SKKN các bài toán giải bằng cách lập phương trình hay hệ phương trình

Phần A
đặt vấn đề
i - lý do chọn đề tài:

! "#!$%!& '
()*$ +, % !# ''
(
/0!"1! 0!#2+3+!
++4%50(617 %8%%&$!1
+9":0(;0 <'<=53 %
>(?@A%BCD"#7%+E=/:
!* "F:"FGA%B@!
H0"I$"#7!
#'4&H(
J*#./+*I D"
=0KH/!&1L-M*N%O
0&H +"@+'@@%O= =$%%8
+* %8+*(-= =$%%8+*%P
8+*+' %%A@N(6'@/!&/%Q
/%R#= S%+"@":+T)*K
PU @&'GB"F:3 4:='&
0EO@&&$ 0(
PV:WW0E04 0D!
!& +#(
PU +' %%A@$#/'4&H
# 0!"(
PS= !&* +*# %8%%
E= 'S+(
-3*$ +N!&AA (
1+%XY% 0G'+*= <Z"#X [
+0'\ =/"@":!&"

#cỏc bi toỏn gii bng cỏch lp phng trỡnh hay h phng trỡnh
B87A8+!$%(100H]'
%O "@":+T.'^%(
J+* 70$K'H/!&
#=$%%8+*%8+* <
'NO''N$%<&&C &8(@3 Z0!
'N (
_
ii - cơ sở lý luận và thực tiễn của đề tài:
1+8+*&HT=$1L-M!&' B/=
=$%%8+*%8+*T/%Q /%R(1
%O"#7N +' %%A= $%M;V
&"'++N '%'@"
` !&++7!!%
*+%8%%E(
1+D:+/"%O= =$%%P
8+*%8+*0`/>BD= $%
"05%%" = "B>
AA`@<&H3B '(
"C%*+ <+D=%%@ %
'+%8%%E(1E'@A%B/@
H=/"@":"S%= (
* @H"#70!&0
<&H"#7&K
P)N&H!K"+&H+$%%07%a%@
'N"G'2%&H(((
P)N'&HK%8+*%8%%%8+*
%a%=#'5'&H(((
P)N*!K0733*%b'\!(((
P)N$ZK07+'<'N073%P

8+*=(((
P)N!K073^'<2^'<c3(((
iii - mục đích yêu cầu và nhiệm vụ nghiên cứu của đề tài:
6 =<M;VQ R&/> <&H&
"D%8%%#A%!&@'%8
%%7*,"%& =##+<
88( /&* :"F:
0"!=/"@": !(
N %":&%+"333,,
,=#%!&E'@A%B30
!(N I@%%O%'5/%8%%
P!0BM;V/(
iv - phơng pháp nghiên cứu:
_(d8%%'N+(
e(d8%%'H7(
f(d8%%(
e
Phần B
nội dung giải quyết vấn đề
i - lý luận chung:
1+/"%+E'+C
''= =$%%8+*g%8+*h
'^4%3ON4'!&C'<
&&C``+i+ (
1. Quy tắc chung giải bài toán bằng cách lập phơng trình (hệ phơng trình)
j/_K U$%%8+*%8+*
P-!k'82k'N"k(
Pjl'=#k ''\=#(
PU$%%8+*g%8+*h=2HD'(
j/eK ;%8+*g%8+*h

(Tuỳ thuộc vào phơng trình, hệ phơng trình mà chọn cách giải cho phù hợp)
j/fK 1+4
V+GB+%8+*g%8+*h
\'N"k "0+^"#$(
2. Các giai đoạn giải bài toán bằng cách lập phơng trình (hệ phơng trình)
1+8&TC E=/* +*
E'A%!&+"= l 8K
;'_Kd3 #"#$= K
P;A%!&= KD"*mO*D*m
Pn0=*o&8'^'"0m
;'eKjl'
P-!k'82k 'N"kg'N"k%%.%
/O= #'4&Hh
P'H'=l==7'&H(
;'fKU$%%8+*g%8+*h
P10H+ =<= E=7'&H
# %8+*g%8+*h
(Ba giai đoạn này chính là bớc 1 trong quy tắc chung)
;'pK;%8+*
P)$"F:%8+*g%8+*hK=#'5N8
=# %8+*g%8+*h%7%(
f
P1*+2k%3@(
;'qK?$'2'"#KV+E*'@
%.%/'N"k"0m
PraGB%8+*g%8+*hE*''\%.%
/= m-@%.%/ZF#"0m
;'sK1+4= 
1+8&T'q+4"#= (
;'tKd3 =$

M"G4O%3&=^]!
&0K
P1'5D"= Dk&H(
P1'5k&HDD" #H"(
P1*"H8(
* VÝ dô minh ho¹ cho c¸c giai ®o¹n gi¶i bµi to¸n:
Bµi to¸nKuJGBE %Hu'# %Hj/$H
+=*pv"c(M_

fvw)GBIE %Hu'#
 %Hj/$H+=*sv"c(u)S%JT3
D\'4EE %Hu'# %Hj(LX %H
="0am
* Híng dÉn häc sinhK
;'_K
LGB.NEu'#j
x4\x3TS%
4"cpv
e
KrB
_
4"csv
_
KrB





=

=
2
gian t
148"aK
e
P
_
y_qg4h
13\'4ujm
;'eK
;!"D %Hu j KGg"hVKGzv
LGBS%T3D\'4{GB''K
e
G
g"h
14GB'#K
e
G
Kpvy
Qv
G
g4h
rB'#K
e
G
Ksvy
_ev
G
g4h
p

;'fK
)*GBG%+/GB_

fvwy
e
f
4@%8+*K
e
f

_ev
G
P
Qv
G
=
g_h
;'pK
g_h
⇔
fGPeGyfsv

⇔
Gyfsv
;'qK
)/Gyfsv\/'N"k+= (
1[K
e
f


_ev
fsv
P
Qv
fsv
=
g'Ah
;'sK
)$ %Hu jfsv"
;'tK
P1'52+3S%00O*"
 %H
P1'54G%g$HGBh*"m
* Lêi gi¶iK
;!"D %Hu j Gg"hg"KGzvh
1B= +GBS%D\'4{GB''
e
G
g"h
6'@4GB'#
e
G
Kpvy
Qv
G
g4h
14GB'#K
e
G
Ksvy

_ev
G
g4h
)*GBG%+/GB_
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fvwy
e
f
4@%8+*K
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f
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P
Qv
G
=

⇔
fGPeGyfsv

⇔
Gyfsv
)/Gyfsv\'N"k
)$"D %Hu j fsv"(
3. Yªu cÇu vÒ lêi gi¶i cña bµi to¸nK
* Yªu cÇu 1:V0&&@g33Gh(
q

LMKO= 'A'C#"0S&*&o"0
'S&(
LMKO##= #"0&o=2=X&@(
-bK=X'82k'N"kg%.%/= 
%.%/#h"'@4"0 (
Lu ýK
PÈ '<$$&%k*'N"8
PÈ D&H\'N"Kv
≤

≤
R(
PÈ 4\'4$H3K'N"8
* Yªu cÇu 2KU4@$%$
PJ+*E=/@0So/(
Pr'2!k%.% 5=$H 'N%*(
P1#$%%8+*g%8+*h*+2k!(
nH$!&!&OG'"#= t
'(
* Yªu cÇu 3KU4% 
PV#= *'%%.%/(
P?#'5'N"''= '#+4%'S=*"#
>%.%(
* Yªu cÇu 4KU4%'8
U4C!l >'=O+(
* Yªu cÇu 5:1+*= 4"!
PnHD=/+= %0So(
Pj/&'"#ED=/+/'\'" 'A(
* Yªu cÇu 6KU4%+i+ 
P-$%$# 4"0^a"0>(

P-=/%3G"!(
PV#%H.# B"(
4. Ph©n lo¹i bµi to¸nK
Pj N'<
Pj @<&H!
Pj @<*!
Pj N:&'<
Pj N`%O=/53`&H%O+:(((
Pj N0  +g= N'82h
s
Pj @<$Z!(
Pj @7&Hgj 5%h(
II - các dạng bài toán giải bằng cách lập phơng trình (hệ phơng trình)
!&CC@H 0+9"|:H %+
0'\% (1+E0.!&
* %8%%5%"#7O/&'@'+= $%
$ = $%'!&+9"|:(
Dạng I: Dạng toán chuyển động
A - Kiến thức cần nhớ:
1. Công thức chuyển động đều:
&y(



&
}

&
==
1+'@K & \'4g"}}(((h

4g}%}h
$Hg"c}c&}(((h
2. Chuyển động trong môi trờng động: (dòng nớc; gió)K

G0
y

~
/


y

P
/
g

z
/
h

G0
$H'<"G0,


$H'<",


$H'<"/S

/
$H,/
3. Bài toán có sự tham gia của nhiều động tử:
M_4"D'<['5(
y
e_
+
#'<N
y
e_

#'<.N(
4. Kĩ năng phân chia thời gian của quá trình chuyển độngK
B - Bài toán áp dụng:
Bài toán 1:
uL.'GB'%E '#L ?<B'4 pQ"(UA
'B'4CC8_f"&/A'$HAN`
=
s
q
$HA'(
t
14N>384A' 
e
_
4(
13$HA'L.m
* Híng dÉn häc sinh:
1.d3= KPL!&+ie+*'<' N(
P-@f'K&}}

PnHe+*K&
N
~_fy&
'
}
N
y
s
q

'
}
N
y
'
P
e
_
2.M[07K&y(}

&
}

&
 ==
3.V#$= K*$HA'm
U$%=K
Mg"h )g"ch 1gh
UA' pQ G
G

pQ
UAN pQP_fyfq
G
s
q
G
pe
* Lêi gi¶i:
;!$HA'L. Gg"chVKGzv
V'@$HAN 
G
s
q
g"ch
14A' 
G
pQ
gh
14AN gpQP_fhK
G
s
q
y
G
pe
gh
1B= +@%8+*K
G
pQ
P

G
pe
y
e
_
⇔RsPQpyG
⇔Gy_e
;+2Gy_e\'N"k(
)$$HA'L. _e"c(
Bµi to¸n 2:
n<0+&0G0,_" ,_"#fq%A(
?#0G0,ev" ,_q"*#_4(13$H
,/ $H0#"0@,/m
Q
* Híng dÉn gi¶i:
PL!&'&'<T'@e+*KG0, P
,(
Pn{+*+= eO(
P-07%K&y(

G0
y

~
/


y

P

/
g

z
/
h
VF:%8+*(
U$%=K
Mg"h g"ch gh
1 G
6,/ 
UO_
r0, _ G~
G
_
+
?, _ GP
G
_
−
UOe
r0, ev G~
y
20
+x
?, _q GP
G
_q
−
* Lêi gi¶i:

;!$H0 $H,/kO Gg"ch 
g"ch(
N"KGzzv
V'@$HG0,0KG~g"ch
)$H,0KGPg"ch
1B= +K0G0_" _"#fq%A=tc_ev4
@%8+*K
120
7

y -x
1

y x
1
=+
+
nS"K0G0_q" ev"#_4@%8+*
_
PG
_q

G
ev
=+
+
R
)$@K








=+
+
=+
+
_
PG
_q

G
ev
_ev
t

PG
_

G
_
S

G
_
•
G
_

r







−
=
+
=
L'\+T 





=+
=+
_•_qrev
_ev
t
•r
;'



=
=

⇔



=−
=+
⇔







=
−
=
+







=
=
q
fqG


fvG
pvG

fv
_
G
_
pv
_
G
_

fv
_
•
pv
_
r
)/Gyfq}yq\'N"k(
)$$H0 fq"c(
)$H,/ q"c(
* Chó ý:V .%8%%'Sk%@'^
>7''N%8+*=$(LS+#%="[G
~ GPgOh
Bµi to¸n 3:
? ?  U.++'4H<&Hs t
"(?#? U'GB'%.<A N*&_q%A
!S%(13$H{4=#+$H?8$H
U p"c(
* Híng dÉn:

Pj @e'H'<N(
Pj#5\'4'H'<(
P14'<A(
PJ$H'<(
P-07$KMy(
PM_4e'<''\'4 K
_
~
e
g"h
_v
U$%=K
Mg"h g"ch gh
?
G
p
_
G
p
_
U

p
_

p
_
* Lêi gi¶i:
5_q%A=_cp4(
;!$H? Gg"ch

;!$HU g"ch'N"KGzzv
M_44''\'4 KG~g"h
?&_q%A\'4='' KgG~h
p
_
g"h
1B= +@%8+*KgG~h
p
_
yt
nS"$H?8$HU p"c@%8+*
GPyp
)$@K



=
⇔



=−
=+
⇔





=

=+
_e
_sG

pG
eQG

pPG
t
p
_
h(gG
)/Gy_s}y_e\'N"k(
)$$H=? UO K_s"c _e"c(
* Chó ýKj @=$%%8+*(Sk#%(
Bµi to¸n 4 gj fv1+eeM;V1RP$%ehK
n<00'Eu '2'#jA_e4+(?#GB/$Hfq
"c*'#j$e4&/'2#GB/$Hqv"c*
'#j&/_4&/'2(13\'4uj 4'G%
00u(
* Híng dÉn:
Pj @_'H'<(
P '<B'2(
P-@eO'5@D4# 4'2(
Pj#4''#'H'<(
P-07$KMy(
__
U$%=
Mg"h g"ch gh
6'2 G 

1#_ G fq
e
fq
G
+=
1#e G qv
_
qv
G
−=
* Lêi gi¶i:
;!'< \'4uj Gg"h 4'200'Eu
'#j ghN"KGzv}z_
14'Eu'#j/$Hfq"c 
fq
G
gh
)*'#j$8e4&/'2@%8+*K
e
fq
G
+=
14'Eu'#j/$Hqv"c 
qv
G
gh
)*'#j&/8_4&/'2@%8+*K
_
qv
G

−=
1@



=
=
⇔



−=−
=
⇔







−=
+=
Q
fqvG

qvqvG
tvfqPG

_

qv
G
e
fq
G
)/Gyfqv}yQ\'N"k(
)$\'4uj fqv"(
1400G%Eu _ePQypg4h(
Bµi to¸n 5gj ft1+epM;VRP$%ehK
L$'<'N+<'4+,'4"3evG%
.<AE<'(?#'<.N*7evA
S%(?#'<N*7pAS%(13
$H{$m
* Híng dÉn:
P?C073'< '4+,gh-ye
π
€y
π
1+'@€ ="3} '4"3(
_e
A
A
B
C
V'<.N7eve$S%"'@$
'8$$_,(
V'<N7pe$S%7 5\
'4''$ _,(
M8'^K
U$%=K

Mg"h g"ch gh
N
)$ evG G ev
)$$ ev  ev
?N
)$ pG r p
)$$ p  p
* Lêi gi¶i:
4+,'4"3ev@'<  K-yev
π
gh
;!$H$'< Ggc&h
;!$H$'<$ gc&h
N"KGzzv
V'<.N7eve$S%"'@$
'8$$_,@%8+*Kev(gGPhyev
π
LGPy
π
V'<N7pe$S%7 5\
'4''$ _,@%8+*KpgG~hyev
π
LG~yq
π
1B= +@



=
=

⇔



=+
=
⇔



=+
=−
π
π
π
π
π
π
2 y
3 x

5
62x

5 y x

yx
yx
-+2G}E*'\'N"k(
)$$H$O Kf

π
c& e
π
c&(
_f
C. C¸c bµi to¸n tù luyÖn tËp:
Bµi to¸n 1:
n<00'EL ?<'#1L/$Hpv"c(Me4`
T1L00E1LNL ?</$Hfv"c(154
'>N _v4pq%Ag"4`T1Lh(13
\'4L ?<P1L(
Bµi to¸n 2:
L4T'2'u jfs""T .<A
'N S%T<'2'u e"(?#.
D$H+4%+4'$8G%+/4
"s%A*!&oS%T3D\'4(13$H{P
4(
Bµi to¸n 3K
n<G^2'E %H- n'#nIB<'4
&0 _ev"(1+'4'G^@`_4T2+?:-:(V
NG^'B'4" 8'4A'q" /$HX8$
HA' q"c(13$HG^A'=#+4N=
4'(
Bµi to¸n 4:
jL% 0U'GB'%E `+\'4 fv"
"T .<A()$HGB=L%/8$HGB0U
 f"c=L%'\'#`+/0U[4(13$HGB{
4(
Bµi to¸n 5:
VD=#&0u j fv"(n<0'E=#u'#

=#j`pv%AT=#j+^=#u(VEA"T '#"N
/=#u#s4(L\*$H0+/S=#+
$H/ f"c(
Bµi to¸n 6:
n<GB['EL ?< j*M8gJ?\h(M'@_4<GB
["'Ej*M8+L ?</$H/8$HGB[7
 q"c(LGBS%<T3D\'4(1*$H
{GB#+\'4L ?<Pj*M8 Rvv"(
_p
Bài toán 7:
J\'4uj^<'H p" <'GHH
q"(n<4'GB'%Eu'#j#pv%A 'EjNu#p_%Ag$
HHGHHA' Nh(13$HAH A
GHH(
Bài toán 8:
n<0G0,E=#u'#=#jp4 ,E=#jN
=#uq4(13"D=#u j=#+$H
,/ e"c(
Bài toán 9:
;%8+*K
92
8
94
6
96
4
98
2 +
+
+

=
+
+
+ xxxx
dạng ii: toán về năng suất lao động
A. Kiến thức cần nhớ:
PJC(
PnHD'KV}?}1
Vy?(1

?yVK1}1yVK?
1+'@K VK"H0(
?K:&'<(
1K4'<(
PM`DV ? $g?#1"0'5h
PM`DV 1 $g?#?"0'5h
PM`D? 1 2g?#V"0'5h
PM%3++*'<(
B. áp dụng:
Bài toán 1:
n<'<"a'2{ pv"{
qe()*$"0D'< G+/4e ,
'pD(133[+< '<% B"#m
* Hớng dẫn học sinh:
P = :&'<HV?1
Pj %+*KB"# #'\ (
Pj'&87D+* (
_q
U$%=K
63_  MH  63+<

6'2 pvgh
pv
G
Ggh
1# qegh
qe
pG +
G~pgh
* Lêi gi¶i:
;!3[+< '<% B"# Ggh(
N"G8(
1*4'<% B"# 
pv
G
g h
1#3'< ' KG~pgh(
14#'< # K
qe
pG +
g h
1B= +@%8+*K
pv
G
P
qe
pG +
ye
⇔_fGP_vgG~phyqev(e
⇔_fGP_vGPpvy_vpv
⇔fGy_vQv

⇔Gyfsv
)/GyfsvX\'N"k(
)$3+<'<'2 B"# fsv(
Bµi to¸n 2:
n<0%  qv&%k+<4'2(
6::&'<q&%k{44'@'\  
&/84'2 _4pv%A(13:&{4B
'20'@m
* Híng dÉn häc sinh theo bµi to¸n 1.
* Lêi gi¶i:
5_pvwy
f
q

;!:&'20'@ Gg&%kch(
N"KGzv}G
1#*:&#0'@ G~qg&%kch
_s
140'@ B"#K
G
qv
gh
-,4#0'@ #K
qG
qv
+
gh
1B= +@%8+*K
f
q


qG
qv
P
G
qv
=
+
⇔fv(gG~qhPfv(GyG(gG~qh
⇔G
e
~qGP_qvyv
;+'G
_
y_v}G
e
yP_q
)/G
_
y_v\'N"k}G
e
yP_q"0\'N"
k(
)$:&'<0'@B'2 _v&%kc(
Bµi to¸n 3:
1+'Oe50. 'pvv#(M&
5•7_v‚5••7_q‚5&&'ppQ#(LX
+'O{5&&'=#m
* Híng dÉn häc sinh:
Pj @e'HgL5&&h

PN$%/:&'<5"g%7%8h
PM::&T%O+:gL!&'_v‚}_q‚h
Pj#"H0A'O "7(
U$%=K
15• 15•• -5
1'O Gg-1nh pvvPGg-1nh pvvg-1nh
1&
G
_vv
_v
G +
g-1nh pvvPG~
( )
Gpvv
_vv
_q
−
g-1nh ppQg-1nh
* Lêi gi¶i:
;!&H#5• '+'O Gg#h(
N"KvƒGƒpvv}G
1#*&H#5•• '+'O pvvPGg#
h
1+&K15• '
G
_vv
_v
G +
y
G

_v
__
g#h
_t
15•• 'pvvPG~
( )
Gpvv
_vv
_q
−
y
( )
Gpvv
_vv
__q
−
g#h
1B= +@%8+*K
G
_v
__
~
( )
Gpvv
_vv
__q
−
yppQ
⇔eeG~efgpvvPGhyQRsv
⇔eeG~RevvPefGyQRsv

⇔Gyepv
)/Gyepv\'N"k
)$'OK15• 'epv#(
15•• 'pvvPepvy_sv#(
Chó ýKj @=$%%8+*(
C. C¸c bµi to¸n tù luyÖn tËp:
Bµi to¸n 1:
j<'<X$%"#"B'@{ %"
qv(V{ '<"'qt(6'@'<'\ 
 "#+/_  ,7_f(LXB"#'<
%"=m
Bµi to¸n 2:
n<G3%"Z%'^<&HB+ev (6
#"|$:&G3%'\:ev‚(jT$`+_Q
 "0DG3%'\  &HO ,'P
epD(13&HB G3%%B%'^(
Bµi to¸n 3:
n<G3%'2&G_qvv&%k+fv (?45
7'<%Z#'\&G{ _q&%k(6'@G3
%'\&G"0D7'2eqq&%k ,  
+/4(LX#G3%'\+AC'= m
Bµi to¸n 4K
?:'82&G0%'tev@(?:
'827 7_q‚'827 7_e‚&/:
(6'@'82'Q_R@(LX{:{'82
'=@(
_Q
Dạng iii: toán về công việc chung riêng
A. Kiến thức cơ bản:
PnHVP?P1

PM8`DVP?P1(
PVF:!kD"N'82(gdOh
Pd3' =2Hk '82'\!(
* Cần lu ý cho học sinhK
P_4 +G4g hG<0*_4g h P
4'@ '_cG0(?#_4g_ h ' _cG0
*4'@ G0+G4g h(
n>=K
H_g'<
_4__
,_ h
He
g'<e4e
e,e h
-'H
ge'<e4
ee, h
14
0g4 h
dO + <
'824g_4
_ h
B. áp dụng:
Bài toán 1:
L,/. <=*&4='O(n{4/
'T,_=/'T,e(LX{,<*
*&=='Om
* Hớng dẫn:
Pd3+i'A/= (
Pj2HDVP?P1(

P-!'N'82!k3%(
U$%=K
),_ ),e -,
14'O= Ggh
q
ep
gh
dO=+<4
G
_
(
e
_
_
g=h
G
_
g=h
ep
q
g=h
_R
2
1
1
5
4
4
* Lêi gi¶i:
;!4,e<*'O= Ggh(N"KGz

q
p
p
n<4,e'
G
_
g%O=h
_4,_'
G
_
(
e
_
_
g%O=h
1B= +@%8+*K
ep
q
Ge
f
G
_
=+
;%8+*'Gy_e
)/Gy_e\'N"k(
)$<*,e'O= _e4(
_,_'yg=h
)$,_<*'O=#Q4(
* Chó ýKj @=$%%8+*(
Bµi to¸n 2gj pq1+etM;V1RP$%ehK

L'<G. <0 '2G+_e
 (L!. /'Q *'<•''N'<' "
'<••#% (6#"F$:&'<:%'0'<
•• G+f +](LX#{'< <**=
 mg)/:&='Oh(
* Híng dÉn:
-!k 4g'82&H hE'< <*G
0(
-! =<"H0 _'82(
U$% %8+*(
* Lu ý häc sinhK?:&:%'0  Gfq */
:&=*4&ofqGeytg h(
ev
8
1
12
1
.
2
1
1
U$%=K
<_ <e -e'<
14  0
(ngµy)
G  _e
dO0+_ 
x
1
gh

y
1
gh
12
1
gh
dO0+Q 
12
8
y
3
2
gh
dO0+fq
 (n¨ng suÊt t¨ng gÊp ®«i)
y
2.5,3
gh
* Lêi gi¶i:
;!4O#''<•'<•• <*G0O
 Gg h g h(
N"KGz_e
n{ '<• '
G
_
g0h
n{ '<•• '

_
g0h

n< '< '
12
1
g0h
1@%8+*
_e
_

_
G
_
=+
 g_h
Q '< '
h0g
f
e
_e
Q
=
6:&:%'0fq '<•• '

t

qf(e
=
g0
h
1B= +@%8+*K
_


t
f
e
=+
geh
1Eg_h geh@%8+*K







=+
=+
)2(1
7
3
2
(1)
12
111
y
yx
e_
⇔








=
=+
3
17
12
111
y
yx
⇔





=
=+
21
12
111
y
yx
⇔






=
=+
21
12
1
21
11
y
x
⇔



=
=
21
28
y
x
-+2Gye_yeQX\'N"k(
)$4'<• <*G0 eQ 
4'<•• <*G0 e_ (
C. C¸c bµi to¸n tù luyÖn tËp:
Bµi to¸n 1:
L,/. <=/g"0@/h*&
5
4
4
4

'O=(?#A'O`T,7 R4&/T,7*
&
5
6
4D/'O=(LX#E'O`T,7*&=
/'O=m
Bµi to¸n 2:
L4. <0+_s4*G(?# 7
 f4 47 s4*`  'eq‚0(
LX# +*{4  0+=m
Bµi to¸n 3:
?#,/. <=/g"0@/h*=&o'O
+_4ev%A(?#T,7+_v%A ,7+_e
%A*`'
15
2
=/(LX#T+E,*4'{,
'O= =m
Bµi to¸n 4:
L'<a&8<0 (?#!. *+p G
(?#! +*'<•  08'<•• s (
LX# +*{'<% += 'Gm
ee
Bài toán 5:
L4.G<=74+t4_e%A*Gg0D
@0"$h(?#47 +q4
47 +s4*G'
4
3
=74(LX{4

<**=G=74m
Bài toán 6:
L0.&8[<0+*+=H *G
(?#47 <*+3 +^47'#.
#%+< D*G0(LX{4 <**
=Gm
Bài toán 7:
L'<GBT'&%<"'(?#'<. *+
_e G(?'<`. +Q (M'@'<7
#%<*+t D*G(LX{'< <**
=G(
Dạng IV: Toán về tỷ lệ chia phần thêm bớt, tăng giảm Tỷ số
các đại lợng, tỷ số phần trăm.
A - Kiến thức cần nhớ:
PV|:=2H'(
Pjl/KdO+:$%%7(((
P-37(
PM:=/=2(
B - Bài tập áp dụng:
Bài toán 1K
L%G\L^-@e"@"7N8"7 _vv
(?#E"&"7sv*A'@&H@T"7
=&H@T"7(13&H@T{"A'O(
ef
_f
_e
Hớng dẫn học sinh lập bảng đối tợng và số liệu, dựa vào bảng hớng dẫn
lập phơng trình:
Kho I Kho II
UA'O G~_vvgh Ggh

UA&" G~_vvPsvyG~pvgh G~svgh
d8+*K
* Lời giải:
;!&H@A'OT"7 GghgGzvh*&H@A'OT"
7 KG~_vvgh(
MH@@'&"T" KG~svgh(
MH@,&"T" KG~_vvPsvyG~pvgh(
1B= +@%8+*K
_fG~qevy_eG~tev
Gyevv
)/Gyevv\'N"k(
)$&H@A'OT" Kevvgh
MH@A'OT" Kfvvgh(
Chú ýKj @=$%%8+*(
Bài toán 2K(Bài 39 trang 25 - SGK toán 9 tập 2)(
n<4 %+5<e_t+'^"
#+2:g)u1h/7_v'H/ 7 Q'H
/ 7e(?##)u1 R'H/e *4'@%
+5<e_Q+'^(LX#"0"#)u1*4'@%+
=NN{ (
* Hớng dẫn học sinhK
mL!&'!'N@C= (
m *m
m?#_ @7#)u1_v@F *m
ep
_f
_e
svG
pvG
=

+
+
_f
_e
svG
pvG
=
+
+
(Nghĩa là: Cha kể thuế, giá hàng đó là 100%, kể thêm thuế 10% thì ngời
mua phải trả 110% giá hàng ban đầu)(
-bK?#N_ "# Gg'^h*+
'@3#)u1_v 4%+5< KGg'^h
* Hớng dẫn học sinh lập bảngK
Loại hàng 1 Loại hàng 2 Cả 2 loại hàng
; N 3
#)u1
Gg+'^h g+'^h
; N 3
#)u1_v

Gg+'^h
e_tg+'^h
; N 3
#)u1Q

g+'^h
; N 3
#)u1R

Gg+'^h

g+'^h
e_Qg+'^h
* Lời giải:
;!&HN 4%+ 7 7
"0"#)u1O Gg+'^hg+'^h(g'N"Gzv}zvh
V'@&HN%+ 7g"#)u1_vh Gg+
'^h}&HN%+ 7g"#)u1Qh g+
'^h(
1@%8+*K
MHN%+g3#)u1Rh 7
7eO Gg+'^h g+'^h(
1@%8+*K
G~yegeh
1Eg_hgeh@%8+*K
__vG~_vQye_t Gyvq
eq

_vv
__v

_vv
__v

_vv
_vQ

_vv
_vR

_vv
_vR
;d8+*'K
'

_vv
__v

_vv
_vQ
g_he_t_vQ__vGe_t
_vv
_vQ
G
_vv
__v
=+=+
_vv
_vR
_vv
_vR
e_Q_vR_vRGe_Q
_vv
_vR
G
_vv
_vR
=+=+