 cvDÛ K‡g hvq hLb 10
†÷vb IR‡bi GKRb †jv‡Ki cwie‡Z© bZzb GKRb
†hvM`vb K‡i| bZzb †jvKwUi IRb KZ? cÖv_wgK
we`¨vjqmnKvixwkÿK(òvUvbv)2008
9 †÷vb 8 †÷vb
8
2
1
†÷vb 7
2
1
†÷vb DËi: M
Avgiv Rvwb, 14 cvDÛ = 1 †÷vb
bZzb †jvK †hvM`vb Kivq IRb K‡g hvq
= 37 = 21 cvDÛ = 1.5 †÷vb
bZzb †jv‡Ki IRb = 10 - 1.5 †÷vb = 8.5 †÷vb|
 ey‡S ey‡S mgvavb:
bZzb †jvKwUi IRb = 10 - 1.5 = 8.5 †÷vb|
†R‡b wbb - 44
 hv‡ì Mo †ei Ki‡eb, Zvi Av‡M Zv‡ì †gvU †ei Ki‡eb|
93. P Ges Q Gi mgwó 72| R Gi gvb 42| P, Q
Ges R Gi Mo KZ? evsjv‡`ke¨vsKA¨vwmm‡U›UwW‡i±i:08
32 34
36 38 DËi: N
(P, Q Ges R Gi Mo †ei Ki‡Z n‡e, GRb¨ P,
Q Ges R Gi mgwó †ei Ki‡Z n‡e)
P, Q Ges R Gi mgwó = 72 + 42 = 114
 P, Q Ges R Gi Mo =
3
114
= 38|
94. A Ges B Gi mgwó 40 Ges C =32 n‡j, A, B
Ges C Gi Mo KZ? evsjv‡`ke¨vsKAwdmvi:01
24 26
28 30 DËi: K
A, B I C Gi mgwó = 40 + 32 = 72 |mgvavb
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Math Tutor  19
 A, B I C Gi Mo =
3
72
= 24 |
95. 𝐱 I y Gi gv‡bi Mo 9 Ges z = 12 n‡j 𝒙, y, z
Gi gv‡bi Mo KZ? 20ZgwewmGm
6 9
10 12 DËi: M
x I y Gi gv‡bi mgwó = 9  2 = 18
∴ x, y, z Gi gv‡bi mgwó = 18 + 12 = 30
myZivs, 𝑥, y, z Gi Mo =
3
30
= ১০ DËit M
96. K, L I M Gi Mo eqm 40 eQi, K I M Gi eqm
GK‡Î 85 eQi L Gi eqm n‡e- cwievi-cwiKíbv
†gwW‡KjAwdmvi 1994
30 eQi 35 eQi
K, L I M Gi eq‡mi mgwó = 403 = 120 eQi
K I M Gi eq‡mi mgwó = 85 eQi|
 L Gi eqm = 120 - 85 eQi = 35 eQi|
97. wZbwU msL¨vi Mo 24| `yBwU msL¨v 21 I 23 n‡j,
Z…Zxq msL¨vwU KZ? evsjv‡`ke¨vsKA¨vwmm‡U›U wW‡i±i:06
20 24
26 28 DËi: N
wZbwU msL¨vi mgwó = 24 3 = 72
`yBwU msL¨vi mgwó = 21 + 23 = 44
myZivs, Z…Zxq msL¨v = 72 - 44 = 28
98. wZbwU msL¨vi Mo 7| hw` `yBwU msL¨v 0 nq, Z‡e
Z…Zxq msL¨vwU KZ? †mvbvjx,RbZvIAMÖYxe¨vsK:08
15 17
19 21 DËi: N
wZbwU msL¨vi mgwó = 7  3 = 21
`yBwU msL¨vi mgwó = 0 + 0 = 0
 Z…Zxq msL¨vwU = 21 - 0 = 21 |
99. wZbRb †jv‡Ki IR‡bi Mo 53 †KwR| G‡ì Kv‡iv
IRb 51 †KwRi Kg bq| G‡ì GKR‡bi m‡e©v”P
IRb n‡e- cwievicwiKíbvAwa.mn.cwiKíbvKg©KZ©v12
53 †KwR 55 †KwR
57 †KwR 59 †KwR DËi: M
wZbRb †jv‡Ki IR‡bi mgwó = 533 †KwR
= 159 †KwR
`yRb †jv‡Ki IRb me©wb¤œ Ki‡j Aci Rb †jv‡Ki IRb
m‡e©v”P Kiv m¤¢e| cÖkœvbyhvqx `yRb †jv‡Ki IRb me©wb¤œ
51 Kiv hv‡e|
`yRb †jv‡Ki me©wb¤œ IR‡bi mgwó = 51  2 †KwR
= 102 †KwR
 3q †jv‡Ki m‡e©v”P IRb n‡e = 159 - 102 †KwR
= 57 †KwR|
100. wZb m`‡mï GKwU weZK©`‡ji m`m¨‡ì Mo eqm
24 eQi| hw` †Kvb m`‡mï eqmB 21 Gi wb‡P bv
nq Zv‡ì †Kvb GKR‡bi eqm m‡e©v”P KZ n‡Z
cv‡i? 33Zg wewmGm wcÖwjwgbvwi
25 eQi 30 eQi
GKRb †jv‡Ki m‡e©v”P ehm = 24 + (32)
= 24 + 6 = 30 eQi|
101. 3 eÜzi IR‡bi Mo 33 †KwR | wZbR‡bi ga¨
†Kvb eÜzi IRbB 31 †KwRi Kg bq| wZb eÜzi
GKR‡bi IRb m‡ev©”P KZ n‡Z cv‡i? evsjv‡`k
e¨vsK Awdmvi : 01
37 35
33 32 DËi: K
GKR‡bi m‡e©v”P IRb = 33 + (22) †KwR
= 33 + 4 †KwR
= 37 †KwR|
05.10
102. wcZv I cy‡Îi eq‡mi Mo 40 eQi Ges gvZv I H
cy‡Îi eq‡mi Mo 35 eQi| gvZvi eqm 50 eQi
n‡j, wcZvi eqm KZ? 14Zg †emiKvwi cÖfvlK
wbeÜb cixÿv 2017
50 eQi 60 eQi
wcZv I cy‡Îi eq‡mi Mo = 40 eQi
wcZv I cy‡Îi eq‡mi mgwó = 40  2 = 80 eQi
gvZv I cy‡Îi eq‡mi Mo = 35 eQi
 gvZv I cy‡Îi eq‡mi mgwó = 35  2 = 70 eQi
(wcZvi eqm †ei Ki‡Z n‡j cy‡Îi eqm Rvb‡Z n‡e)
cy‡Îi eqm = (70 - 50) eQi = 20 eQi
 wcZvi eqm = (80 - 20 ) eQi = 60 eQi|
 ey‡S ey‡S kU©KvU 01:
wcZv + cyÎ  Mo 40 eQi  †gvU 402 = 80
gvZv + cyÎ  Mo 35 eQi  †gvU 352 = 70
wcZv  gvZv = 10
wcZvi eqm gvZvi †P‡q 10 eQi †ewk|
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20  Math Tutor
 wcZvi eqm = 50 + 10 = 60 eQi|
 ey‡S ey‡S kU©KvU 02:
wcZvi mv‡_ cy‡Îi Mo, gvZvi mv‡_ cy‡Îi M‡oi
†P‡q 5 †ewk , ZvB wcZvi eqm gvZvi eq‡mi †P‡q
†ewk n‡e|
 wcZvi eqm = 50 + (52) = 50 + 10
= 60 eQi|
 g‡b ivLyb: gvZvi eqm †`qv Av‡Q, wcZvi eqm †ei
Ki‡Z n‡e| gvZvi Zzjbvq wcZvi eqm hZUzKz †ewk
†mwU gv‡qi eq‡mi mv‡_ †hvM Ki‡Z n‡e| Avevi hw`
wcZvi eqm †`qv _vKZ, Zvn‡j wcZvi eqm †_‡K
gvZvi eqm hZUzKz Kg n‡e †mwU wcZvi eqm †_‡K
we‡qvM Ki‡Z n‡e|
103. iwng I Kwi‡gi eq‡mi Mo 35 eQi| iwng I
nvgRvi eq‡mi Mo 20 eQi| nvgRvi eqm 11
eQi n‡j Kwi‡gi eqm KZ? 16Zg†emiKvwiwkÿK
wbeÜb(K‡jR/mgch©vq) 2019
40 eQi 41 eQi
nvgRvi eqm †`qv Av‡Q, Kwi‡gi eqm †ei Ki‡Z
n‡e| nvgRvi eq‡mi mv‡_ Kwi‡gi eqm hZUzKz †ewk
†mwU †hvM Ki‡Z n‡e|
Kwi‡gi eqm = 11 + (152) = 41 eQi|
104. wcZv I 3 cy‡Îi eqm A‡cÿv gvZv I 3 cy‡Îi
eq‡mi Mo 1
2
1
eQi Kg| gvZvi eqm 30 n‡j
wcZvi eqm KZ? cÖvK-cÖv_wgKmnKvixwkÿK(†gNbv)2013
30 eQi 31
2
1
eQi
wcZvi eqm = 30 + ( 1.54) = 36 eQi|
105. wcZv I `yB cy‡Îi eqm A‡cÿv gvZv I D³ `yB
cy‡Îi eq‡mi Mo 2 eQi Kg| wcZvi eqm 30 eQi
n‡j gvZvi eqm KZ? cÖv_wgK we`¨vjq mnKvwi wkÿK
(emšÍ) 2010
20 eQi 22 eQi
wcZvi eqm †`qv Av‡Q, gvZvi eqm †ei Ki‡Z
n‡e| ZvB wcZvi eq‡mi †_‡K gvZvi eqm hZUzKz
Kg †mwU we‡qvM Ki‡Z n‡e|
 gvZvi eqm = 30 - (23) = 24 eQi|
106. wcZv I Pvi cy‡Îi eq‡mi Mo, gvZv I Pvi cy‡Îi
eq‡mi Mo A‡cÿv 2 eQi †ewk| wcZvi eqm 60
n‡j gvZvi eqm KZ? Rbkw³ Kg©ms¯’vb I cÖwkÿY
ey¨‡ivi Dc-cwiPvjK 2007
48 eQi 52 eQi
gvZvi eqm = 60 - (25) = 50 eQi|
107. eya, e„n¯úwZ I ïµev‡ii Mo ZvcgvÎv 400
c Ges
e„n¯úwZ, ïµ I kwbev‡ii Mo ZvcgvÎv 410
c |
kwbev‡ii ZvcgvÎv 420
c | kwbev‡ii ZvcgvÎv 420
c n‡j eyaev‡ii ZvcgvÎv KZ? bvwm©s I wgWIqvBdvwi
Awa`ß‡ii wmwbqi ÷vd bvm© 2018
380
c 390
c
410
c 420
c DËi: L
kwbev‡ii ZvcgvÎv †`qv Av‡Q, eyaev‡ii ZvcgvÎv
†ei Ki‡Z n‡e| cÖkœvbymv‡i, eyaev‡ii (e„n¯úwZ I
ïµevimn) ZvcgvÎvi Mo kwbev‡ii (e„n¯úwZ I
ïµevimn) ZvcgvÎvi M‡oi †P‡q Kg, ZvB
kwbev‡ii ZvcgvÎv †_‡K we‡qvM Ki‡Z n‡e|
 kwbev‡ii ZvcgvÎv = 420
c - (13) = 390
c
108. K, L I M Gi gvwmK Mo †eZb 500 UvKv| L, M I
N Gi gvwmK Mo †eZb 450 UvKv| K Gi †eZb
540 UvKv n‡j N Gi †eZb KZ? cÖvK-cÖv_wgK mn.
wkÿK (cÙv) 2013
375 UvKv 380 UvKv
385 UvKv 390 UvKv DËi: N
K Gi †eZb †`qv Av‡Q, N Gi †eZb †ei Ki‡Z
n‡e| K Gi ( L I M mn) M‡oi Zzjbvq N ( L I M
mn) Mo Kg, ZvB K Gi †eZb †_‡K we‡qvM Ki‡Z
n‡e|
 N Gi †eZb = 540 - (503) = 540 - 150
= 390 UvKv |
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m¤ú~Y© Aa¨vqwU‡Z †Kv_vI †Kvb fzj ÎæwU cwijwÿZ n‡j Avgv‡K `qv K‡i AewnZ Ki‡eb|
 †dmey‡K fb/ kabial.noor A_ev B‡gB‡j kabialnoor@gmail.com AewnZ Kiæb|

Math Tutor  21
109. 20 Rb QvÎ MwYZ cixÿvq AskMÖnY Kij| Zv‡ì
g‡a¨ `yBRb M‡o 60 b¤î, †Zi Rb M‡o 65 b¤î I
Aewkó mK‡j M‡o 55 b¤î †cj| QvÎiv M‡o KZ
b¤î †cj? 24ZgwewmGm(wjwLZ)
2 Rb Qv‡Îi Mo b¤î = 60
 2 Rb Qv‡Îi †gvU b¤î = 60  2 = 120
13 Rb Qv‡Îi †gvU b¤î = 65  13 = 845
Aewkó QvÎ msL¨v = 20 - 2 - 13 Rb = 5 Rb
 5 Rb Qv‡Îi †gvU b¤î = 55  5 = 275
20 Rb Qv‡Îi †gvU b¤î = 120 + 845 + 275
= 1240
 20 Rb Qv‡Îi Mo b¤î =
20
1240
= 62| (DËi)
110. 5 Rb evj‡Ki eq‡mi Mo 10 eQi| H `‡j AviI 2
Rb evjK †hvM w`j| Zv‡ì mK‡ji eq‡mi Mo nq
12 eQi| †hvM`vbKvix evjK `ywU hw` mgeqmx nq
Z‡e Zv‡ì cÖ‡Z¨‡Ki eqm KZ? cywjkmve-B¯ú‡c±i
(wbi¯¿)wb‡qvMcixÿv(cyiæl/bvix)2012 (wjwLZ)
5 Rb evj‡Ki eq‡mi Mo 10 eQi
 5 Rb evj‡Ki †gvU eqm = 105 eQi
= 50 eQi
Avevi,
(5+2) ev 7 Rb evj‡Ki Mo 12 eQi
 7 Rb evj‡Ki †gvU eqm = 127 eQi
= 84 eQi|
2 Rb evj‡Ki †gvU eqm = 84 - 50 eQi
= 34 eQi
`yRb evjK mgqeqmx nIqvq Zv‡ì eq‡mi Mo n‡e
Zv‡ì cÖ‡Z¨‡Ki eqm|
 Zv‡ì cÖ‡Z¨‡Ki eqm =
2
34
eQi = 17 eQi|
(DËi)
111. K I L Gi gvwmK Av‡qi Mo 300 UvKv, L I M Gi
gvwmK Av‡qi Mo 350 UvKv Ges K I M Gi gvwmK
Av‡qi Mo 325 UvKv| K Gi gvwmK Avq KZ? cywjk
mve-BÝ‡c±iwb‡qvMcixÿv1999 (wjwLZ)
K I L Gi gvwmK Mo Avq 300 UvKv
K I L Gi gvwmK †gvU Avq = 300  2 UvKv
= 600 UvKv
L I M Gi gvwmK Mo Avq 350 UvKv
L I M Gi gvwmK †gvU Avq = 350  2 UvKv
= 700 UvKv
K I M Gi gvwmK Mo Avq 325 UvKv
K I M Gi gvwmK †gvU Avq = 325  2 UvKv
= 650 UvKv
GLb,
K + L + L + M + K + M Gi
†gvU Avq = 600 + 700 + 650 UvKv
ev, 2 ( K + L + M) Gi †gvU Avq = 1950 UvKv
 K + L + M Gi †gvU Avq =
2
1950
UvKv
= 975 UvKv
myZivs, K Gi gvwmK Avq = (K + L + M Gi gvwmK
Avq) - ( L + M Gi gvwmK Avq) = (975 - 700)
UvKv = 275 UvKv| (DËi)
112. KwZcq Qv‡Îi MwY‡Zi cÖvß b¤^‡ii mgwó 1190 Gi
mv‡_ 88 b¤î cÖvß GKRb Qv‡Îi b¤î †hvM nIqv‡K
QvÎ‡ì cÖvß b¤^‡ii Mo 1 †e‡o †Mj| KZRb Qv‡Îi
cÖvß b¤^‡ii mgwó 1190? ¯^v¯’ÏcwieviKj¨vY gš¿Yvj‡qi
¯^v¯’¨wkÿvIcwieviKj¨vYwefv‡MiZ…Zxq†kÖwYiRbej-19(wjwLZ)
g‡b Kwi, †gvU Qv‡Îi msL¨v = x Rb
Ges Zv‡ì b¤^‡ii Mo =
x
1190
1 Rb Qv‡Îi 88 b¤î †hvM nIqvq †gvU QvÎ = x+1
Rb Ges Zv‡ì Mo =
1
881190


x
kZ©g‡Z,
1
881190


x
-
x
1190
= 1
ev,
)x(x
xx
1
119011901278


= 1
ev, x2
+ x = 88x  1190 = 0
ev, x2
+ x  88x  1190 = 0
ev, x2
 87x  1190 = 0
ev, x2
 70x  17x + 1190 = 0
ev, x( x - 70) - 17 ( x - 70) = 0
ev, (x - 70) (x - 17) = 0
x - 70 = 0 A_ev x - 17 = 0
 x = 70  x = 17
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ïay wjwLZ Av‡jvPbv 

22  Math Tutor
myZivs QvÎ msL¨v 70 A_ev 17 Rb|
113. GKRb †`vKvb`vi 12 w`‡b 500 UvKv Avq
Ki‡jv Avi cÖ_g 4 w`‡bi Mo Avq 40 UvKv n‡j
Aewkó w`b¸wji Mo Avq KZ? evsjv‡`k†ijI‡q
mnKvix†÷kbgv÷vi2018(wjwLZ)
†`qv Av‡Q, 12 w`‡bi †gvU Avq = 500 UvKv
cÖ_g 4 w`‡b †gvU Avq = 40 4 = 160 UvKv
Aewkó (12 - 4) ev 8 w`‡bi Avq = 500 - 160
= 340 UvKv|
114. 7wU msL¨vi Mo 40| Gi mv‡_ 3wU msL¨v †hvM Kiv
n‡jv| msL¨v 3wUi Mo 21| mgwóMZfv‡e 10wU
msL¨vi Mo KZ? cwi‡ekAwa`ß‡iiAwdmmnKvix Kvg
Kw¤úDUvigy`ªvÿwiK2018(wjwLZ)
7wU msL¨vi Mo 40
 7wU msL¨vi mgwó = 40 7 = 280
3 wU msL¨vi Mo 21
 3wU msL¨vi mgwó = 21  3 = 63
(7 + 3) ev 10 wU msL¨vi mgwó = 280 + 63
= 343
 10 wU msL¨vi Mo =
10
343
= 34.3 | (DËi)
115. hw` 5wU µwgK we‡Rvo msL¨vi Mo 55 nq Z‡e
†kl `ywU msL¨vi Mo KZ? cÖv_wgKIMYwkÿvgš¿Yvj‡qi
mvuUgy`ªvÿwiKKvgKw¤úDUviAcv‡iUi2017(wjwLZ)
g‡bKwi, 1g µwgK we‡Rvo msL¨vwU x
2q Ó Ó Ó x + 2
3q Ó Ó Ó x + 4
4_© Ó Ó Ó x + 6
5g Ó Ó Ó x + 8
†`qv Av‡Q, 5wU µwgK we‡Rvo msL¨vi Mo 55
5 wU µwgK we‡Rvo msL¨vi mgwó 555 = 275
cÖkœg‡Z,
x + x + 2 + x + 4 + x + 6 + x + 8 = 275
ev, 5x + 20 = 275
ev, 5x = 275 - 20
ev, 5x = 255
 x =
5
255
= 51
†kl `ywU msL¨v h_vµ‡g x + 6 = 51 + 6 = 57
Ges x + 8 = 51 + 8 = 59 |
 †kl `ywU msL¨vi Mo =
2
5957
=
2
116
= 58
cy‡Îi Mo eqm 27 eQi n‡j cy‡Îi eqm KZ? e¯¿ I
cvUgš¿Yvj‡qiDc-mnvKvixcvUDbœqbKg©KZ©v2012(wjwLZ)
wcZv I gvZvi Mo eqm 35 eQi
 wcZv I gvZvi †gvU eqm = 352 eQi
= 70 eQi
wcZv, gvZv I cy‡Îi Mo eqm 27 eQi
 wcZv, gvZv I cy‡Îi †gvU eqm = 273 eQi
= 81 eQi
cy‡Îi eqm = 81 - 70 eQi = 11 eQi| (DËi)
117. wcZv I Zv‡ì Pvi mšÍv‡bi eq‡mi Mo 22 eQi 5
gvm| gvZv I Zvu‡ì H Pvi mšÍv‡bi eq‡mi Mo 21
eQi 2 gvm| wcZvi eqm 45 eQj n‡j, gvZvi eqm
KZ? wb¤œ gva¨wgKMwYZ(1996wkÿvel©)cÖkœgvjv3
wcZv I Pvi mšÍv‡bi eq‡mi mgwó = 22 eQi 5
gvm  5 = 112 eQi 1 gvm
 Pvi mšÍv‡bi eq‡mi mgwó = 112 eQi 1 gvm -
45 eQi = 67 eQi 1 gvm
Avevi, gvZv I Pvi mšÍv‡bi eq‡mi mgwó = 21 eQi
2 gvm  5 = 105 eQi 10 gvm
 gvZvi eqm = gvZv I Pvi mšÍv‡bi eq‡mi mgwó
- Pvi mšÍv‡bi eq‡mi mgwó = 105 eQi 10 gvm -
67 eQi 1 gvm = 38 eQi 9 gvm| (DËi)
118. †Kvb we`¨vj‡qi GKwU †kÖwYi 25 Rb Qv‡Îi eq‡mi
Mo 12 eQi| H †kÖwY‡Z 14, 15 I 21 eQi eq‡mi
3 Rb bZzb QvÎ fwZ© nj| H †kÖwY‡Z eZ©gv‡b
QvÎ‡ì eq‡mi Mo KZ? wb¤œ gva¨wgKMwYZ(1996wkÿvel©)
cÖkœgvjv3
25 Rb Qv‡Îi †gvU eqm = 12  25 = 300 eQi
3 Rb bZzb Qv‡Îi †gvU eqm = (14 + 15 + 21 ) eQi
= 50 eQi|
(25 + 3) ev 28 Rb Qv‡Îi †gvU eqm
= 300 + 50 eQi
= 350 eQi|
28 Rb Qv‡Îi eq‡mi Mo =
28
350
eQi
= 12.5 eQi
AZGe, H †kÖwY‡Z eZ©gv‡b QvÎ‡ì eq‡mi Mo = 12.5
eQi| (DËi)
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