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ARITMETICA 5to
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ARITMETICA 5to
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Quinto Año
|
Aritmética
1
Quinto Año
INDICE
NÚMERO FRACCIONARIO Y SU CLASIFICACIÓN
03
NÚMERO DECIMAL
15
POTENCIACIÓN Y R ADICACIÓN
25
R AZONES Y PROPORCIONES
34
PROMEDIOS
41
M AGNITUDES PROPORCIONALES
47
REPARTO PROPORCIONAL
56
REGLA DE COMPAÑA
63
REGLA DE TRES Y COMPUESTO
6!
T ANTO POR CIENTO
76
Aritmética
2
Quinto Año
INDICE
NÚMERO FRACCIONARIO Y SU CLASIFICACIÓN
03
NÚMERO DECIMAL
15
POTENCIACIÓN Y R ADICACIÓN
25
R AZONES Y PROPORCIONES
34
PROMEDIOS
41
M AGNITUDES PROPORCIONALES
47
REPARTO PROPORCIONAL
56
REGLA DE COMPAÑA
63
REGLA DE TRES Y COMPUESTO
6!
T ANTO POR CIENTO
76
Aritmética
2
Quinto Año
TEMA: NÚMERO FRACCIONARIO Concepto L"#$#%&' (#$)*+, -F.#//*&,' -).#%&' -,$.&' ).#%&' & -.#//*&,' .#/*&,#"' *,, # '. "#' /#,(*%#%' , "#' /#"' "# ,*%#% ' %**% , 8#.(' *9#"' % "#' /#"' ' (&$#, ,# & $:' % ,#; <.#//*&,' 8&'*(*#'= Notación f
A B
>
S "? -A '&). @ -A ,(. @ & -A @ #&'
Forma General
A FB B
A; B ; b 0
Nota? 0
a
L#' &.$#' 0 0 > # '&, &.$#' ,& %(.$*,#%#' , '( ,*" 8&. "& *(#$&' ' '&;
Estructura
# N$.#%&. ) D,&$*,#%&.
N$. N$.#%& #%&.? .? *,%*/# *,%*/# " ,$ ,$.& .& % 8#.( 8#.(' ' ' /&,'* /&,'*%. %.#, #, % "# "# ,* ,*%#% %#%;; D,& D,&$* $*,# ,#%& %&.? .? I,%* I,%*/# /# " " ,$ ,$.& .& (&( (&(#" #" % % 8#.( 8#.(' ' , , ' # # %** %**%* %*%& %& "# ,*%#% (&%#' ""#' *9#"';
Nota? L&' ,$.&' .#//*&,#.*& %#, "9#. # , /&,,(& % ,$.&' /&,(*, #" /&,,(& % ,$.&' ,#(.#"'
N$.&' F.#//*&,#.*&'
ON
Q
Z
N Z Q
Clasificación Aritmética
3
Quinto Año
L&' ,$.&' .#//*&,#.*&' ' /"#'**/#,?
1 !or las rrelaci elaciones ones entre entre sus t"rminos t"rminos a# Frac accción ión !ro !ropia: pia: A""# $,&. "# "# ,*%#% a
< b 1 # )= 3 7
;
E;
H !
, etc
$# Frac accción ión Impr Impro opia: pia: A""# A""# $#&. "# ,*%#%; a
< b 1 # )=
17 37
E;
5
;
H
, etc
O% L#' .#//*&,' *$8.&8*#' /&,'(*(, "&' ,$.&' $*(&' ' %/*. #""&' 8&', 8#.( ,(.# 8#.( .#//*&,#.*#; 17
E; 5
17 5 2 3
17 5
3
2 5
c# Frac acccione ioness i'u i'uales ales a la uni( ni(a(: a(: A""#' %&,% " ,$.#%&. ' *9#" #" %,&$*,#%&. %,&$*,#%&. 8&. "& " #"&. % "# .#//*J, ' *9#" # ,&;
a 1 a b b E; 1H > 27 (/; 1H 27
) *as *as a'rupac a'rupacion iones es (e fracci fraccion ones: es: a# Frac acci cion onees +om +omo'"n o'"nea eas: s: S&, #""#' 8&', " $*'$& %,&$*,#%&. %,&$*,#%&.; E;
3
;
27
;
9
16 16 16 >K '&, .#//*&,' &$&9+,#'
$# Frac acccione ioness +et +etero ero'"ne '"neas as:: S&, S&, # #""""#' #' 8&' 8&', , %* %*. .,( ,(' ' %,&$*,#%&.' ,(. '; 7 26 8 ; ; ; K '&, .#//*&,' (.&9+,#'; 5 3 81
E; , !or la natura naturale-a le-a (el (enomina( (enomina(or: or: /&..* *,( ,(' ' &.%* &.%*,# ,#.* .*#' #'?? '&, '&, # #""""#' #' a# Fracciones comunes. /&.. 8&', , %,&$*,#%&. " /#" ' 8&(,/*# % 10; Aritmética
4
Quinto Año
E;
5 30 5 ; ; ; K '&, .#//*&,' /&$,'; 8 20 2000
$# Fracciones (ecimales: '&, %,&$*,#%&.' '&, 8&(,/*# % 10;
#""#'
.#//*&,'
/&'
73 42 ; E; 100 10000 ; K '&, .#//*&,' %/*$#"'
/ !or su car0cter como nmero racional a# Fracciones re(uci$les o re(ucti$les: '&, #""#' .#//*&,' 8&', %**'&.' /&$,' (#,(& , " ,$.#%&. /&$& , " %,&$*,#%&. %*'(*,(&' % "# ,*%#%; F&.$# 9,.#"? S*? a ak f ' .#//*J, .%/(*)" b bk
E;
30 B 3<10= 50 B 5<10=
>
f B MCD <#)=
N 1 C&$& 10 ' MCD <30 > 50= 30 ' .#//*J, .%/(*)" 50
$# Fracciones irre(ucti$les o irre(uci$les: S&, #""#' .#//*&,' ,& 8&' %**'&.' /&$,' ,& '# "# ,*%#% ' %/*. "&' "$,(&' % "# .#//*J, '&, ,$.&' PRIMOS ENTRE '
E;
a b ' .#//*J, *..%/(*)" # ) '&, PESI 5 3 8 ; ; ; K '&, .#//*&,' *..%/(*)"' 7 5 9
c# Fracciones e2uimltiplo: ' #""# #9.8#/*J, % .#//*&,' , "#' " ,$.#%&. % "# 8.*$.# .#//*J, ' *$"(*8"& /&, " ,$.#%&. % "# '9,%# " %,&$*,#%&. % "# 8.*$.# ' *$"(*8"& /&, " %,&$*,#%&. % "# '9,%#; E; 156 2472; 15 B 3<5= > 24 B 3
15 > 24 Aritmética
'&, *$"(*8"&' 5
Quinto Año
6
72
(# Fracción e2ui3alente: ' #""# .#//*J, /&,(*, , ,$.& % /' # "# &(.#; E; 1H 54 '&, .#//*&,' *#",(' 8&. H 24 1H<3= B 54 H<3= B 24
!rincipios a Si 1= b ' .#//*J, 8.&8*#
<# )= < Z=
a a k b k b
k b k a
a b
a a a k Si 2= b ' .#//*J, *$8.&8*# b k b
<# )=
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a
b k b
3= T&%# .#//*J, '(# , ."#/*J, %*./(#$,( 8.&8&./*&,#" /&, ' ,$.#%&. , ."#/*J, *,.'#$,( 8.&8&./*&,#" /&, ' %,&$*,#%&. #'?
P#.# " N$.#%&.
# <#= ) )
< =
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P#.# " D,&$*,#%&.
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< =
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4) S* 2 .#//*&,' '&, *..%/(*)"' ,(&,/' (&%#' "#' 8&(,/*#' ,(.#' 8&'*(*#' % %*/#' .#//*&,' <#" $*'$& 8&,,(= (#$)*+, '.:, .#//*&,' *..%/(*)"';
5= S* # 2 .#//*&,' *#",(' ,& *..%/(*)"' ' "' %**% 8&. " MCD % "&' 4 (+.$*,&' % %*/#' .#//*&,' ,(&,/' ' &)(,%.:, 2 .#//*&,' *..%/(*)"'; E$8"&'? Aritmética
6
Quinto Año
I=
3 4
3 1 4 3 1 0,18 0,75 > S 4 1 5 4 1
3 1
S 4 1 II=
4
1,33 > S 3
4 1 3 1
4 1
S 3 1 3
2
5
3
3 4
2
0,66 1,25
1,50
3 4
3 1
3
4 1 4 4 1
4
4 1
4
3 1 3
3 1 3
4
3
2
9
4
2
III= 4 5 '&, .#//*&,' *..%/(*)"' 4 16 > 5 "9&?
9
16
16
25
16 25
'&, .#//*&,' *..%/(*)"'
32
16
I= 24 80 ,& '&, .#//*&,' *..%/(*)"'> MCD <16> 24> 32> H0= B 1 1 .#//*&,' HK; D**%*,%& "#' .#//*&,' ,(. H? > *..%/(*)"'; 3
5 = 6
5.3 6
12
20
15 6
5 2
J
5 6
5 1
5
6 3 18
3
6= T&%# .#//*J, ' *#",( # '* $*'$#; 7= MCD MCM % .#//*&,'? MCD < # > / > = B QMCD <,$.#%&.'=QQ ) % MCM <%,&$*,#%&.'= MCM < # > / > = B Q QQMCM <,$.#%&.'=Q ) % MCD <%,&$*,#%&.'=
Aritmética
7
Quinto Año
E$8"&? I=
MCD < 12 > 4 > 20 = B MCD < 12> 4> 20 = B Q4Q B Q1Q H 3 5 MCM < H> 3> 5 = 120 30
2= MCD < 5 > 6 > 14 = B MCD < 5> 6> 14 =QQ B Q230 B 105 4 H 6 MCM < 4> H> 6 = 2
4 5i3isi$ili(a( (e fracciones: # )
0 / %
# )
/ %
"
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d
O% 1 M6nimo Comn 5enomina(or (e Fracciones: ' " MCM % "&' %,&$*,#%&.' % .#//*&,' *..%/(*)"' /&, ' &$&9,*# # .#//*&,' (.&9+,#'; ) 5i3isión Racional: E' "# %**'*J, , ' &)(*, " /&/*,( .%#%.& & #/(& % 2 ,(.&'; F&.$# 9,.#"? D .
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.
B % >
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Aritmética
8
Quinto Año
Operaciones con fracciones#7 1 A(ición 8 sustracción a b
#
c d
ad # bc bd
E$;
2 3
4 5
10 12 15
22 15
5 4
2 5
15 8 12
7 12
) Multiplicación ace... a c e ... bdf ... E$; b d f
2 4 2.4 8 3 5 3.5 15
, 5i3isión
a c d a ad E$; b d c b bc
2 4 2.5 5 3 5 3.4 6
O$s: 1= Nmero mi9to: ' #" &.*9*,#%& # .# % "#' .#//*&,' *$8.&8*#' % "# '$# ,(. , ,$.& ,(.& ,# .#//*J,; E$; 2
5
8
21 8
2= Fracción (e fracción: E' #""# , %&,% " ,$.#%&. " %,&$*,#%&. '&, .#//*&,'; 3= Fracción reiterati3a: A""# ' " .'"(#%& % , 8.&%/(& % .#//*&,';
Fracción Continua &imple FC S&, .#//*&,' /&' %,&$*,#%&.' '&, '$#' '/'*#' % .#//*J, ,$.&' ,(.&'; F&.$# 9,.#"?
Aritmética
9
Quinto Año
fcs a1 a2
1
a3
1 a4
1 a5
1 1 a
> %&,%? #1 Z #2> #3> K#<,1= Z #12 B C&/*,( I,/&$8"(&
F.#//*J, *,(9.#,(
#1> #2> #3> K #, B R%/*%#' /&,.9,(' & /&/*,(' *,/&$8"(#' N&(#/*J,?
F/' B
Aritmética
F/' B a 1 ; a 2 ; a 3 ;...; a n 1 a
1 a2
1
1
1
a 3 a 4 a n 1
10
Quinto Año
!RO%*EMA& !ARA *A C*A&E 1=
U, &$). (,# 30L % #9# "&' 45 "&' ,#'& , )&(""#' % % "*(.& " .'(& , )&(""#' % V "*(.&; W#""#. "# /#,(*%#% (&(#" % )&(""#'; R8(#;?
2=
A,(&,*& ""9& (#.% # ,# /&,.,/*# ' 8.%*J 17 % ""# 3 $*,(&' $:' (#.% ""9& X&'+ '//J "&' 56 % "# /&,.,/*#; S* "# /&,.,/*# $8J # "#' 10 #;$; A &.# (.$*,J R8(#;?
3=
S* ' ##,& "&' % , (.#/(& "9& .(.&/%& "# $*(#% % "& ##,#%& $ ,/&,(.#.# # 10$; #,(' %" 8,(& $%*& %" (.#/(&; W#""#. "# %*'(#,/*# (&(#" %" (.#/(& , $(.&'; R8(#;?
4=
E, ,# #'*# "",# % #9# ' #9.9# /*.(# /#,(*%#% % '#" %*'&"*+,%&"#; S (.# 47 %" /&,(,*%& ' .$8"## /&, #9#> "9& ' .(*.#, 711 %" ./*8*,( ' ", # /&$8"(#. /&, #9# *,#"$,( ' .(*.#, "&' 516 %" &"$, (&(#"; S* #" *,#" %#, 21 9.; D '#" #'$*,%& "# /#,(*%#% % #9# ' .(*.# + /#,(*%#% % '#" ' .(*.& *,*/*#"$,( R8(#;?
5=
6=
U, /&$./*#,( ,% '' 8#,(#"&,' % "# '*9*,( $#,.#? %" (&(#" (,# 13 $#' 4 # S; 50 /#%# ,& *,#"$,( ,% "# $*(#% % "&' " %#)#, $#' 4 # S; 30 /#%# ,& /&, "& ' " #/#)#, "&' 8#,(#"&,'; W#""#. "# '$# % /*.#' % "# /#,(*%#% % 8#,(#"&,' "# /#,(*%#% % '&"' ./#%&; R8(#;? P#.# /#,(&' #"&.' % -N $,&.' % 1000 <, Z = ' #/ .%/(*)" "# .#//*J,;
N 4 FB Aritmética
81 N 2 80 N 81 N 3 1
7=
R8(#;? E,(. %&' /&"//*&,*'(#' (,#, 224 $&,%#' #,(*9#'; E" 8.*$.& " %*J #" '9,%& "# '(# 8#.( % '' $&,%#' $#' (#.% /%*J " '9,%& #" 8.*$.& "# /#.(# 8#.( % "#' $&,%#' (,#, *,*/*#"$,( /&, "&' "&' %&' (,#, " $*'$& ,$.& % $&,%#' C:,(#' $&,%#' (,# " 8&'# $:' $&,%#' R8(#;?
H=
W#""#. ,# .#//*J, *#",( # 712 '#)*,%& '* #" (+.$*,& $,&. ' " '$# 70 8#.# " #"&. % "# .#//*J, 8.$#,/# *,#"(.#)" ,(&,/' " &(.& (+.$*,& %) (.*8"*/#.'; D#. /&$& .'8'(# " %&)" % "# '$# %" ,$.#%&. /&, " %,&$*,#%&.; R8(#;?
!=
P#.# #"&.' % N "# 8.'*J,? 3Q 1 f Q 5 E' , ,$.& ,(.&; D#. /&$& .'8'(# "# '$# % "&' #"&.' % - R8(#;?
10= C*.(# ("# %'8+' % "##%# ' ,/&9 15 % ' "&,9*(% 116 % ' #,/&; C:,(&' $(.&' %), /&$8.#.' 8#.# %'8+' % "##%# ' %*'8&,9# % 100 $ 2 '#)*,%& " #,/& &.*9*,#" ' % H0 /$ R8(#;? 11= C#%# , #"$,& ,(.# # "# /#(.# 9#'(# , (./*& % "& (*, $#' 40 '&"'; A" '#"*. 8&. (./.# ' %# /,(# ' %& '*, %*,.&; C:,(& %*,.& (,*# #" /&$*,& R8(#;? 12= U,# 8"&(# /# %'% ,# #"(.# % 1;20 $; , /#%# .)&( ' "# ,# #"(.# *9#" # "&' 23 % #"(.# % "# /#" /#&;
11
Quinto Año
W#""#. " '8#/*& (&(#" .(*/#" ./&..*%& 8&. "# 8"&(*(# #'(# %#. , .8&'&; R8(#;? 13= U, 9#%&. 8*.% 25 % ' %*,.& " # #8&'(#. 9#,# 17 % "& " %#)#> "9& 8*.% 16 % "& (*, 8&. "(*$& 9#,# S; 7140; '* "# 8+.%*%# %" 9#%&. 1H % ' %*,.& &.*9*,#"; C&, /#,(# /#,(*%#% % %*,.& %*'8&,# #" $8#. # 9#. R8(#;?
14= W#""#. " ,$.#%&. % "# .#//*J, *..%/(*)" *#",( #?
M
3 7
2
1 7
3
3 7
5
1 7
6
3 7
8
1 7
9
D#. /&$& .'8'(# "# '$# % '' /*.#'; R8(#;?
15= W#""#. $ , '* "# .#//*J, n n ' *#",( # 57152; %#. /&$& .'8'(# " 8.&%/(& % $ ,; R8(#;? 16= S* "# '$# % "#' .#//*&,' *..%/(*)"'? 47) #13 .'"(# , ,$.& ,(.& $"(*8"& % 7 #""#. " $,&. #"&. 8&'*(*& % -# %#. /&$& .'8'(# " 8.&%/(& % '' /*.#'; R8(#;? 17= I,%*/#. .%#%.& <= & #"'& <= '9, /&..'8&,%#? I;
E" ,$.& *..#/*&,#" .8.',(#%& 8&. "# F;C;S; I,*,*(# a , 4,8 '
$
%
S* & ' *..#/*&,#" & B ,(&,/' ' ' *..#/*&,#";
Aritmética
$
%
36
I; S* $, ,(&,/' *$8.&8*#;
$
%
1
, 2 $2 144
45
2 ' ,# .#//*J, 1 n
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
18
II;
III; L# .#//*J, /&,(*,# 8.'#%# 4,3,2,5 /&$&? 4,3,2,4,1 .8.',(# " $*'$& ,$.& .#/*&,#";
1
1
1
1
7
' B
1
1 7
1
12
Quinto Año
W#""#. "#' 2 '&"/*&,' #%$*( %#. /&$& .'8'(# "# '$# % '(#';
R8(#;?
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
E,(&,/' <2R 3= '? #= M#&. 5 )= M,&. 4 /= U, ,$.& .#/*&,#" %= U, ,$.& *..#/*&,#" = N& ' 8% %(.$*,#. 5=
U,# #,*%# '(# 8"#,(#%# , #$)&' "#%&' % :.)&"'; L# %+/*$# 8#.( % "# "&,9*(% "& &/8#, /.& "&' 2! %" .'(& /*."&'> V " ,& .'(& 8.#"'> 13 %" ,& .'(& $#,#,#' "&' 16H $(.&' .'(#,(' %.#,&'; A /#,(& #'/*,% " ,$.& % :.)&"' 8"#,(#%& '* "# %*'(#,/*# ,(. :.)&" :.)&" ' *9#" # 12 $(.&' #= 60 )= 61 /= 121 %= 124 = 122
6=
U, /&$./*#,( ,% "&' % , "&( % )&"'#' % #/#. $#' 174 % )&"'#> ,'9*%# ,% "&' %" .'(& $#' \ % )&"'# %'8+' % &(.#' %&' ,(#' /#' , "#' $*'$# /&,%*/*&,' " /&$./*#,( ' %# '*, #/#.; C:,(#' )&"'#' #)#, "9& % "# 8.*$.# ,(# #= 105 )= !0 /= 30 %= 26 = 21
7=
E,/&,(.#. "# .#//*J, *..%/(*)" A@ '#)*,%& ? @ (*, 6 %**'&.'; S*? A@ B DE /&, E /#%.#%& 8./(& D (*, H %**'&.'> " 8.&%/(& D;E (*, 4H %**'&.'; A ' " $,&. ,$.& 8&'*)" "& $*'$& &/.. /&, @ , # "9*%& A; *,%*/#. " #"&. %
H=
U, 8..& '*9 # ,# "*). " ""# HH '#"(&' % ,(## " 8..& %# H '#"(&' $*,(.#' "# "*). % !> 8.& 5 '#"(&' %" 8..& *#", # 7 '#"(&' % "# "*).; C:,(&' '#"(&' %) %#. " 8..& 8#.# #"/#,#. # "# "*).
S*? R
1
3
1
1
1
3 1 Aritmética
1 3
1
13
Quinto Año
!=
#= 2H0 )= 320 /= 340 %= 300 = N;A; U, '(#, 8&' 3 9.*&' "& "",#, , 10 15 20 &.#' .'8/(*#$,( ,# /#.(# ""# %'#9# (&%& ' /&,(,*%& , , %#; S* '(#,%& #/*J " '(#, ' #). " 8.*$. 9.*& ,# &.# %'8+' #" '9,%& # "#' '*9*,(' 2 &.#' " (./. 9.*&; E, (*$8& ' "",#.# " '(#, '* ' #). "# ""# %" %'#9] ,# &.# %'8+' % #). #)*.(& " (./. 9.*& #= 3 &.#' %= 6
)= 4 = 7
/= 5
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
)= 5340 = N;A;
/= 7160
11= P#.# #"&.' ,(.&' % < N= "# 8.'*J,? 3! 1 F ! 5 E' , ,$.& /&$8.,%*%& ,(. 3 7 #= 5
Aritmética
/= H = N;A;
%= 4
12= D&' 9.*&' A @ "",#, ,(&' " (#, % #9# % , %**/*& , 30 &.#'; S* " 9.*& @ ' % %'#9] ' (#.%#.# , "",#. " (#, 60 &.#'; E, /#,(& (*$8& "",#.# " 9.*& @ " (#, '* *,*/*#"$,( '( ' ,/,(. #/& #= 100 &.#' )= !0 %= 120 = N;A;
/= H0
13= M#.*'&" ' ,# *,9,*.# % 8.'(*9*& 9#'(# (&%& " %*,.& 9#,# , , $' % "# '*9*,( $#,.#? "&' 23 , #"*$,(&' "&' 37 %" .'(& , $&*"*%#% "&' 2136 %" ,& .'(& , .&8# "& " %# '&, S; 2000 "&' #&..#; C:,(& %*,.& 9#,# #" #[& #= 302 400 %= 400 000
)= 572 000 /= 300 000 = N;A;
14= U, &$). , 8.&$%*& %.$ H &.#' %*#.*#' '* , &$). (*' 5 #[&' $,&' " (*$8& )*.# 8.$#,/*%& %.$*,%& '.# "# *,(# 8#.( %" (*$8& )*.# 8.$#,/*%& %'8*.(& '* ' (*' 10 #[&' $:'; C:,(&' #[&' ""# %.$*,%& #= ! %= 6
)= H = 5
/= 7
15= E, ,# #'*# % 3 "*(.&' % /#8#/*%#% ' 8&,, 2 "*(.&' % *,& 1 "*(.& % #9# # /&,(*,#/*J, ' "*$*,# 13 % "# $/"# ' "",# /&, #9# "# #'*# %'8+' ' "*$*,# 13 % "# $/"# ' "",# /&, #9# "# #'*# 8&. "(*$& ' "*$*,# "# $*(#% % '(# "(*$# $/"# ' "",# ,#$,( "# #'*#; /#,(*%#% % *,& /&,(*, , "*(.& % "# "(*$# $/"#
)= H
14
Quinto Año
#= 17 L
Aritmética
)= 45 L
/= V L
%= 21! L
= 1213 L
15
Quinto Año
TEMA: NÚMERO& 5ECIMA*E& Concepto#7 E, "#' 8.'*&,' "*,#"' % "#' .#//*&,' .#/*&,#"' &$& .'"(#%& % %**%*. #" ,$.#%&. % "# .#//*J, ,(. '' %,&$*,#%&.=; F&.$# 9,.#"?
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5ecimales Racionales#7 I
5ecimal e9acto o limita(o: A" /# $#,(*'# (*, , ,$.& *,*(& % (+.$*,&'; S &.*9*,# # 8#.(*. % ,# .#//*J, &.%*,#.*# *#",( % ,# .#//*J, %/*$#"; ; Fracción 'eneratri- f': E' "# .#//*J, *..%/(*)" %# &.*9, # "&' %/*$#"' .#/*&,#"'; F&.$# 9,.#"?
- ab...n$...# * ( <-1 2 10 ... 0 , )
#) K $ ,8 K Z B +
/'= -1 /' -2 /'
Aritmética
<- 2 /'=
16
Quinto Año
E, &(.#' )#'' E, &(.#' <)#' ##"= /'=
- ab...n$...# ( # ) * #) K $ ,8 K Z< = B + ( <- $ 10 ... 0 +, () ( k ) - /' -$ /'
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<-$ /'=
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O@S? 1) U,# .#//*J, /&$, %# %/*$#" #/(& '* ' %,&$*,#%&. (*,
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II
5ecimal Ine9acto: A" /# $#,(*'# 8&' , ,$.& *"*$*(#%& % (+.$*,&'; P% '. # ' ; a 5ecimal !erió(ico< !erió(ico &imple o !erió(ico !uro: E' #" %/*$#" &.$#%& 8&. ,# .#//*J, /&$, /# $#,(*'# 8&' (+.$*,&' ' .8*(, , %(.$*,#%&' /*/"&' & 8.*&%&' <'.*' % ,# & $:' /*.#' .8(*%#' *,*,*(#$,( , "# $#,(*'# % , %/*$#"=; " "
"
...% . a , k&...% . a , ' & ...% N&(#/*J,? a, '&
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...# F&.$# 9,.#"? abc ... , n$ & eces
Aritmética
' eces $e*i+d +
eces &
abc...# abc... 999 ... 9 ' eces
17
Quinto Año
E, &(.#' )#'' veces E, &(.#' )#''& ' & veces <)#' ##"=abc... z ( K ) abc...m ( K )
ab ...m , np... z ( z ) & veces
K P 1
' veces
E$? 033 B 33!! B 13 O@S? 1=
U,# .#//*J, %#.: %/*$#" 8.*J%*/& 8.& '* " %,&$*,#%&. ,& 8&' ,* "&' #/(&.' 2> 5 & ,& % ""&; E$?
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121 133
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E" ,$.& % /*.#' % "# $#,(*'# "& %#.: " $,&. ,$.& &.$#%& '&"& 8&. /*.#' ! /&,(*,, #" %,&$*,#%&. < # '' #/(&.' 8.*$&'= ':,%&' 8#.# ""& "# -(#)"# % ! ! !! !!! !!!! !!!!!
B B B B B
32 32;11 33;37 32;11;101 32;41;271
$ 5ecimal semiperió(ico< !erió(ico Compuesto o !erió(ico Mi9to: ' #""# %+/*$# /# $#,(*'# 8.',(# 8.*&%& , 9.8& % ,$.& ""#$#%&' -8#.( ,& 8.*J%*/#; T#$)*+, ' &/#'*&,#%& 8&. .#//*&,' &.%*,#.*#'; F&.$# 9,.#" & ' / eces
abc... ,
/i ... ,
n$...#
& ' eces
abc...# abc P#.( P#.( 8.*J%*/# -&... /' N& 8.*J%*/# <-/ /'= 999 99 00 00 ... ... / eces
Aritmética
'eces
<-' /'=
18
Quinto Año
O@S? U,# .#//*J, '.: %/*$#" P.*J%*/& M*(& '* ' %,&$*,#%&. 8&' &(.&' #/(&.' 8.*$&' #%$:' %" 2 & %" 5; 2= E" ,$.& % /*.#' % "# $#,(*'# %" %/*$#" P.*J%*/& M*(& .'8&,% #" /.*(.*& %" ,$.& % /*.#' %" %/*$#" #/(& <8#.# "# 8#.( ,& 8.*J%*/#= " %" %/*$#" 8.*J%*/& 8.& <8#.# " 8.*&%&=; 1=
!ropie(a(es 'enerales (e los nmeros (e los (ecimales#7 1 S* # , %/*$#" ' " /&.. "# /&$# - ,*%#%' # "# **.%# & %./# ,(&,/' #" %/*$#" %#.# %**%*%& & $"(*8"*/#%& 8&. ,# 8&(,/*# % 10 /& 8&,,( '# , ,$.& ,(.& *,%* " ,$.& % /#'*""#' ' %'8"## "# /&$# .'8/(& % ' 8&'*/*J, *,*/*#"; ) S* " '$#$&' & " .'(#$&' /.&' # , %/*$#" ' #"&. #)'&"(& ,& ' #"(.#.#
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= S* f +
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Aritmética
19
Quinto Año
"
G B G#''*#,& & I,%*/#%&. % G#''? E' " $,*$& 8&,,( % , ,$.& ' (&$# 8#.# '( '# /&,.9,( /&, "# ,*%#% .'8/(& % /#"*. ,$.& #..&#, " $*'$& .'(& 8&(,/*#"; N&(#/*J,? G<10= 3 B -G#''*#,& % )#' 10 .'8/(& #" $&%"& 3
Aritmética
20
Quinto Año
> 5escomposición !olinómica (e 5ecimales#7 F&.$# 9,.#"? abc ... , n$...# ( k ) ' eces
& eces
ak ( & 1) bk ( & 2 ) ... k 1 ,O 1 8O 2
?
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Complemento Aritm"tico (e 5ecimales#7 F&.$# 9,.#"? 2 abc ... , n$...# ( k ) 10 ... 0 ( k ) abc ... , n$...# ( k ) & cif*as & cif*as ' cif*as & 1 cif*as ' cif*as
E$? O@S? 1=
CA<07= B 1 07 B 03 CA<125= B 100 125 B H75
A, %/*$#" '(# , &(.#' )#'' ' " %,&$*,#? ,$.& - ##" (#" - *,%*/# "# )#' %" '*'($#; E$? ; 372
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Aritmética
21
Quinto Año
!RO%*EMA& !ARA *A C*A&E O
1= S*?
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4000 2 217 2 350
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22
Quinto Año
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R8(#;? != D(.$*,#. " ,$.#%&. % "# .#//*J, 8.&8*# *..%/(*)" %" %,&$*,#%&. 3!6 '#)*,%& %# &.*9, # , %/*$#" 8.*J%*/& $*(& % "# &.$# 0$,8 %&,% ' /$8" 8 B 2 ; $, R8(#;? 10= E,?
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Aritmética
13= S .8#.( ,# .,/*# ,(. 3 8.'&,#'; A "# 8.*$.# " /&..'8&,%, S; 124567> # "# '9,%# " (.*8" % "# 8.*$.# $#' S; 56H!> # "# (./.# S; 76!7 $,&' # "# '$# % "& % "#' &(.#' %&'; S* #%$:' ' #, '8#.#%& S; 30173 8#.# 9#'(&' A /:,(& #'/,%# "# .,/*# R8(#;? 14= S %#, "&' '*9*,( ,$.&'?
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23
Quinto Año
N B 0343<6= > '* ' /$8"?
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Aritmética
db ca
f
3 150 148
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R8(#;?
24
Quinto Año
!RO%*EMA& !ARA *A CA&A 1=
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Aritmética
2
5
4
7
.'8/(*#$,( %&,% $ , B 8 ; /#"/" <$ , 8 => '#)*,%& #%$:' $ ' ' & ' % & n & *..%/(*)"; #= 1H5 )= 1H0 /= 1!0 %= 200 = 1!5
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H=
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2
nn 25
Quinto Año
#= 054
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11=
Aritmética
)= S; 314 %= S; 112
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12=
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E" #9# /&,(,*%# , /#(.& %8J'*(&' 8'# H7!002 9; E" 8.*$. %8J'*(& /&,(*, 1H132 9; M,&' " '9,%& " '9,%& 43 016 9; $:' " (./.& " (./.& 7H15 9; $:' " /#.(&; W#""#. " 8'& %" #9# /&,(,*%# , " %8J'*(& D#. /&$& .'8'(# " 8'& % ,& % ""&' #= 2471!7 )= 266H3!
26
Quinto Año
/= 22231H = N;A; 13=
%= 15017H
P%.& #%*. /*.(& ,$.& % "*).&' 8&. S; 466H; '* )*.# /&$8.#%& 4 $#' " #).#, /&'(#%& S; 77H0 C:,(&' "*).&' # /&$8.#%& /:,(& 9#,#.: '* /#%# "*).& "& ,% 8&. S; !63 #= 6> S;1110 )= H> S;10 /= 7> S; !H %= 10> S; 466H = 30> S; 77H0
14=
U, %8J'*(& ' 8% "",#. 8&. 2 ""#' "# 8.*$.# *.( 2523 "*(.&' , 5 $*,(&'; C:,(& (*$8& (#.%#.# , "",#.' "
Aritmética
'(#, '* '(#,%& #/*J ' #)., # , (*$8& "#' 2 ""#' '#)*,%& ' /#8#/*%#% ' % 42543 "*(.&' #= 0001 L /= 0003 L = 0005 L
15=
)= 0002 L %= 0004 L
C&$8.& 100 "*).&' 8&. S; H5;00; ,%& "# *,(# 8#.( # S; 050> "# $*(#% %" .'(& # S; 175 "& .'(#,( # S; 2 / C:" ' $* ),*/*& #= S; 25 /= S; 75 = S; 125
)= S; 50 %= S; 100
27
Quinto Año
TEMA: !OTENCIACI@N RA5ICACI@N !otenciación n B#7
I;
E' ,# &8.#/*J, %*./(# , "# /#" , ,$.& <8&(,/*# 8./(#= .#" 8&'*(*& .'"(# "# $"(*8"*/#/*J, % &(.& ,$.& .#" 8&. '* $*'$& , ,$.& ,(.& % /'; a 3 a 3a 3...3 a F&.$# 9,.#"? B # P ! eces D&,%? a )#' <# R= k 8&,,( *,%*/# " ,$.& % /' -# ' $"(*8"*/J 8&. ' $*'$&; < R= $ P&(,/*#
!ropie(a(es#7 1= *e8 (e uniformi(a(: -*'( '&"& , ,$.& (#" %*/& ,$.& ' "# 8&(,/*# 8./(# .'"(#,( % "#. , %(.$*,#%& ,$.& # , %(.$*,#%& ,$.& # , %(.$*,#%& 8&,,(;
a b a b4 k
2= 3=
k
k 4 *e8 (e la monoton6a: S* bk a k b a 4 5 a 3 b 3 c a 3 b *e8 (istri$uti3a: k
$
k
3 c $
a k a k 5 b b
O%&:
L# 8&(,/*#/*J, '&"& '(# %*,*%# /#,%& " 8&,,( #" /#" ' "# "# )#' ' , ,$.& ,(.&; E, "# 8&(,/*#/*J, ,& *'( 8.&8*%#% /&,$(#(*# ,* 8.&8*%#% #'&/*#(*#; L % '*9,&'; <5= <=PAR B <5= <=IMPAR B O@S? 1= # #
<5= <=PAR B <5= <=IMPAR B
E" 8&,,( - '&"& #/(# # "# )#' ,& #" '*9,& E" 8&,,( - #/(# # "# )#' #" '*9,& 8&. '. <# #= &8.#%&. % #$)&'; !ropie(a( Fun(amental#7
Aritmética
28
Quinto Año
a & 3 n ' 3 $ / 3 ... 3 16 a k &k 3 n 'k 3 $ /k 3 ... 3 16k
# ' 8&(,/*# 8./(# % - '*?
; n; ...1 fact+*es $*i+s de )&) D&,%?
&; '; ...6 di(is+*es de ) ! )
1 Cua(ra(o !erfecto o !otencia !erfecta (e $ase ) N ! )B E' #" ,$.& 8&' .# /#%.#%# #/(# 8&' , ,$.& *$8#. % %**'&.'; Mo(os (e inclusión (el cua(ra(o perfecto#7 1= T&%& ,$.& '.: /#%.#%& 8./(& '* '&"& ' (&%&' '' #/(&.' 8.*$&' 8.',(#, 8&,,(' $"(*8"&' % 2; E$? 144
2 4 3 32 2 144 2 k
2= T&%& ,$.& '.: /#%.#%& 8./(& '* '&"& ' %*/& ,$.& ' $"(*8"& % 4
0
2
100 B
0
4
S* N B P2
8 1 <'&"& , *$8#.'= N B 9 1; 9 4 ; 9 7 ; 9 <'&"&
NB
0
,& % ""&'=
0
0
0
5= S* N 2 k k ...cds ' /$8"?
Aritmética
29
Quinto Año
0 S*
N
...cds
d
2; c
2 <'&"& ,& % ""&'=; E$? 625 B
6 252 6= S* , ,$.& (.$*,# , 1> 4> 5> 6> J !> ,(&,/' 8% '. , ,$.& /#%.#%& 8./(&; Mo(os (e e9clusión (el cua(ra(o perfecto#7 1= T&%& ,$.& (.$*,# , 2> 3> 7 J H ,,/# ' /#%.#%& 8./(&; 2= T&%& ,$.& (.$*,# , , ,$.& *$8#. % /.&' ,& ' /#%.#%& 8./(&; 3= T&%& ,$.& /" #" /#%.#%& 8./(& % , #/(&. 8.*$& ,& ' /#%.#%& 8./(&; 4= T&%& ,$.& ,& /$8"# /&, "# 8.*&.*%#% ,%#$,(#" % "# 8&(,/*#/*J, ,& '.: /#%.#%& 8./(&;
; Conclusión: T&%& ,$.& ' /#%.#%& 8./(& /$8" /&, (&%&' "&' $&%&' % *,/"'*J, /&, "# 8.&8*%#% ,%#$,(#" % 8&(,/*#/*J,; ) Cu$o !erfecto o !otencia !erfecta (e Gra(o o $ase , N ! ,B E' #" ,$.& 8&' .# /)*/# #/(#; M&%&' % I,/"'*J, %" C)& P./(&? 1= T&%& /)& 8./(& (.$*,# , /#"*. /*.#; 2= T&%& /)& 8./(& 8.',(# (&%& #/(&. 8.*$& "#%& # 8&,,(' $"(*8"&' % 3; 3= T&%& ,$.& (.$*,# , /.& ' 8./(& '* '&"& ' 8&' /#,(*%#% % /.&' $"(*8"&' % 3 " ,$.& &.$#%& 8&. "#' /*.#' %*'(*,(#' # /.& ' /)& 8./(&; 4= T&%& /)& 8./(& /$8"? P3 B B
Aritmética
0
0
0
0
0
4; 4 1 4 1 9; 9 1 9 1 0
30
Quinto Año
0
8 <'&"& 8#.'=
B
5= S* (&%& /)& 8./(& (.$*,# , S? N B K /%' %B2J7 / B 3> 6> H J 1
;Mo(os (e e9clusión (el cu$o perfecto: 1= U, ,$.& ,& ' /)& 8./(& '* ,& /$8" /&, #" $,&' ,& % "&' $&%&' % *,/"'*J,; 2= U, ,$.& ,& ' /)& 8./(& '* ,& /&,(*, #" /)& % ' #/(&. 8.*$&; II; Ra(icación
k
n ; E' "# &8.#/*J, *,.'# # "# 8&(,/*#/*J, , "# /#" " ,%*/ % "# .# *#" #" 8&,,( "# .# *#" # "# )#' " .#%*/#,%& # "# 8&(,/*#; F&.$# 9,.#"?
!
n
! n >
D&,%? B ,%*/ < 1= , B R#%*/#,%& & C#,(*%#% S).#%*/#" <, R= R B R#
O%&: S&"& '(# %*,*%# "# .#%*/#/*J, % ,%*/ ,(.& 8&'*(*#; !ropie(a(es#7 1) M&,&(&,#? S* a b k a ! b Si a b ! a ! b 2) D ,*&.$*%#%? 7 : k
3) D*'(.*)(*#?
a & ' / ...16
k
n
Si ! ; a b
k a & 3 k ' 3 k / 3 k 16 ;
k
!
a
a& '
b
k
a&
k
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*e8 (e si'nos#7
Aritmética
31
Quinto Año
<= <=
#
"M
n 2
3
1;1;R# C#%.#%# #/(#? , R 0
n = 4
7
n
n 3 2;1 R# /)*/# #/(#? 3
2
1;2;R# /#%.#%# *,#/(#?
, R 0
n
3
4
2;2;R# /)*/# *,#/(#? 3
2
, R , R . .
1;3;P&. E/'&? , R .
)# Ra6- C$ica#7 F&.$# 9,.#"?
1# Ra6- Cua(ra(a#7 Forma 'eneral n
"M
<= <=
, R .
n
4
3
*
2;3;P&. /'& n n
4 > 2 * > 4 1 > 2
3
*>
, R .
n
n
4 > 3 * 4 1 3
5on(e? , B R#%*/#,%& RI B R# 8&. /'& R B R# <8&. %/(&= . B R'(& & .'*%& <8&. %/(&= . b B R'(& & .'*%& 8&. /'& !ropie(a(es; 1 !ara Ra6- Cua(ra(a: <= . . I B 2R 1 <= E, .# *,#/(#?
Aritmética
32
*>
Quinto Año
* in 1
* 2 4 1 * a' 24 ) !ara Ra6- C$ica: <= . . I B 3R
* 34 4 1 1 34 4 1
* in 1 * a'
Aritmética
33
Quinto Año
!RO%*EMA& !ARA *A C*A&E 1=
I,%*/#. " #"&. % "# .%#% % "#' '*9*,(' 8.8&'*/*&,'?
I;
S* 3 6 n$+ ' , /#%.#%& 8./(& $"(*8"& % 3 % 7 ,(&,/' $ , 8 B 1H
II;
S* N
2
0
18 ,(&,/' N 9
;
D , /)& 8./(& % 5 /*.#' ' '#) " 8.&%/(& % "#' /*.#' % ' .# /)*/# (.$*,# , 5 ,(&,/' "# '$# % /*.#' % %*/# .# ' 8#.;
2=
S* " ,$.& % "# &.$# 39ab 4 ' , /#%.#%& 8./(& " ,$.& 39cb 4 ' , /)& 8./(& #""#. "# '$# % <# ) /=;
3= S* ? n ( 4 )( n 2)( n 2)( n : 2) (% 0n ) 2 A%$:' $, B 2 8 B , 2 2> *,%*/#. " #"&. % $ , 8
5=
S#)*,%& " ,$.#" '/#/)& ' , /#%.#%& 8./(& %**'*)" 8&. 77 #%$:' 2 dabae ; I,%*/#. " ab(c 4) ,$.& % %**'&.' /#%.#%& 8./(&' (*, " ,$.#" %) '#, PESI /&, /#; D(.$*,#. , ,$.& % 6 /*.#' /#%.#%& 8./(& (#" "# %*.,/*# ,(. " ,$.& &.$#%&
Aritmética
6=
S* #)#/ ' , /#%.#%& 8./(& " MCD % )# /# ' H; #""#. <# ) /=
7=
C:,(&' ,$.&' *'(, /# .# /#%.#%# ' 52 /# .# /)*/# ' 13
H=
A" (.#. "# .# /)*/# % , ,$.& 8&. /'& 8&. %/(& ' &)(*, /&$& '$# % .#/' 13; #""#. "# '$# % "#' /*.#' % %*/& ,$.& '* #%$:' " .'(& 8&. %/(& ' 123 $#&. " .'(& 8&. /'&;
0
III; L# /#,(*%#% % %**'&.' % , ,$.& /)& 8./(& 8% '. ab 2a 2 b I; N& ' /*.(& '* ab ' " $#&. ,(.& (#" " /&/*,( % 3024 ,(. ab ' , /#%.#%& 8./(& "# /#,(*%#% % %**'&.' % ab ' 12;
4=
8&. '' 4 8.*$.#' /*.#' " ,$.& &.$#%& 8&. '' 2 "(*$#' /*.#' ' 1275; %#. /&$& .'8'(# "# '$# % "#' /*.#' % ' .# /#%.#%#;
!= E, ,# .# /)*/# *,#/(# %&,% " .'*%& ' $:*$& ' '#) "# '$# %" .#%*/#,%& "# .# " .'(& ' 1626; %(.$*,#. /#,(#' ,*%#%' ' " %) #9.9#. " .#%*/#,%& /&$& $,*$& 8#.# &)(,. , /#%.#%& 8./(& (.$*, , "# /*.# 1; 10= C#"/"#. " #"&. %? 3,1777... 0,0655...
D#. /&$& .'8'(# " ,$.& ' ,/,(.# , "# 8&./*J, % "&' 10 $*"+'*$&; 11= L&' '&"%#%&' % , )#(#""J, '*$8. &.$#, #/*,%& , /#%.#%&> /#,%& 13 % '(&' '&"%#%&' '(:, % 9#.%*#; S* 6H '&"%#%&' '&, "9*%&' 8#.# *,(9.#.' # %*/& )#(#""J, ,(&,/' #" &.$#.' (&%& " )#(#""J, &.$#, , /#%.#%& C:,(&' '&"%#%&' #)#, *,*/*#"$,( , " )#(#""J, '* '&, $,&' % 300; %#. /&$& .'8'(# "#
34
Quinto Año
'$# % /*.#' %" ,$.& % '&"%#%&' 12= W#""#. <# ) / %= S* ab
a
acadb
13= A" (.#. "# .# /)*/# %" ,$.& 405ab7 ' &)(& /&$& .'(& 2 a 3 b 3 #""#. <# )=
14= S*?
5 555 555 5555 5555 55555 555 6 5 D(.$*,#. "# '$# % /*.#' % "# .#; 15= E" .'*%& % (.#. "# .# /#%.#%# % , ,$.& % 3 /*.#' ' 3; '* *,.(*$&' " &.%, % "#' /*.#' %" ,$.& " .'*%& #$,(# , 35 "# .# /#%.#%# #$,(# , 3; '* ' *,*.( "#' 2 /*.#' % $#&. &.%, %" ,$.& C:" ' " .'*%& % (.#. ' .# /#%.#%#
Aritmética
16= S*
2 11 2
3
/#"/"#. "# .# /#%.#%# 8&. %/(& %" ,$.#" ? 1 3 3 2 %#. /&$& .'8'(# "# '$# % /*.#'; 17= S* aaa 0cs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
35
Quinto Año
!RO%*EMA& !ARA *A CA&A 1= S (*, ,#' 3 )&"'#' /&,(*,, "#8*/.&' #"' .&&' ,9.&' .'8/(*#$,( ' &)'.#? 5 L# 4(#; P&(,/*# %" (&(#" % "#8*/.&' ' '/.*) /&, 6 /*.#'; 5 E" /)& % "# /#,(*%#% % "#8*/.&' #"' ' '/.*) /&, 3 /*.#' "#' /#"' '$#, 10; 5 L# '$# % "&' "#8*/.&' .&&' " /#%.#%& % "# /#,(*%#% % "&' "#8*/.&' ,9.&' ' H!; C:" ' " ,$.& % "#8*/.&' #= 1H %= 27
)= 22 = 30
/= 24
2= U, #,/*#,& /&$,(#)#? -Y& /$8" #[&' , " 2 - ' ,*(& .'8&,%*J? -Y& (#$)*+, /$8"*.+ #[&' , " 2; C:" ' "# %*.,/*# ,(. "# %#% %" #)"& %" ,*(& #= 64 %= 7H 3=
%/(& % ' .# /#%.#%# '* # ' .# /#%.#%# 8&. %/(& ' " .'(# 1 ' &)(*, " (.*8" % ' .# /)*/# 8&. /'&; D#. /&$& .'8'(# "# '$# % '' /*.#'; #= 3H )= 1 /= 47 %= 74 = 20 6= W#""#.
"# .# /#%.#%# % 2 1 ; D#. /&$& .'8'(# 2 1 " #"&. " /)& % ' /*.# /&..'8&,%*,( #" "9#. $*"+'*$&; #= 11 )= 1 /= 27 %= 64 = H
7= W#""#. " #"&. % # ) / % ,?
# ) / # % ) %4 H
)= 42 /= !0 = #"(#, %#(&'
-N" 3 -N" 2 - N " D (.$*,#. /#" ' #" '$# % "&' %**'&.' % DNI #= 727 %= !H1
)= H42 = 364
)= 337 = 3!6
/= 3!1
5= #""#. , ,$.& % 3 /*.#'> (#" " .'*%& 8&. /'& % ' .# /)*/# ' " %&)" %" .'*%& 8&. Aritmética
10 4 !
/= 10!3
4= ' (.# "# .# /#%.#%# % , ,$.& 8&. %/(& 8&. /'& &)(,*+,%&' /&$& '$# % '' .#/' 3!; #""#. %*/& ,$.& '#)*,%& " .'*%& 8&. %/(& ' " %&)" %" .'*%& 8&. /'&; #= 23! %= 3H7
3 4
#= 30 %= 24
)= 2H = 22
/= 27
H= T&%#' "#' $#[#,#' , .9*$*,(& % '&"%#%&' ' 8.',(# # ' '8.*&. &.$#,%& ,# *9.# ./(#,9"#. %&,% , "#%& ' " %&)" %" &(.&; D'8+' % "# )#(#""# $., 125 '&"%#%&' /&, "& /#" #&.# ' %) '9*. &.$#,%& " ./(:,9"&; D(.$*,#. /:,(&' '&"%#%&' #, %#%& #= 64 )= 75 /= 72 %= 6H = 7H
36
Quinto Año
!= S*? a a 2 b 2 a b 1 0 a 3 a 2
C#"/"#.
a b
1
#= 173 /= 27H = 314
4
)= 637 %= 20H
10= 10= U, &$). &$). (*, (*, 400$ 400$ % #"#$ #"#$). ). 8#.# 8#.# /./ /./#. #. ' (.. (.., ,& & % &.$ &.$# # /#%.# /#%.#%# %# /&$& %'8+ %'8+'' % /).*. /).*. (&%& " 8.$(.& " '&) '&).# , # (,'*J, % #"#$). %/*% 8&,. ,# ,# '9 '9,% ,%# # /./ /./# # , $(. $(.& & $#' $#' #%, #%,(. (.& & (*"* (*"*# #,% ,%& & #' #' " .'( .'(& & %" %" #"#$).; C:" ' " :.# %" (..,& #= 2601 / = 2401 = 255075
)= 245025 %= 2500
11= W#"" W#""# #. aa > '* abcdaa aa 3 ; D#. /&$& .'8'(# " .'*%& % (.#." ' .# /# /#%.#%# 8&. /'&; #= 5 /= 3 = 1
)= 4 %= 2
Aritmética
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
12= C C:" ' " ,$ ,$.& .& /& /& /#% /#%.# .#% %& $"(*8"*/#%& 8&. 3 #[#%*,%& 6 # '( 8.&%/(& %**%*,%& "# '$# ,(. 3 ' &)(*, 8&. .'"(#%& 2!1 #= 13
/= 15 %= 16 = 17 13= U, /&$./*#,( /&$8.J /*.(& ,$ ,$.& .& % (.# (.#' ' " 8./ 8./*& *& 8#9J 8#9J 8&. 8&. /#%# /#%# (.# (.# .# .# "# /#. /#.(# (# 8#.( %" ,$. $.& % (.#' /&$8.J; S* 9#'(J S; 30!76 C: C:,( ,(&' &' (.# (.#' ' /&$8 /&$8.J .J /:,( /:,(& & 8#9& 8&. /#%# ,&
#= 17 /= 11 = 2!
)= 13 %= 7
)= 14
37
Quinto Año
Aritmética
38
Quinto Año
TEMA: RAZONE& !RO!ORCIONE& I
Ra-ón o Relación#7 E' "# /&$8#.#/*J, ,(. 2 /#,(*%#%' 8&. $%*& % "#' &8.#/*&,' *,.'#' ):'*/#' <''(.#//*J, %**'*J,= %**'*J,= Clases (e ra-ones o relaciones#7 1 RaRa-ón arit aritm m"tic "tica# a#77 C#,%& "# /&$8#.#/*J, ,(. "#' 2 /#,(*%#%' ' .#"*#, 8&. $%*& % "# %*.,/*#; N&(#/*J,? # ) B . <-# ' $#&. ) , . ,*%#%'=; A a antecedente 4 a; b; * ? A b c+n sec ente A * (a@+* de @a *a#9n 4 ) RaRa-ón Geom"t om"trric ica# a#77 C#,%& "# /&$8#.#/*J, /&$8#.#/*J, ,(. "#' 2 /#,(*%#%' ' .#"*#, 8&. $%*& % "# %**'*J,; N&(#/*J,; # ? ) B . <-# ' 8.&%/(& % ) 8&. .= A a antecedente 4 a; b; * ? A b c+n sec ente A * (a@+* de @a *a#9n 4 , Ra-ón Armónica#7 E' "# /&$8 /&$8#. #.#/ #/*J *J, , 8&. 8&. ''( ''(.# .#// //*J *J, , ,(. ,(. "#' "#' *,.'#' % 2 ,$.&' &.$#, .#J, #.*($+(*/#; N&(#/*J,?
1 1 % a b
A 1 antecedente a 4 4 1 1 1 A c+n sec 8ente ; ; % Q b a b 4 A % (a@+* de @a *a#9n4
IIII
!roporciones#7
Aritmética
39
Quinto Año
E' "# *9#"%#% % 2 (*8&' /&$,' % .#&,' <% "# $*'$# /"#'= & $#&.' % 2; C"#'' % 8.&8&./*&,';
1) Proporción Aritmética
a 1 )ca 8
1
N Ó I C A T O N
2)Proporcirción
Geométrica
c 3d 1d9 a 2d9 b d 4d
3) Proporción Armónica B
1c
-# ) '&, 8.&8&./*&,#"' # / %
1 a
1 b
1 c
1 d
3c -1# ' # 1) /&$& 1/ ' # 1%
D&,%? # / < *,.'#'= B A,(/%,(' ) % < *,.'#'= B /&,'/,('
D A D E I P O R P
a d
a b
b c
a 3d
cd
a b
1 1 1 1 a d c b
b 3 c
c d
D&,%? D&,%? #> )> /> %> B 4(# D*.,/*#" A> )> /> % B 4(# %+/#%# ,& .'8/(& # "&' P.&8&./*&,#" .'8/(& # "&' &(.&' 3 <, ' ' $*'$ *'$& &(.&' 3; &.%,=
a
b c d ?
Aritmética
a
b c d ?
1 a
1 b
1 c
1 d
D&,%? 1#> 1)> 1/> 1% B 4(# A.$J,*/# .'8/(& % "&' &(.&' 3 <' ' $*'$& &.%,=
1 1 1 1 Q a b c d
40
Quinto Año
O A U N I T N O CP#.# "& 8.&)"$#' "# S I '9,%& /&,'/,(; D N Ó I C A R T E O P R C O S R I P D
a b
/#.(# %*.,/*# 8.&8&./*&,#" & #.$J,*/# ' /&,'*%.#%& /&$& "
b c
D&,%? A) B M%*# T A.*($+(*/#; E
D*.,/*#" &
R C#> / B T./*# S I %*.,/*#" & A.*($+(*/# D .'8/(& % "&' &(.&' 3 S N I (+.$*,&'; O A a c U N I T N O C N Ó I C R O P O R P
b
ac 2
a b
b
1 a
c
D&,%? ) B M%*# P.&8&./*&,#" & G&$+(.*/#; #> / B T./*# 8.&8&./*&,#" & G&$+(.*/# .'8/(& % "&' &(.&' 3 (+.$*,&';
1 b
1 b
1 c
D&,%? ) B M%*# A.$J,*/# #> / B T./*# #.$J,*/# .'8/(& % "&' &(.&';
a c b
3a
c
1
2ac
b
ac
a c
P#.# "&' 8.&)"$#' "# (./.# & (./*# A.*($+(*/# G&$+(.*/# & A.$J,*/# ' /&,'*%.#%& /&$& " '9,%& /&,'/,(;
O%&? S* ,& ' %(.$*,# (*8& % .#J, & 8.&8&./*J, ' '(#)"/ , , 8.&)"$# ' #'$ ' GEOMeTRICA; &erie (e Ra-ones E2ui3alentes R#E#7 1 &erie Aritm"tica: S;R;E;A C&,(*,#? F&.$# G,.#"? # ) B ) / B / % B % BQ S;R;E;A; D*'/.(#? F&.$# G,.#"? # ) B / % B B K ) &erie Geom"trica: a
b
c
d
S;R;E;D; C&,(*,#? F&.$# G,.#"? b c d e k
Aritmética
41
Quinto Año
a
S;R;E;G; D*'/.(#? F&.$# G,.#"? b
c d
e f
k
NOTA? P.&8*%#%' % "#' S.*' G&$+(.*/#'? a c e D#%&? b d f k a #c#e
k
1=
b # d # f
3=
# # e ! b # d # f
a 3c3e
2=
b 3 d 3 f
! N ; N B Nd % .#&,'
a
Aritmética
42
Quinto Año
!RO%*EMA& !ARA *A C*A&E 1=
L# .#J, % 2 ,$.&' ' % 7 # 3; C:" '.: "# .#J, ,(. "# '$# % /#%.#%&' "# %*.,/*# % /#%.#%& % %*/&' ,$.&'
2=
L# %#% % P8 ' # "# %#% % L*' /&$& 5 ' # 6 %'8+' % /*.(& (*$8& '' %#%' '(:, , "# ."#/*J, % ! # 10; E, ."#/*J, '(:, " (*$8& (.#,'/..*%& "# %#% *,*/*#" % L*'
A @ %*'(#, 550 $ /&, "&/*%#%' *,*/*#"' '&, ,(. '* /&$& 4 ' # 7 .'8/(*#$,(; S* *,$%*#(#$,( %'8+' %" /./ "# ."#/*J, % "&/*%#%' /#$)*#? ' % 5 # H '*,%& P#)"& " $:' "&; C#"/"#. "# %*'(#,/*# % A #" 8,(& , " /#" "9& %" /./ P%.& ' ,/,(.# '8#.#%& % P#)"& 1!5 $; 7=
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H=
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3= A ,# *'(# #'*'(, 400 8.'&,#' ,(. &$).' $.' #'*'(*,%& 3 &$).' 8&. /#%# 2 $.'; L9& % 2 &.#' 8&. /#%# 2 &$).' # ,# $.; C:,(#' 8#.#' ' .(*.#.&,
4= E" 8.&%/(& % 3 ,$.&' ' 5H32; '* " 8.*$.& ' #" '9,%& /&$& " '9,%& ' #" (./.&; W#""#. " '9,%& ,$.&; 5=
D&' $&(&/*/"*'(#' 8#.(, % , $*'$& 8,(& , %*.//*&,' &8'(#' (.#,'/..*%&' "&' 8.*$.&' 45 $*,(&' "# .#J, % "# %*'(#,/*# # ' 8,(& % 8#.(*%# ' % 3 # 5 # "&' 30 $*,(&' '*9*,(' ' ,/,(.#, %*'(#,/*#%&' H0 $; C:" ' "# %*.,/*# % '' "&/*%#%' , $&.#
6= P%.& P#)"& 8#.(, '*$"(:,#$,( ,& #" ,/,(.& %" &(.& % %&' 8,(&' Aritmética
43
Quinto Año
% "# '9,%# ""9# # A; "# %*'(#,/*# ,(. A @ ' 24 $; C:,(&' $' ./&.. /#%# ,# % "#' 8.'&,#' , 1 &.# D#. /&$& .'8'(# "# '$# % %*/#' %*'(#,/*#'; 10= E, ,# #/#%$*# "# ."#/*J, % &$).' $.' ' 2 ? 5 "# ."#/*J, %" '$'(.#" ' 7 ? 3 C:" ' "# ."#/*J, % "&' &$).' '(:, , " '$'(. " (&(#" % #"$,&'
16= L&' ,$.&'? # ) . '&, a * ,#(.#"' ' /$8"? b a b * 1 a b %(.$*,#. 4 3 "# $#&. '$# -# ) . $,&. 20; 17= S*? 10 a
13= S#)*,%& ?
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a
c
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20= S*?
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Aritmética
100 c
10 a 11 b 100 c <. 1= # ) / 1 B . 2;
11= L&' :,9"&' % , /#%.*":(.&' '(:, ,(. '* /&$& "&' ,$.&' 4 7 ! 10; C:" ' " #"&. %" $#&. % ""&' 12= L# %*.,/*# ,(. " $#&. $,&. (+.$*,& % ,# 8.&8&./*J, 9&$+(.*/# /&,(*,# ' 25 " &(.& (+.$*,& ' 30; #"#. "# '$# % "&' (+.$*,&' '* "&' 4 '&, 8&'*(*&';
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a b
b
c
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a 3 b 3c 3d ... 3E 2D a D
44
Quinto Año
!RO%*EMA& !ARA *A CA&A 1= S*?
a b
2
4= S* ' #$,(# ,# $*'$# /#,(*%#%
n
a c
2
$
n
b c
2
$
a b
2
$
a c b c 2
2
# "&' ,$.&' 20> 50> 100 ' &.$# ,# 8.&8&./*J, 9&$+(.*/# /&,(*,# /# .#&,'? #= 12 %= 13
)= 43 = 35
/= 2
5= E, "# '*9*,( '.* % .#&,' *#",('? a 3 b3 a 2 b 2
a 2 b2
E,(&,/' 82 $2 ' *9#" #?
182 25 7 ab /#"/"#. # ) <, '( &.%,= 12
#= ,2 %= #2 )2
#= 4 6 %= H 6
n
)= , = )2 /2
/= #2
2= D&' ,9&/*#,(' % *,&' *,9.'#, #" P. 8&.(#,%& ,& % ""&' 64 )&(""#' % *,& " &(.& 20; /&$& ,& (*,, '*/*,( %*,.& 8#.# 8#9#. "&' %./&' % A%#,# " 8.*$.& 8#9# /&, 5 )&(""#' % *,& S; 40;00 " '9,%& /&, 2 )&(""#' % *,& 8.& '( ./*) % "(& S; 40;00; C:" ' " 8./*& % /#%# )&(""# % *,& #= S; 120 )= S; 50 %= S; 105 = S; !5
/= S; 110
3= E" 8.&%/(& % "&' 4 (+.$*,&' % ,# 8.&8&./*J, 9&$+(.*/# ' 50625; '#)*,%& "&' $%*&' '&, *9#"' ,& % "&' (.$&' ' 75> *,%*/#. "# '$# % "&' /#(.& (+.$*,&'; #= 1H0 )= 10H /= 216 %= 25H = !5 Aritmética
)= 3 4 = 4 3
/= H 6
6= T.'
./*8*,(' /*",%.*/&' % &"$,' %*.,(' 8.& 9&$+(.*/#$,( '$#,(' /#' #"(.#' '&, ,(. '* /&$& 1 2 3 /&,(*, #9# , /#,(*%#% 8.&8&./*&,#" # ' #"(.#; S (.#'#'# " #9# % $&%& " ,*" '# " $*'$& , "&' 3 ./*8*,('; S #/ 8#'#. #&.# ! "*(.&' % #9# %" (./.#" 8.*$. ./*8*,( ' &)'.# " '9,%& /&,(*, 2 /' " &"$, /&,(*, " 8.*$.&; S 8*% " &"$, *,*/*#"$,( (,*# " (./. ./*8*,(; #= H1 L %= 20 L
)= 42 L = N;A;
/= 63 L
45
Quinto Año
a
7= S* ' (*,?
Z= '* n "#'
C
n
c
c d a b c 5n
!
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#= 1 %= 4
b
b
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ab bc cd b 2
c2 d 2 )= 2 = 6
/#"/"#.?
/= 3
H= S (*, ,# '.* % 3 .#&,'
9&$+(.*/#' *#",('; E" 8.&%/(& % "#' '$#' % / % "#' .#&,' ' 64; #""#. "# '$# % "# $%*# 9&$+(.*/# % #,(/%,(' % "# 9&$+(.*/# % /&,'/,('; #= 2 %= 6
)= 4 = 10
8.&9.'*J, #.*($+(*/# #$,(#, , 3 3 7 .'8/(*#$,( &.$#, ,# 8.&9.'*J, 9&$+(.*/# % '$# 2H '&,? )= 1 5 ! /= 5 ! 13 = N;A;
10= U, /&,,(& % $'*/# $&%.,# &./*J 3 /&,/*.(&' '*,%& " (&(#" % #'*'(,(' , /#%# ,# % '' 8.',(#/*&,' A @ C; '#)*,%& A ' # @ /&$& 4! ' # 64 C ' , /)& 8./(& $"(*8"& % 5 #%$:' $%*# 8.&8&./*&,#" ,(. A @; C#"/"#. " $,*$&? B A @ C %#. "# '$# % '' /*.#';
Aritmética
)= 1H = N;A;
/= 17
11= E" 8.&%/(& % "&' 4 (+.$*,&' % ,# 8.&8&./*J, %*'/.(# ' 15H76; '* " 8.*$.& % '(&' (+.$*,&' ' 7; /#"/"# " 8.&%/(& % "&' (+.$*,&' $%*&'; #= 120 )= 122 /= 126 %= 127 = N;A; /##%&.' %*'8#.#, 12= T.' '*$"(:,#$,( /&,(.# ,# "*).; E" 8.*$.& /&,'*9 #/. )"#,/& 2 /' % /#%# 5> " '9,%& 4 /' % /#%# 10 " (./.& H /' % /#%# 10; C:,(#' /' % /#%# 500 8&. "& $,&' ,& % "&' 3 /##%&.' #"/#,# "# "*). #= 374 %= 464
)= 352 = N;A;
/= 3!6
/= 6
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#= 6 5 4 %= 5 7 !
#= 10 %= 1!
13= S#)*,%& ?
15
24
33
4A 2@ 5C B 2!5; #""#. A @ C #= !0 %= 150
)= 10H = N;A;
/= 120
14= L# '$# %*.,/*# " 8.&%/(& % -N ,$.&' '(:, , "# $*'$# ."#/*J, 4 2 15 .'8/(*#$,(; C:" ' " $#&. % "&' ,$.&' #= 15 %= 4
)= 10 = 14
/= 16
46
Quinto Año
15= L&'
/#%.#%&'
%
1 , 1 & 1 2 4 8 8.&8&./*&,#"'
'&, #
,$.&' '$#,
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#= 76 %= 67
)= 447 = 744
/= 71H
(.'
147 ; 176
U,& % %*/&' ,$.& '?
Aritmética
47
Quinto Año
TEMA: !ROME5IO&
Concepto: E' , #"&. *"*).# '*,((*# " #"&. % /#%# ,& % "&' (+.$*,&' % , /&,,(& '&"/*J, #%$:' 8% & ,& 8.(,/. # %*/& /&,,(& *,*/*#" <"# '/'*J, % ,$.&'=; T#$)*+, ' " /&,&/ /&$& -M%*#; N&(#/*J,? #1> #2> #3> K > #, B f#,g #"&. % "# '/'*J,; P.&$%*& % f# ,g B <#1> #2> #3> K > #,= D&,%? #1 : <#1> # 2> # 3> K > # ,= : #, # 1 #2 #3 K #, C"#''; ; M : I !rome(io Aritm"tico a 1 ; a 2 ;....; a n
a1
n
II !rome(io
a 1 ;
a 2 ...a n
; M :
Geom"trico
a 2 ;.... ; a n
n a1 a 2 ... a n
III !rome(io Armónico B ; MB : ' "# *,.'# % "# '$# % "&' ./8.&/&' % /#%# (+.$*,& % "# '/'*J, &.*9*,# #" ; 1
B a 1 ;
a 2 ;.... ; a n
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1 a2
1 a3
n
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1 an
1 a1
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1 an
n 3 a 1a 2 a 3 ...a n a 1 a 2 ...a n
n
ID !rome(io !on(era(o : ' " 8.&$%*& &.*9*,#%& 8&. ,# '/'*J, %&,% /#%# ,& % '' (+.$*,&' <& 8&. "& $,&' ,& % ""&'= ' .8*(, , ,$.& ,(.& % /' $#&. % ,&; #;P.&$%*& P&,%.#%& A.*($+(*/&
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a 1 a 2 ...a n
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Aritmética
48
Quinto Año
B
D&,%?
' & # ... ; 1 ' 1 & ... 1 a1 a2 an
;
, B h % (+.$*,&' % "# '/'*J,; #1> #2> #3> K > #, B (+.$*,&' % "# '/'*J,; > > K > ; B h % /' , ' .8*( /#%# (+.$*,& % "# '/'*J,;
A"9,#' 8.&8*%#%' 9,.#"'; 1= S /$8"? M 0 M 0 MB <$#&. '* "&' (+.$*,&' '&, %*'(*,(& ,(. '> *9#" '* "&' (+.$*,&' '&, *9#"' ,(. '=; 2= S&"& 8#.# 2 ,$.&'? M ( a
; b )
A%$:'? M ( a
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M ( a
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a b 2
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M ( a
; b )
; b )
ab ; MB ( a ; b )
a b
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M ( a ; b )
A,#"J9*/#$,(?
M ( a
; b )
2ab
M ( a ; b )
a b 4 M ( a
; b )
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2
2
M (a ; b )
O%&: 1=
P#.# "&' 8.&)"$#' ' #'$ /&$& 8.&$%*& #" 8.&$%*& #.*($+(*/&;
2=
L# M%*# G&$+(.*/# % , 9.8& % ,$.&' ' (#$)*+, "# $%*# G&$+(.*/# % "# M%*# A.*($+(*/# "# M%*# A.$J,*/# "# M%*# A.$J,*/# %" $*'$& 9.8& % ,$.&;
Aritmética
49
Quinto Año
!RO%*EMA& !ARA *A C*A&E 1= E" 8.&$%*& % 100 ,$.&' 8#.' /&,'/(*&' ' 12!; E,(&,/' "# %*.,/*# ,(. " $#&. " $,&. % %*/&' ,$.& '?
8.&$%*& 9#,#, "&' 100 (.#)##%&.' -, " 8.&$%*& 9#,#, "&' (.#)##%&.' % "# '//*J, -C; W#""#. $ ,;
2= L# %+/*$# 8#.( % "# /#..(.# ,(. "#' /*%#%' A @ ' %'#..&""# # (.#+' % 8&)"#%&' " .'(& , /#$8& #)*.(&; S* "# "&/*%#% "$*( , "&' 8&)"#%&' ' % 30 $ , /#$8& #)*.(& ' % !0 $; C:" ' "# $:*$# "&/*%#% 8.&$%*& # 8% /&,%/*.' , /"& ,(. A @ '*, /%. "# "&/*%#% "$*(
6= E, , 9.8& % 30 8.'&,#'? " 8.&$%*& % "#' %#%' % "&' 15 $#&.' ' 42 " 8.&$%*& % "&' .'(#,(' ' 2H; '* " 8.&$%*& % "&' 10 $#&.' ' 45 " 8.&$%*& % "&' 10 $,&.' ' 22; C:" ' " 8.&$%*& % "&' &(.&' 10
3= E" *,9.'& 8.&$%*& %" '/(&. &)..& , ,# $8.'# ' % S; 3000 $,'#"'; E, , $' , /.'& # , *,/.$,(& % #).' %" 10 i %" #). #,(.*&. $#' ,# )&,**/#/*J, 9,.#" % S; 600 8.& ' %/.(# , %'/,(& %" 5i %" #). #/(#"*#%& 8.&&,%&' , ./&,'(.//*J,; E" 8.&$%*& #/(#" '? 4= E" 8.&$%*& % "#' %#%' % 5 '[&.#' ' % 42 #[&' ,*,9,# % ""#' ' $,&. % 36 #[&'; C:" ' "# $:*$# %#% 8&%.# (,. ,# % ""#' 5= U,# /&$8#[# % 100 $8"#%&' , '' 3 '//*&,'? -A /&, 70 (.#)##%&.' 9#,#, S; 2;50 8&. &.# -@ /&, 20 (.#)##%&.' 9#,#, S; 3;25 8&. &.# -C /&, 10 (.#)##%&.'; E" 8.&$%*& % 9#,#,/*# % "&' (.#)##%&.' % "#' '//*&,' -@ -C ' % S; 3;50 8&. &.#; S# -$ " Aritmética
7= E" 8.&$%*& ,$.&' ' #.$J,*/& % /#"/"#. " ,$.&';
#.$J,*/& % 40 16 " 8.&$%*& &(.&' 30 ' 72; B % "&' 70
H= E" '#"#.*& $,'#" 8#9#%&' # (&%&' "&' (.#)##%&.' % ,# /&$8#[# % S; 576; "&' '#"#.*&' $%*&' $,'#"' 8#9#%&' # &$).' # $.' % "# /&$8#[# .&, S; 600 S; 4H0 .'8/(*#$,(; S* " ,$.& % (.#)##%&.' &$).' #$,(# , , 25i " % "#' $.' #$,(# , , 40 i '.*#, , (&(#" 4H0 (.#)##%&.'; C#"/" " ,$.& *,*/*#" % &$).'; != E" % 4 ,$.&' ,(.&' 8&'*(*&' %*.,(' ,(. '* ' 2 2 ; W#""#. " % %*/&' ,$.&'; 10= L# M % "&' /#%.#%&' % 2 ,$.&' ,(.&' /&,'/(*&' '
50
Quinto Año
30; L# MB % %*/&' ,$.&' '? 11= P#.# 2 ,$.&' # ) /# %*.,/*# % /#%.#%&' ' 144 ' /$8" "# ."#/*J, ,(. ' B ' ' *#",( # 1516; #""#. " $#&. % "&' ,$.&'; 12= S* ' '#) ?
M a
; b
M a
; b
M a
; b
6 2 6 3 2
W#""#. "# MB % A @ C 13= U,# ",# % $*/.&)'' (*, , /*.(& ./&..*%& " /#" &.$# , (.*:,9"& *":(.& , ,# /*.(# /*%#%; S* , "#%& "& ./&.. # 100 $ " '9,%& # 75 $ " (./.& # 50 $; C:" ' "# "&/*%#% 8.&$%*& % "&' $*/.&)'' #" /(#. , ./&..*%& /&$8"(&> '(# )*/#%& , ,& % "&' +.(*/' %" (.*:,9"& 14= L# %#% 8.&$%*& % -, &$).' ' -# #[&' ,*,9,& % ""&' ' $,&. # -$ #[&'; C:" ' "# %#% $:*$# 8% (,. ,& % ""&' 15= D#%& , /&,,(& % -, ,$.&' /& ' -P; S* # "# (./.# 8#.( % ""&' ' "' #$,(# -# ,*%#%' /#%# ,& # "&' 35 %
Aritmética
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
51
Quinto Año
!RO%*EMA& !ARA *A CA&A 1= E" 8.&$%*& % 30 ,$.&' ' 2H; S*,%& 40 44 % "&' ,9.& "*$*,#,%& '(&' 21 ,$.&' " 8.&$%*& % "&' .'(#,(' '? #= 27 /= 26 = 25 J 25;5
)= 2H %= 26J 26;5
2= E" 8.&$%*& % 50 ,$.&' ' 62 1 ' .(*.#, 5 ,$.&' /& 8.&$%*& ' 1H; E, /#,(& #.*# " 8.&$%*& #= 5 /= 4! = 3!
)= 47 %= 57
3= E" 8.&$%*& % %#% % 1H &$).' ' 16 #[&' "# %#% 8.&$%*& % 12 $.' ' % 14 #[&'; C#"/"#. " 8.&$%*& %" '#"J,; #= 15 /= 152 = 161
)= 162 %= 151
4= E, ,# '/"# "#' ,&(#' ' /#"**/#, % 0 # 20 <(&%#' "#' ,&(#' '&, ,$.&' ,(.&'= X#, (*, $#' % "# $%*# L*' (*, $,&' % "# $%*#; S* # "#' ,&(#' % #$)&' ' "' .'(# "# (./.# 8#.( % "# $,&. ,(&,/' "# %*.,/*# $#&. ' 3 /' "# %*.,/*# $,&.; W#""#. "# ,&(# $#&.; #= 13 /= 15 = 17
)= 14 %= 16
5= L#' %*'(#,/*#' C&'*/# T*/"*&> T*/"*& L# O.&# '&, % !0 $ 60 $ .'8/(*#$,(> , (., 8#'# 8&. L# O.&# # "#' 10?20 #;$ %) 8#'#. 8&. C&'*/# # "#' 2?05 '* " /#$*,& #/*# T*/"*& "& ./&..*J # "&/*%#% 8.&$%*& % 30 $; A "&/*%#% 8.&$%*& <#8.&*$#%#$,(= %) *. "9& 8#.# /$8"*. /&, " *(*,.#.*& #= 514 $ /= 30 = 45
)= 621 %= 20
6= S* ' ,%*.&, 100 $8"#.' % -E" C&$./*& # S; 3 /#%# ,& 150 $8"#.' % -E" C*,& # S; 2 '&"' /#%# ,&; C:" ' " 8./*& 8.&$%*& % "&' %*#.*&' $*(*%&' #= S; 250 /= S; 230 = S; 2H0
)= S; 240 %= S; 260
7= E" 8.&$%*& % "#' %#%' % 3 8.'&,#' ' *9#" # -; '* ' #9.9# ,# /#.(# 8.'&,# " 8.&$%*& %*'$*, , 2; ' 8% #*.$#. ? *=
L# %#% %" /#.(& ' $#&. " 8.&$%*&;
**=
L# %#% %" /#.(& ' $,&. " 8.&$%*&; ***= P&. "& $,&' ,# 8.'&,# ' $#&. " /#.(&; #= S&"& * /= '&"& *** = * **
)= '&"& ** %= ** ***
H= U, #(& *## % "# /*%#% A # "# /*%#% @ %*'(# 2H0 $ %" Aritmética
52
Quinto Año
'*9*,( $&%&? L&' 8.*$.&' 120 $ "&' ./&..*J # 40 $ 8&. &.# "&' '*9*,(' H0 $ "&' ./&..*J # 60 $ " .'(& *#& # H0 $; W#""#. "# "&/*%#% 8.&$%*& %" *#; #= 72 $ /= 60 $ = 64 $
)= 4! $ %= 525 $
!= E" 8.&$%*& % "&' 8#'&' % 60 &)(&' ' 50 *"&9.#$&'; C#%# ,& % "&' &)(&' 8'# , ,$.& ,(.& % *"&9.#$&'; C:,(& %) 8'#. /&$& $:*$& ,& % ""&' '* ,*,9,# 8'# $,&' % 4H 9 #= 160 9; /= 16! 9; = 165 9;
)= 16H 9; %= 161 9;
10= E" 8.&$%*& #.*($+(*/& % 3 ,$.&' ' 14 " 8.&$%*& 9&$+(.*/& ' 8#. *9#" # , % ""&' ' 8.&$%*& #.$J,*/& ' 727; #""#. " ,$.& $,&.; #= H /= 24 = 4
)= 6 %= 30
11= L# MB "# M % 2 ,(.&' '(:, , ."#/*J, % 4H # 4!; #""#. "&' ,$.&' '#)*,%& " $#&. '(# /&$8.,%*%& ,(. 41 47; #= 44 23 /= 46 25 = 44 33
)= 45 30 %= 44 13
12= C#"/"#. " B % "&' (+.$*,&' % "# '/'*J,? 2> 6> 12> 20>K> !!2; #= 30 /= 32 = 34
13= U, ,$.& .#/*&,#" *..%/(*)" $ ' (*, "#' '*9*,(' % /#.#/(.'(*/#'? <= <35= <45= <= S* ' %**% " *,(.#"& $35 > 45% , 5 8#.(' *9#"' " 8,(& '(# , " 8,(& $%*& %" (./. *,(.#"&; W#""#. 8 #= 5 /= 17 = 4!
)= 12 %= 25
14= E" 8.&$%*& % 30 ,$.&' ' 40; A /#%# ,& % 10 % ""&' ' "' #$,(# 5 ,*%#%' %" .'(& ' '/&9 ,# /*.(# /#,(*%#%; A $&%& % "&' /#"' # /#%# ,& ' " #$,(# 10 ,*%#%' /&, "& /#" " ,& 8.&$%*& % "&' ,$.&' #$,(# , "# %+/*$# 8#.(; D(.$*,#. /#,(&' ,$.&' #, #.*#%& ' #"&.; #= 13 /= 15 = 17
)= 14 %= 16
15= E" 8.&$%*& % "#' %#%' % 4 &$).' ' % 45 #[&'> ,*,9,& % ""&' ' $,&. % 40 #[&'; C:" ' "# $:*$# %#% 8&%.# (,. ,& % ""&' #= 55 /= 57 = 65
Aritmética
)= 31 %= 33
)= 60 %= 54
53
Quinto Año
Aritmética
54
Quinto Año
TEMA: MAGNIT5E& !RO!ORCIONA*E& Concepto#7 E' "# /&$8#.#/*J, ,(. "&' #"&.' #%&8(# , 9.8& % $#9,*(%' <(&%& #""& (*, # '.*. #"9, (*8& % #.*#/*J, 8.&8&./*&,#"; E$;? "# $#'# "# "&,9*(% " (*$8&=> *,%*/#,%& ' ."#/*J, 8.&8&./*&,#"; Clases (e Ma'nitu(es !roporcionales M ; 1= Ma'nitu(es (irectamente proporcionales: C#,%& "#' 2 /#,(*%#%' '., "# $*'$# #.*#/*J, 8.&8&./*&,#" .'8/(& % ' #"&. &.*9*,#" , ,$.& ,(.& % /'; N&(#/*J,? A DP @ > A & @ S* A @ B A B @ < B /&,'(#,(= G.:*/&? P,(&' % ./(# /&"*,#"' %*'/&,(*,#' b b b
*
3 2 1
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a1 a
2
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Ejemplo de Magnitudes directamente proporcionales: El nmero de o!reros " $<9.#6 =1.o!ra (9 2. #elocidad " $istancia @ A 3. %iempo " di&icultad de o!ra' etc.
<1
F,/*J, % 8.&8&./*&,#"*%#%?
& a* iab@e de$endiente 4 ' a* iab@e inde$endiente B <= B <= 4 (a@+* de anF@isis)
2=
Ma'nitu(es in3ersamente proporcionales: C#,%& "#' 2 /#,(*%#%' '., "# #.*#/*J, /&,(.#.*# % , #"&. &.*9*,#" "# ,# % "# &(.#> , ,# $*'$# 8.&8&./*J,; L# #.*#/*J, % ,# $#9,*(% *,.'#$,( 8.&8&./*&,#" 8% '. , ,$.& ,(.& % /' .'8/(& % ' #"&. &.*9*,#"; N&(#/*J,? A I8 @ > A< 1 & = @ A@ B b@ A B b <1@= @
a1 ; b n
b
n
b 3 b 2 G.:*/& b 1
S Aritmética
S1 S2 ;;; S, 4 a ; b Nj S Oj $<9.#6 = 4 n
a1 a
2
a
3
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n
1
W*8+.)&"# E*":(.# A',(&(# # "#' )#'' <,,/# .&# # "&' '=
A
55
Quinto Año
F,/*J, % 8.&8&./*&,#"*%#%? B b B b <1= <= B b <1= P&(,/*# % *,.'*J, B #.*#)" %8,%*,( E$8"&?
B #.*#)" I,%8,%*,( <#"&. #,:"*'*'=
%
R,%*$*,(& T*$8& E*/*,/*#
1; N$.& % &)..&' 2; "&/*%#% T*$8&
3; N$.& % &)(&' /&'(& (&(#";
O%&: 1= S #'$ /&$& 8.&8&./*J, # "# 8.&8&./*J, %*./(# 2=
G-de(8@tas*-Gdeints* -Gde(8@tas*-Gdeintsde*
+ (+ ( + (+ ( 4, 1 ), 41 ) , 42 ), 42 )
4
2
1
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*1
R%#' ,*%#' 8&. ,# /#%,#
h "(#' R 1 3 h % %*,(' R 1 h "(#' R 2 3 h % %*,(' R 2
*1 4
*2 1
4
2
E,9.#,#' ,*%&'
Aritmética
56
Quinto Año
3=
4
2
4
1
R%#' ,*%#' 8&. ,# ## S* R2 R1
h % "(#' 1 B h % "(#' 2
P.&8*%#%'; 1= 2= 3= 4=
S* A IP @ A DP <1@= S* A DP @ A IP <1@= S* A DP @ @ DP A <= A IP @ @ IP A
@> C? A DP @ 3 C B /( S* A DP @
5=
S* A DP @ A, DP @, <, Z =
Aritmética
57
Quinto Año
!RO%*EMA& !ARA *A C*A&E 1= L# % &.$#/*J, 8.&%/*%# 8&. , .'&.( #" #8"*/#.' ,# .# ' DP # %*/# .#; S* # , .'&.( % 30 /$ % "&,9*(% ' " #8"*/# ,# .# % 3N ' ,# "&,9*(% ' 36 /$; C:" '.: "# ,# "&,9*(% %" .'&.( '* ' " #8"*/# ,# .# % 4N
2= E" /&'(& % , (..,& ' IP #"
/#%.#%& % "# %*'(#,/*# % L*$# #" (..,& DP # ' :.#; U, /*.(& (..,& /'(# 500 $*" &(.& (..,& % %&)" :.# '*(#%# # ,# %*'(#,/*# /#%.8" "# #,(.*&. /&'(#.:?
#).*/#.&, 1H 000; C:,(&' ' #).*/#.&, " #[& !5
5= S
# /&$8.&)#%& '(#%'(*/#$,( " .,%*$*,(& % , $8"#%& ' DP # ' /&*/*,( *,("/(#" ' *,.'#$,( 8.&8&./*&,#" #" /#%.#%& % ' %#%; S* N#,& % 20 #[&' #/ , 8.&$%*& % 25 000 '#,%`*/' '$#,#"' ' %'# '#). # %#% N#,& #/ 24 000 '#,%`*/' '$#,#"' '* ' /&*/*,( *,("/(#" #$,(# #,#"$,( , , 10i %" (,*# # "&' 20 #[&';
6= L# '*9*,( *9.# $'(.# "# 3= S '#) -A ' %8 #" /#%.#%& % -@ " /)& % -C ' *,.'#$,( 8.&8&./*&,#" /&, "# .# /#%.#%# % -F; D" /#%.& %(.$*,#. " #"&. % ;
M#9,*(%' A @ C F
C#,(*%#%' k 10H 324 5 2 4 2 3 Y 25 ! 16
9.:*/# % 2 $#9,*(%' %*./(#$,( 8.&8&./*&,#"'? "# 8.&%//*J, % ,# #).*/# .'8/(& #" ,$.& % &)..&'; L# 8.*$.# ./(# ' # &)(,*%& /&, &)..&' 8.*$,(#%&' "# '9,%# /&, &)..&' ,&'; L# 9.,/*# %'# #.*9#. , 8.*$. "9#.; C:" '.*# ' 8.&%//*J, % 60 &)..&' 8.*$,(#%&' , '9,%& "9#. /#,(&' &)..&' ,&' ,/'*(#.# 8#.# 8.&%/*. /&, 1760 #.(/"&'
4= E" ,$.& % /#%.,&' ' 8.&%/ , ,# :).*/# ' 8.&8&./*&,#" #" ,$.& % /#%.#%&' #).*/#%& " #[& #,(.*&.; S* , " !3 ' #).*/#.&, 30 000 /#%.,&' , " !4 '
Aritmética
58
Quinto Año P.&%4//*J, O ) . . & ' 8 . *$ , (# % & '
!= C:" % "&' '*9*,(' 9.:*/&' *,%*/# ,# ,/*J, % 8.&8&./*&,#"*%#% %*./(# & *,.'#
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1100
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< III=
50
10= S (*, 2 $#9,*(%' A @ (#"' A ' IP # @ 2; '* /#,%& @ #$,(# 20 i A #.*# , 22 ,*%#%'; E, /#,(& #.# A '* @ %*'$*, , , 25i
7= L# *9.# $'(.# "# 9.:*/# % "&' #"&.' (&$#, 2 $#9,*(%' A @; /#"/"#. A @; A @
H 12
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11= C:" % "&' '*9*,(' 9.:*/&' *,%*/# ,# ,/*J, % 8.&8&./*&,#"*%#% %*./(# & *,.'#
< k =
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H= U,# .%# A % H0 %*,('
,9.#,# /&, &(.# .%# @ % 50 %*,('; F*& #" % @ # &(.# .%# C % 15 %*,(' ,9.#,# /&, ,# .%# D % 40 %*,(' '* A %# 120 "(#' 8&. $*,(&; C:,(#' %#.: "# .%# D
12= S '#) " 8./*& % , %*#$#,( #.# % &.$# DP /&, " /#%.#%& % ' 8'&; S* ,# &# ' %**% , 4 8#.(' *9#"'; A 8&./,(# % ' #"&. *,*/*#" %# .%/*%& " #"&. % %*/# &#
13= C:" ' " 8./*& % , %*#$#,( #" 55 000 '&"' '* ,& % 6 *"#(' /'(# 1! H000 " 8./*& ' 8.&8&./*&,#" #" /#%.#%& % ' Aritmética
59
Quinto Año
'& <(J$' 1 *"#( *9#" # 025 9.';=
14= S#
" &"$, % , 8#.#""88%& ./(#,9"#. % #,/& ^ % "#.9& A #"(.# C "#' /#"' '&, #.*#)"'; C ' *,%8,%*,( %" #"&. % ^> A ' IP %" #"&. % ^;
15= S#
,# ,/*J, % 8.&8&./*&,#"*%#% (#" <3= <7= B 20 ,(&,/' " #"&. %" 8.&%/(& <215= 3 <5= 3<7= '?
16= E" /&'(& % , #.(/"&
W B 6 ,(&,/' CB 12 B 4> W B ! ,(&,/' C B 16> C:,(#' &.#' ' %) (.#)##. 8#.# C B 23> '* B 6
17= S#, A @ 2 $#9,*(%' (#"' A IP @ 2 <'* @ : 24= A DP @ 2 <'* @ 0 24= S* A B 360> /#,%& @ B H C:,(&' '.: " #"&. % A /#,%& @ '# 600
@ C ,9.#,#, C D (*,, , /&$,> '* "# .%# A %# 75 .&"/*&,' 8&. '9,%& ' &)'.# "# .%# D 9*.# 25 .&"/*&,' 8&. '9,%&; D(.$*,#. " ,$.& % %*,(' % "# .%# @ '* '(# (*, 20 %*,(' $,&' "# .%# C;
1!= L# 8&'*)*"*%#% % *,9.'& # "# U,*.'*%#% N#/*&,#" % S#, A9'(, ' %*./(#$,( 8.&8&./*&,#" # "&' #[&' % 8.$#,,/*# , "# #/#%$*# *,.'#$,( 8.&8&./*&,#" #" 9.#%& ,.*&' (*, , ' $&$,(& (#$)*+, *,.'#$,( 8.&8&./*&,#" # "# %#% ' (*,; S* X#, (*, , 9.#%& % ,.*&' /&$& 5 $*,(.#' P#)"& , 9.#%& /&$& 2 X#, (*, ,# 8&'*)*"*%#% %" 50i /&, '' 1H #[&' (*, 2 #[&' , "# #/#%$*#; l 8&'*)*"*%#% (,%.: P#)"& '* (*, 15 #[&' , #[& 8.8#.:,%&'
20= P&.
/(& %" /"*$# "# ($8.#(.# 8.&$%*& , " .#,& %" 2001 '.: $%*# $:' "# %" .#,& %" 200 '* "# 8.&%//*J, #9./&"# ' *,.'#$,( 8.&8&./*&,#" #" /#%.#%& % "# ($8.#(.#; C:" '.: "# 8.&%//*J, %" 2001 '* , " 2000 % 36 (&,"#%#'
1H= S (*, "#' .%#' A> @> C> D %&,% A @ (*,, , /&$,
!RO%*EMA& !ARA *A CA&A Aritmética
60
Quinto Año
1= L# *9.# $'(.# "# 8.&%//*J, % ,# ().# % 1bb % %*:$(.& # &(.# % bb % %*:$(.&; S 8*%? A= P&. /#" % "#' 2 '//*&,' 8#'# $#' #9# @= E , ."#/*J, ' ,/,(.#, "#' "&/*%#%' , "#' '//*&,' ' $'(.#, <1 2=
1
1
2
2
#=P&. 1> ! # 16 )=P&. 2> 16 # ! /= P&. "#' 2 8#'# "& $*'$&> 3 # 4 %=P&. "#' 2 8#'# "& $*'$&> ! # 16 =P&. "#' 2 8#'# "& $*'$&> 5 # 2 2= E" ,*" % 8&"/*J, % #*. , ,# /*.(# /*%#% #.# /&, " /#%.#%& % "# 8&)"#/*J, /&, " ,$.& % :).*/#' /&, "# .# /#%.#%# %" :.# % 8#.' #.%*,'; S* "&' *,/.$,(&' % 8&)"#/*J, ,$.&' % #).*/#' :.#' .%' '&,? 20i> 30i 44i .'8/(*#$,(; E, 8&./,(# ' *,/.$,(# "# 8&)"#/*J, #= 56 i /= 40 i = N;A;
Aritmética
)= 30 i %= 17 i
3= E" ,$.& % &)..&' ..*%&' 8#.# /&,'(.*. , %**/*& %8,% %" &"$, % /&,/.(& *,(.*, , "# &).#> %" /&/*,( .#/*&,#" % %**/"(#% %" ,$.& % %#' " ,$.& % &.#' %*#.*#' % "#)&.; S* " &"$, % /&,/.(& ' .%/ # '' 2 (./.#' 8#.(' "# %**/"(#% ' .%/ # '' 8#.(> " ,$.& % %#' ' (.*8"*/# " ,$.& % &.#' %*#.*#' % "#)&. ' %8"*/#; E, ."#/*J, ' ,/&,(.#.#, "&' ,$.&' % &)..&' ..*%&' , #$)&' /#'&' #= 1 # 15 /= 1 # 30 = N;A;
)= 1 # 20 %= 1 # 5
4= -A ' %*./(#$,( 8.&8&./*&,#" # <@ C= -@ ' 8.&8&./*&,#" # -D -C ' 8.&8&./*&,#" # mDb 2; /#,%& D B 2> A B 4H /#,%& D B S> A B 30; 8#.# #"&.' -D2 -A ' *9#" # /.&; #= 0> 6 /= 5> 6 6> 4
)= 1> 3 %= 7> H
5= S* "# $#9,*(% -A ' %*./(#$,( # "# $#9,*(% -@ "# $#9,*(% -C ' %*./(#$,( 8.&8&./*&,#" # <# )=> '* /#,%& -A #$,(# , 2 ,*%#%'> -@ #$,(# , 5 ,*%#%' -C , 3 ,*%#%'; W#""#. -C /#,%& -A (&$# " #"&. 20 #= 30 /= 40
)= 20 %= 21
61
Quinto Año
= 32 6= L#' $#9,*(%' A> @ C 9#.%#, "#' '*9*,( ."#/*&,'? C&, C B /&,'(#,( A @
1 1
H 05
27
0,3
64 025
3 225
4 4
C&, @ B /&,'(#,( A @
1 025
2 1
S* /#,%&? A B 4 @B !> CB 16; W#""#. A /#,%& @ B 3 C B 4 #= 27 /= !2 = 36
)= H1 %= 54
7= L# $#9,*(% A ' IP # @ 2; "#' #.*#/*&,' % A @ '(:, %#%&' , "# '*9*,( (#)"# % #"&.'? A @
3# 6
144 )
/ 2
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W#""#.? -# ) / #= 15 /= 33! = 340
)= 12 %= 335
H= L# .# % ''(,(#/*J, '&). " #"# % , #*J, ' DP # ' :.# #" /#%.#%& % ' "&/*%#%; S* ' #$,(# " :.# , , 20i ' .%/ "# "&/*%#% , , 10 i + '/% /&, "# .# % ' (,(#/*J,
)= A$,(# 2Hi /= D*'$*, 2H i %= A$,(# 37 i = N;A; != E" 8./*& % , ("*'&. # /&"&. #.# , &.$# %*./(#$,( 8.&8&./*&,#" #" /#%.#%& % ' (#$#[& *,.'#$,( 8.&8&./*&,#" # "# .# /#%.#%# % "# ,.9# /&,'$; S* /#,%& ' (#$#[& ' % 14 8"9#%#' /&,'$ -E % ,.9# /&,'$; S* /#,%& ' (#$#[& ' % 14 8"9#%#' /&,'$ -E % ,.9# ' 8./*& ' % S; 360; C:,(& /&'(#.# , ("*'&. /& (#$#[& ' % 21 8"9#%#' '* /&,'$ E4 % ,.9# #= S; 520 /= S; 640 = S; 3240
)= S; 720 %= S; 1620
10= S '#) ,# $#9,*(% A ' IP # @2; #""#. " #"&. % A '#)*,%& '* %*'$*,, , 36 ,*%#%' " #"&. % -@ #.*# , , 25i #= 1H0 )= 10H /= 200 %= 360 = 100 11= D&' ,9.#,#' % 24 3H %*,(' '(:, /&,/#(,#%&' , " (.#,'/.'& % 4 $*,(&' ,& %# 70 "(#' $:' " &(.&; W#""#. "# "&/*%#% $,&. , .&"/*&,' 8&. $*,(&;
#= D*'$*, 41i
Aritmética
62
Quinto Año
#= 25 /= 35 = 45
)= 30 %= 40
12= E, ,# $8.'# " '"%& ' DP # "# %#% # "&' #[&' % '.*/*& %" $8"#%& IP #" /#%.#%& % "# /#(9&.#; X#, $8"#%& % '9,%# /#(9&.# /&, 10 #[&' % '.*/*& % "# $8.'# 56 #[&' % %#% 9#,# S; 2000; X&'+ ,(.& # "# $8.'# 3 #[&' %'8+' X#, 9#,# S; 500 ' $8"#%& % 3d /#(9&.#; W#""#. "# %*.,/*# % "#' %#%'; #= 11 /= H = 6
)= ! %= 7
13= S* " 8./*& % , %*#$#,( ' %*./(#$,( 8.&8&./*&,#" #" /#%.#%& % ' 8'&; C:,(& ' 9#,#.# & 8.%.: , , %*#$#,( #" S; 720 ' 8#.( , 2 8%#&' ,& " %&)" %" &(.&
#= )= /= %= =
N& ' 9#,# ,* ' 8*.% S 9#,# S; 120 '&"' S 9#,# S; 2H0 S 8*.% S; 240 S 8*.% S; 320
14= L# /#,(*%#% % %$#,%# % , /*.(& )*, ' %*./(#$,( 8.&8&./*&,#" #" /)& % "# *,.'*J, , 8)"*/*%#% *,.'#$,( #" /#%.#%& %" 8'& ,*(#.*&; S* " #[& 8#'#%& ' ,%*.&, 64 $*""&,' % #.(/"&' # S; 2 *,*.(*,%& , 8)"*/*%#% ,&' S; 4000; C:,(& # *,.(*. '( #[& , 8)"*/*%#% '* ' *. ,%. 12H $*""&,' % #.(/"&' # S; 4 /#%# ,& #= S; 5000 /= S; 6000 = S; 10000
15= -A ' "# '$# % %&' /#,(*%#%' % "#' /#"' ,# #.*# /&, -@ '* -A ' 1! /#,%& -@ ' 2 J 3; #""#. -A /#,%& -@ ' 6 #= 15 /= 35 = 40
Aritmética
)= S; 4000 %= S; H000
)= 31 %= 36
63
Quinto Año
TEMA: RE!ARTO !RO!ORCIONA* Concepto ; E' "# .9"# , "# /#" # ,# /#,(*%#% ' .8#.( , &(.#' '&, %*./(# & *,.'#$,( 8.&8&./*&,#"' /&, &(.#' ""#$#%#' -,%*/' & #/(&.' % 8.&8&./*&,#"' % $&%& (#" (&%#' &.$, ,# -'.* % .#&,' *#",(' Notación; R8#.(*. -N , 8#.(' 8.&8&./*&,#"' # -> -.> -'> K> - N B # ) / K < = # <&= > ) <&= .> / <&= '> K> < &=
a %
b *
c s
#
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D&,%? N B C#,(*%#% .8#.(*. #> )> /> K B 8#.(' % N *,(.*,, , " .8#.(& > .> '> K > B ,%*/' & #/(&.' % 8.&8&./*&,#"*%#%; b B /&,'(#,( % 8.&8&./*&,#"*%#%;
Clases (el Reparto !roporcional#7 Reparto !roporcional &imple R!
I= 1.
R! 5irecto o Reparto 5irectamente !roporcional#7 C#,%& /#%# 8#.( %" ,$.& ' .8#.( ' %*./(#$,( 8.&8&./*&,#" .'8/(*#$,( /&, /#%# ,%*/ % 8.&8&./*&,#"*%#%; Re'la pr0ctica: <= S '$#, "&' ,%*/' % 8.&8&./*&,#"*%#% <= S #""# "# /&,'(#,( -b b B ,$.& # .8#.(*. '$# % ,%*/'; <= S #""#, "#' 8#.(' 8.&8&./*&,#"' $"(*8"*/#,%& "# /&,'(#,( -b 8&. "&' ,%*/' 8.&8&./*&,#"'; E$? R8#.(*. 750 , ,$.&' <3= %8 # 6> 7> 12
Aritmética
64
Quinto Año
(A) 6 7 12 25
4 (AA) k 750 : 25 k 30 4e $a*t+ de 750 180; 210; 360 (A A A) 30 (6) 180; 30 (70) 210; 30 (12) 360 4 R#!# In3erso o Reparto In3ersamente !roporcional#7 /#,%& /#%# 8#.( %" ,$.& & /#,(*%#% # .8#.(*. ' *,.'#$,( 8.&8&./*&,#" /&, /#%# -,%*/ % *,.'*J, <,%*/ % "#' $#9,*(%' *,.'#$,( 8.&8&./*&,#"'=;
2.
Re'la !r0ctica: S *,*.(, "&' ,%*/' % *,.'*J, ' 8.&/% /&$& " .8#.(& P.&8&./*&,#" S*$8" D*./(&; E$? R8#.(*. 5!4 , 8#.(' IP # 2> 3> 6> 10
"#' 8#.(' '&, DP # V> 13> 16> 110 DP # 3> 5 > 10> 15 . : B 33 <= 5!4 33 B 1H <= 1H <10= B 1H0> 1H<3= B 54> 1H<5= B !0> 1H<15= B 270 E" .8#.(& <5!4= B 54> !0> 1H0> 270 <=
II Reparto !roporcional Compuesto R!C
E' " (*8& % .8#.(& /&$)*,# "#' 2 &.$#' %" .8#.(& 8.&8&./*&,#" '*$8" 8&. "& (#,(& '' 8#'&' % .'&"/*J, '&, "&' $*'$&' '9, /#%# /#'&; E$? R8#.(*. 12H 8#.(' DP # 1 4 IP # 5 3 <= <= <=
12H
4> 15> 13 3015> 6015> 7515> 4515> "9& 30 60 75 45 B 210 128 64 k > 210 105 128
128 7
256 7
320 7
192 7
; e $a*t+ de 128
!RO%*EMA& !ARA *A C*A&E
Aritmética
128 256 3 ; ; 7 7
65
Quinto Año
1= T.' &)..&' A> @> C (.#)##, , /*.(# &).#> " 8.&8*(#.*& % "# &).# &(&.9# *,/,#"$,( ,# 9.#(**/#/*J, % n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
Aritmética
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
66
Quinto Año
,/*J, % "&' #[&' % '.*/*& # '' .,%*$*,(&' # '' $*,(&' % (#.%#,# % #/.%& #" /#%.#%& '*9*,(? AÑOS
REND;
TARDANZA
15 12 10
H0i !0i 70i
40 $*, 30 $*, 35 $*,
A? @? C?
C:,(& ./*)*J @ != U, &$). $. %#,%& # ' '8&'# $)#.##%# , ('(#$,(& % 130 000 ,&' '&"' ' .8#.(*.: % "# '*9*,( &.$#? 25 # "# $#%. 35 # "# /.*#(.# '* ,#/ ,*[&> 47 # "# $#%. 37 # "# /.*#(.# '* ,#/ ,*[#K 8.& '/% "# '[&.# %*& # " # , ,*[& # ,# ,*[# ,(&,/' "& "' (&/# # "# ,*[# #" ,*[& , ' &.%, '? 10= S .8#.( " ,$.& 145 H00 , 8#.(' 8.&8&./*&,#"' # (&%&' "&' ,$.&' 8#.' %'% 10 #'(# !H; C:,(& " (&/# # "# ' 8.&8&./*&,#" # 72 11= U, &$). %/*% .8#.(*. ,# .,/*# , &.$# 8.&8&./*&,#" #" &.%, , ,#/*.&, '' *&'; L# .,/*# (&(#" ' S; 4H0 000 #%*/*&,#"$,( %# 160 000 '&"' 8#.# " $#&.> % (#" $&%& " 8.*$.& " "(*$& ./*), *9#" .,/*#; C:" ' " $#&. ,$.& % *&' (*, " 8.'&,#
Aritmética
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
67
Quinto Año
/#,%& ' .8#.(*.&, "# (*"*%#% # '( "(*$& " (&/& "# $*(#% 17= P#(.*/*# R)/# %/*%, .8#.(*.' S; 240 ./*)*.&, /&$& 8.$*& 8&. '' '(%*&' 8.& "& #, # #/. *,.'#$,( 8.&8&./*&,#" # '' %#' % #"(# '&, 5 7 %#' .'8/(*#$,(; C#"/"#. / % "#' 8#.(';
1H= U, /#$8'*,& (*, 275 #' ,(. 9#""&' 9#""*,#' 8#"&$#'; E" ,$.& % 9#""*,#' ' #" % 9#""&' /&$& 7 ' # 3 " ,$.& % 8#"&$#' #" % 9#""*,#' /&$& 5 ' #" 2; C:,(#' #' % /#%# '8/* (*, 1!= E, , /&"9*& # 130 #"$,&' % "&' /#"' # /#%.8" ,$.& % #$.*/#,&' % '8#[&"' %&)" ,$.& % /)#,&' % #$.*/#,&'; C:,(&' #"$,&' % /#%# ,#/*&,#"*%#% # 20= D**%*. 74 ,(. A> @> C D> (#" "# 8#.( % A '# # "# % @ /&$& 3 ' # 4> "# % @ '# # "# % C /&$& 1 ' # 3 "# % C ' # "# D /&$& 2 ' # 3; ,&$).#. ,& % "&' .8#.(&';
Aritmética
68
Quinto Año
!RO%*EMA& !ARA *A CA&A 1= S .8#.(, 24 /,(#&' , 8#.(' 8.&8&./*&,#"' # "#' %#%' % 3 ,*[&' % 2> 4> 6 #[&'> .'8/(*#$,(; C:,(& (&/# # /#%# ,& #= 2> 4> H /= 40> 1H> 30 = 4> H> 12
)= 12> 16> 20 %= 3> 4> 5
2= D&' &)..&' #'(#, ,# &).# 8&. S; 110; " &.,#" %" 1d ' % S;3 " '9,%& S; 250; C:,(& 8./*)*.: /#%# ,& % "# /#,(*%#% (&(#" #= H0> 65 /= 100> 75 = 70> 60
)= 30> 40 %= 60> 50
3= T.' .$#,&' #%*. ,# 8.&8*%#% , S; H5 000 #"9, (*$8& %'8+' "# ,% , S; 100 000; '* "#' 8#.(' *$8'*.#, '&, 8.&8&./*&,#"' # "&' ,$.& 3> 4> H; C:,(& 9#,& /#%# ,& #=S; 1000> S; 2000> S; 3000 )=S; 7000> S; H000> S; !000 /= S; 3000> S; 4000> S; H000 %=S; 4000> S; 6000> S; 10000 =S; 10000> S; 12000> S; 14000 4= R8#.(*. 1100 , ,$.& *,.'#$,( 8.&8&./*&,#"' #? S; 1010> 1011> 1012; "# $#&. 8#.( '? #= 10 /= 100 Aritmética
)= 1000 %= 10 000
= 100 000 5= A" %**%*. 36 8#.(' '#, *,.'#$,( 8.&8&./*&,#"' # "&' ,$.&' 6> 3 4 <, ' &.%,=> &)(,*+,%&' 3 ,$.&' #> ) /> ,(&,/' # 3 ) 3 / '? #= 1356 /= 1563 = N;A;
)= 1536 %= 1635
6= R8#.(*. 4H0 , 3 8#.(' DP # 3> 4> 5 IP # 6> 12 1H; #""#. "# $,&. % "#' 3 8#.('; #= 216 /= 126 = 210
)= 144 %= 120
7= S .8#.( N , &.$# IP # 3 ,$.&' '*,%& " 8.*$.& % ""&' " (.*8" %" (./.&; S*,%& "# '9,%# 8#.( "# $*(#% % "#' &(.#' 2 ,(#'; W#""#. " $,&. (.& % ,$.&' /&, "&' /#"' ' 8%& /(#. " .8#.(&; #= 6> 4> 1 /= 3> 2> 1 = 6> 5> 2
)= 6> 3> 2 %= 4> 4> 3
H= S .8#.( ,# /#,(*%#% % %*,.& ,(. 3 8.'&,#' , &.$# DP # '' %#%' '&, 3 ,$.&' ,(.&'; L# $*'$# /#,(*%#% % %*,.& 8.& , &.$# IP 8.&8&./*&,#" # %*/#' 3 %#%'; L9& "# '9,%# 8.'&,# ./*) , " '9,%& .8#.(&; #=M#' , " 8.*$. .8#.(& )=M,&' , " 8.*$. .8#.(& 69
Quinto Año
/= I9#" , " 8.*$. .8#.(& %=N& ' 8% '#). =D8,% % "# /#,(*%#% .8#.(*%# != A" .8#.(*. 22050 DP # "#' .#/' /#%.#%#' % "&' ,$.&' 72> !H 12H ' &)(*, "# $#&. 8#.( /% # "# $,&. 8#.( ,? #= 1250 /= 2000 = 2400
)= 1H00 %= 2100
10= S # .8#.(*%& ,# /#,(*%#% , 3 8#.' DP # 3 ,$.&' '*,%& "# 8.*$.# 8#.( 13200> "# '9,%# 33000 "# (./.# 52H000; '* "# %**'*J, ' #/ , &.$# IP # '&' 3 ,$.&'; C:" '.*# "# '9,%# 8#.( #= 60 000 /= 15 000 = 35 000
)= 24 000 %= 33 000
11= A" %'/&$8&,. " ,$.& 1134 , 4 '$#,%&' /&' /#%.#%&' '#, 8.&8&./*&,#"' # 12> 27> 4H 75; W#""#. " $,&.; #= 100 /= 162 = 173
)= 160 %= 152
12= R8#.(*. S; 1H14 , 4 8#.(' 8.&8&./*&,#"' # H11> 32> 56> 3H; D#. /&$& .'8'(# "# '$# % "# 8#.( $#&. $,&. #= S; 4!5 /= S; !H0 = S; 1010
)= S; H45 %= S; !!0
13= L# '[&.# C#.$, %J ,# .,/*# % S; 14400 8#.# .8#.(*."# 8.&8&./*&,#"$,( # "#' %#%' % '' *&'? M*9" C#."&' X#*$; S* "# %#% % X#*$ ' " %&)" % "# M*9" C#."&' &)(& S; 4200 #%$:' "# '$# % "#' %#%' ' % 72#[&'; C:" ' "# %#% X#*$ #= 21 #[&' /= 32 = 44
)= 11 %= 33
14= S .8#.(, S; 2210 , 4 8#.(' (#"' "# '9,%# ' # "# (./.# /&$& 7 # 11 "# 3d ' # "# /#.(# /&$& 4 ' # -$ "# 8.*$.# ' # "# '9,%# /&$& 3 ' # 5; '* # "# /#.(# " (&/& S; 1100; C:" ' " #"&. % -$ #= H /= 12 = 44
)= 11 %= 33
15= T.' /*,&' *., 8*,(#. "#' #/#%#' % '' /#'#' '*,%& " /&'(& (&(#" 136! '&"'; L# (,'*J, % "# #/#%# %" 8.*$.& ' "&' 35 % "# %" (./.& "# %" 2d ' "&' 72 % "# %" 1d; S* " 9#'(& ' .8#.( , &.$# DP # "# (,'*J, % "#' #/#%#'; C:,(& " (&/& #)&,#. #" 3d #= S; 222 /= S; 370 = S; 777
)= S; 22H %= S; 725
TEMA: REG*A 5E COM!AA Aritmética
70
Quinto Año
Concepto ; E' , /#'& 8#.(*/"#. %" .8#.(& 8.&8&./*&,#" %&,% ' .8#.(, "#' 9#,#,/*#' & 8+.%*%#' % "#' (.#,'#//*&,' '9, " /#8*(#" *,.(*%& 8&. /#%# '&/*& , , 8.*&%& *& % (*$8&> %,(.& % ,# '&/*%#% $./#,(*"; E, "# R9"# % C&$8#[# ' /&,'*%.# #" /#8*(#" #" (*$8& /&$& %*./(#$,( 8.&8&./*&,#"' # "# 9#,#,/*# & # "# 8+.%*%# % ,# (.#,'#//*J, /&$./*#";
5 ! 5 & 5 & H H
D&,%? <5= ? G#,#,/*# & 8+.%*%#
Clases#7 1) Re'la (e CompaH6a &imple< /#,%& *'( , /#8*(#" ,*/&' 8#.# /#%# '&/*& 8.',(#. 2 /#'&'?
i#
Capital Constante: L# #.*#/*J, % "# 9#,#,/*# & 8+.%*%# ' DP #" (*$8&; ii# Tiempo constante: L# #.*#/*J, % "# 9#,#,/*# & 8.%*%# ' DP #" /#8*(#" # %./& <% /#%# '&/*&= 2) Re'la (e CompaH6a Compuesta< /#,%& *'(, %*'(*,(&' /#8*(#"' , %*'(*,(&' (*$8&' 8.',(# 2 /#'&'? i# Capital Constante en tiempo 3aria$le: "# 9#,#,/*# & 8+.%*%# ' DP #" /#8*(#" $"(*8"*/:,%&' /&, " (*$8& % /#%# '&/*&; ii# Capital Daria$le: G#,#,/*# & 8+.%*%# ' %8 #" 8.&%/(& %" /#8*(#" ,*/& 8&. " (*$8& (&(#";
O%&: 1 Capital Único: ' # '$# % (&%&' "&' /#8*(#"' <8.'#%&' , ,# $*'$# ,*%#% % (*$8&=; ) *a Ganancia neta Gn: ' "# 9#,#,/*# ),*/*& & (*"*%#% R#" %'8+' % "# *,.'*J, %" /#8*(#" *,%*/# "# /#,(*%#% ./8.#%# .'8/(& #" /#8*(#" *,*/*#"; G#,#,/*# ,(# B
Aritmética
anancia "nica@
a$ita@ "ne*tid+
71
Quinto Año
!RO%*EMA& !ARA *A C*A&E 1= X#, P%.& 9#,#.&, , 1!66 1!67 1200 '&"' /#%# #[& , , ,9&/*& (*,,; E, 1!66 X#, .# %[& % "&' %" ,9&/*& ' '&/*& %" .'(& , 1!67 X#, G#./# %[& % "&' 25 %" ,9&/*& ' '&/*& %" .'(& 8&. " 8.*$.& ,%*J #" '9,%& ,# 8#.(; W#""#. "# 9#,#,/*# (&(#" % /#%# '&/*& , "&' 2 #[&'; R8(#;? 2= A> @> C $8.,%, , ,9&/*& *$8&,*,%& A B S; !00> @ B S; H00 C B S; 750; #" /#)& % , #[& A ./*) /&$& 9#,#,/*# S; 1H0; C:,(& # 9#,#%& @ C R8(#;? 3= T.' '&/*&' #)#, *,(.'#%& S; 25000 " 8.*$.&> S; 24000 " '9,%& S; 16000 " (./.& (*,, .8#.(*.' ,# 8.%*%# % S; 1!500; C:,(&' %#, # /#%# ,& R8(#;? 4= C#(.& '&/*&' #, 9#,#%& , "&' 3 #[&' 8"&(#.&, ,# *,%'(.*# "& '*9*,(? " 8.*$.& S; 5000> " '9,%& "&' 25 % "& 9#,& " 8.*$.&> " (./.& "&' % "& 9#,& " '9,%& " /#.(& "&' 5H % "& 9#,& " (./.&; S* " /#8*(#" '&/*#" .# % S; 44000> C&, /#,(& /&,(.*)& /#%# ,& R8(#;? 5= T.' /&$./*#,(' .,*.&, S; !0000 8#.# "# 8"&(#/*J, % , ,9&/*& 9#,#.&,? " 1d 1000> " 2d 600 H00 " 3d; C:,(& *$8'& /#%# ,& R8(#;? 6= E, ,# *,%'(.*# (.#)#& %.#,( 4 #[&' $%*& /#(.& '&/*&' *$8'*.&,? " 8.*$.& S; 500 $#' " '9,%& " Aritmética
'9,%& S; 600 $,&' " (./.&> " (./.& "# $*(#% % "& 8'& " /#.(& '( *$8"'& S; 3000; '* # #.&,(#. ,# 8.%*%# % S; 3400; C:,(& 8.%.: /#%# ,& R8(#;? 7=
T.' #$*9&' ' #'&/*#, 8#.# $8.,%. , ,9&/*& *$8&,,? S; 2500> " '9,%& "# $*(#% % "& 8'& " 8.*$.& $#' 600> " (./.& 400 $,&' "&' #,(.*&.' ,(&'; A" /#)& % 3 #[&' ' .8#.( , ),*/*& % 16600; C:,(& (&/# # /#%# ,& R8(#;?
H= A $8.,% , ,9&/*& /&, S; 3000 # "&' 3 $'' $#' (#.% ,(.# % '&/*& C /&, S; 3000; '* # , ),*/*& % S; 2700 #" /#)& %" #[& % $8.,%. A " ,9&/*&; C:,(& ./*) /#%# ,& R8(#;? !=
A $8.,% , ,9&/*& % S; 2000; A" /#)& % 6 $'' ,(.# /&$& '&/*& @ /&, S; 2000 11 $'' $#' (#.% ,(.# /&$& '&/*& C /&, S; 2000; '* # "&' 2 #[&' % /&$,#. A ' ,9&/*& # , ),*/*& % S; 630; C:,(& ./*) /&$& 9#,#,/*# /#%# ,& R8(#;?
10= A> @> C *$8'*.&, S; 300 /#%# ,& 8#.# "# 8"&(#/*J, % , ,9&/*&; A 8.$#,/*J , " $*'$& , #[& @ /#(.& $'' $,&' A C> 4 $'' $,&' @; S* # ,# 8+.%*%# #'/*,% #" 20i %" /#8*(#" '&/*#"; C:,(& 8*.% /#%# '&/*& R8(#;? 11= R,*,%& , /#8*(#" % 10 000 '&"' 8&. 8#.(' *9#"' (.' '&/*&' $8.,%, , ,9&/*& 8&. 2 #[&';
72
Quinto Año
12=
13=
14=
15=
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d 8&. 4 " 3d 8&. 5 #[&'; C:,(& (&/# # /#%# ,& % , ),*/*& % S; 44H R8(#;? T.' *,%**%&' .,, 25 000 )&"#.' % "&' /#"' " 8.*$.& # *$8'(& H000> " 2d> 3000 $#' " 8.*$.& " 3d "& .'(#,(; E" 8.*$.& # 8.$#,/*%& , " ,9&/*& 8&. H $'' " (./.& 8&. 5 $''; S* # #.&,(#. ,# 8.%*%# % 1143; C:,(& %) 8.%. /#%# ,& R8(#;? E, ,# *,%'(.*# 3 '&/*&' #, *$8'(&? " 1d /&, 6000 '&"' $#' " '9,%&> " '9,%& /&, 3000 $#' " (./.& '( H000; E" 8.*$.& 8.$#,/*J , "#
Aritmética
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d 8&. 2 V #[&' " 4d 8&. 2 #[&'; S* # .8#.(*. ,# 9#,#,/*# % 4350; C:,(& /&..'8&,% # /#%# ,& R8(#;? 1!= D&' *,%**%&' $8.,%, , ,9&/*& 8&. 1 #[&; E" 8.*$.& $8*# /&, S; 500 7 $'' %'8+' #[#% S; 200> " '9,%& $8*# /&, S; 600 3 $'' %'8+' #[#% S; 300> C:,(& /&..'8&,% # /#%# ,& % , ),*/*& % S; 33H R8(#;? 20= E, , ,9&/*& # %.#%& 3 #[&' , '&/*& *$8'& 4000 )&"#.' > # "&' H $'' .(*.& "# $*(#%> " '9,%& *$8'& 6000 #" #[& #[#%*J 3000> " (./.& $8J /&, 6000> # "&' 2 #[&' .(*.& 1500; C:,%& /&..'8&,% # /#%# ,& , ),*/*& % 5740 R8(#;?
73
Quinto Año
!RO%*EMA& !ARA *A CA&A 1) D&' .$#,&' &.$#, , ,9&/*&
#8&.(#,%& /#%# ,& , $*'$& /#8*(#" A , $' % *,*/*#%& " ,9&/*& " 8.*$.& #$,(# , '' 23 % /#8*(#"> 4 $'' $:' (#.% " '9,%& .%/ # '' 23 % ' /#8*(#"; S* " ,9&/*& %.& 6 $'' #" *,#" ' &)(& ,# 9#,#,/*# `##`> C:" ' "# %*.,/*# % "#' 9#,#,/*#' '* '(#' '&, /#,(*%#%' ,(.#' #= 2661 /= 1221 = 3113
)= 1331 %= 2112
2= E, "# *$8.,(# ^*""o' ' &)'.# " '*9*,( #*'&? ( de tarjetas impresas
Medida
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)= 40H %= 413
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Aritmética
$'' %'8+' F*$&.* (#$)*+, ' .(*.& %" ,9&/*& 4 $'' %'8+' T&"%& "**%#.# ' ,9&/*& .8#.(*,%& "#' (*"*%#%'; S* A,%.#% )*' 8.$#,/*%& , $' , " ,9&/*& #).# ./*)*%& S; 64 $:'; C:" "# (*"*%#% (&(#" &)(,*%# , " ,9&/*& #= 2536 /= 21H2 = 212H
)= 2H12 %= 221H
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74
Quinto Año
#= 20 /= 1H = 6
)= 10 %= !
6= T.' *,%**%&' ' #'&/*#, , , ,9&/*& %.# 2 #[&'; E" 8.*$.& *$8&, S; 2000 #" /#)& % H $'' S; 1500 $:'; E" '9,%& *$8&, #" 8.*,/*8*& S; 5000 %'8+' % , #[& '#/# "# $*(#%; E" (./.& #)# *$8'(& #" 8.*,/*8*& S; 2500 '#/# # "&' 5 $'' S; 1000 2 $'' $#' (#.% #9.9# S; 500; '* # ,# 8.%*%# % S; 500; C:,(& /&..'8&,% 8.%. # /#%# ,& #= 170 1112> 212 13> 117 33H )= 170 7!> 212 V> 117 1517 /= 170 25> 212 3435> 117 \ %= 170 16> 212 3435> 117 23 = 170 1047> 212 3647> 117 147
8.*$.& 8.$#,/*J , "# $*'$# %.#,( , #[&> " '9,%& %.#,( 7 $'' $:' " 8.*$.& " (./.& %.#,( H $'' $:' " '9,%&; E" 8.*$.& #)# *$8'(& S; H00 " '9,%& 400 $,&' " '9,%&; S* # ,# 8.%*%# % 224 '&"'; C:,(& /&..'8&,% 8.%. # /#%# ,& .'8/(*#$,( #= 16> 30> 4H /= 27> 3!> 51 = 30> H7> !6
)= 12> 15> 23 %= 4H> H5> H1
!= T.' *,%**%&' ' #'&/*#, 8#.# *,*/*#. ,# $8.'#; E" 8.*$.& *$8&, S; 2000 %.#,( 3 #[&'> " 2d S; 1H00 %.#,( 4 #[&' " 3d S; 3300 8&. H $''; C:,(& /&..'8&,% # /#%# ,& '* # , ),*/*& % 2500 '&"' D#. "# #8.&*$#/*J, % "# 8#.( ,(.#; #= 7!!> 276> 402 )= 612> 400> 10
7= C*,/& '&/*&' #, *$8'(&? " 8.*$.& S; 2000 8&. 2 #[&' 4 $''> " '9,%& S; 2500 8&. "&' 37 %" (*$8& #,(.*&. " (./.& S; 3000 8&. &' 56 %" (*$8& %" '9,%&> " /#.(& S; 4000 8&. , #[& H $'' " *,(& S; 500 $,&' " /#.(& 8&. % #[&; W#)*,%& S; !100 '&"' % (*"*%#%; C:,(& 9#,# /#%# ,& D#. /&$& .'8'(# "# '$# % "# '$# % "#' /*.#' % /#%# #"&.; #= 25 )= 26 /= 27 %= 2H = 2! 8) D "&' (.' *,%**%&' /&,(.*).&, ,# '&/*%#% "
Aritmética
/= !00> 1200> 300 %= !H6> 11H4> 32H = 57H> 1207> 610 10= A $8.,% , ,9&/*& /&, /#8*(#" % S; 2000 # "&' 4 $'' (&$# /&$& '&/*& # @ #8&.(# S; 2000 3 $'' $#' (#.% #%$*(, /&$& '&/*& # C #8&.(# &(.&' S; 2000; C#,%& ' /$8" , #[& # /&,(#. %" %# , A $8.,%*J " ,9&/*& # ,# (*"*%#% % S; 1250; C:,(& ./*) /#%# '&/*& <.'8/(*#$,(= #= 600> 400> 250 )= 300> 120 /= 460> 500> 300 75
Quinto Año
%= 700> 600> 500 = 250> 120> 212 11= T.' *,%**%&' $8.,%, , ,9&/*& *$8&,*,%& A B S; !00> @ B S; H00 C B S; 750 #" /#)& % ,# #[& A ./*) /&$& 9#,#,/*# S; 1H0; C:,(& #, 9#,#%& @ C #= 120> 130 /= 140> 150 = 160> 150
)= 130> 140 %= 170> 1H0
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76
Quinto Año
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77
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78
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79
Quinto Año
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80
Quinto Año
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
.#/*&,' %*#.*#'> #" /#)& 20 %#' ./*) 40 '&"%#%&' /&, .' 8#.# 36 %#' # .#J, % 4 .#/*&,' %*#.*#'> '* ' ,(#, "&' .' /&,'$, # .#J, % 2 .#/*&,' %*#.*#'; C#"/"#. 8#.# /#,(&' %#' #"/#,# "&' .'; 15= E, ,# 9.#,# " 20i %" (&(#" % #' '&, 8#(&' " 45i 9#""*,#' " 35i 8#&'; S* " ,$.& % 8#(&' .# " (.*8"; + 8&./,(# %" (&(#" '.*#, "&' 8#&' 16= S* ' ,% , #.(/"& /&, , %'/,(& %" 12i ' &)(,%.: ,# 9#,#,/*# % S; H0 8.& '* ' ,% %*/& #.(/"& /&, , %'/,(& %" 30i ' )*.# 8.%*%& " 125i %" /&'(&; W#""#. " 8./*& % "*'(# <" /&'(&= %" #.(/"&; 17= A" *#. " 8./*& , , #.(/"& ' #$,(& ' /&'(& , ,i #" $&$,(& % ,%."& ' *& , %'/,(& %" 10i; S* "# 9#,#,/*# &)(,*%# .8.',(# " 17 8&. 117 %" 8./*& % ,(# ,/&,(.#. " #"&. % -N; 1H= U, ."& ' #(.#'# 10 $*,(&' /#%# %#; D,(.& % /#,(&' %#' &".: # $#./#. "# &.# #/(# 1!= L# .#8*% % A ' *9#" # 3 /' "# .#8*% % @ '( ' # ' 4 /' "# .#8*% % C; '* A #/ ! $*,(&' 15 '9,%&' % (.#)#&; E, (*$8& C "& #.# 20= U, *,%**%& ./&.. 4H $ , 7 V &.#' %#,%& 54 000 8#'&'; S* '' 8#'&' '&, (&%&' %% "# $*'$# "&,9*(%; C:,(&' 8#'&' %#.: , 12 .' 8#.# ./&... 60 $
!RO%*EMA& !ARA *A CA&A Aritmética
81
Quinto Año
1= S (*, 50 /#%.,&' % "&' /#"' 15 '&, .##%&' " .'(& /#%.*/"#%&'; C:,(&' /#%.,&' .##%&' ' %), #[#%*. 8#.# 8&. /#%# 40 /#%.,&' .##%&' ##, 5 /#%.,&' /#%.*/"#%&' #= 2H0 %= 275
)= 265 = 2!5
"",#%# 8&. 2 /#[&' , 3 &.#' 30 $*,(&' #= 14HH L %= 1740 L
6=
/= 256
2= U, #(& # 60 $ /). "# %*'(#,/*# % L*$# # T$)' , 16 &.#' A "&/*%#% %) ./&... 8#.# /).*. %*/# %*'(#,/*# , "# $*(#% % (*$8&
3=
P#.# ./&... , (.#/(& , /.'*&,*'(# /#$*,# 425 $ # $8"#%& 6 &.#'; C:,(& (*$8& #).# $8"#%& '* )*.# #,%#%& H50 $(.&' $:' 8&. &.# #= 5 &.#' )= 4 %= H = N;A;
/= 3
4= U, &)..& (#.%# , #/. , /)& /&$8#/(& % /&,/.(& % 30 /$; D #.*'(# ,&' 50 $*,(&'; + (*$8& (#.%#.: , #/. ! /)&' /#%# ,& /&, 50 /$; % #.*'(# #= 34 %= 35
)= 34 1H13 /= 34 131H = N;A;
5= U, /#[& 8% "",#. , /*"*,%.& % #9# % 120 "*(.&' , 30 $*,;> $*,(.#' &(.& "",# " $*'$& /*"*,%.& , 5 $*,(&' $,&'; + /#8#/*%#% (,%.: ,# (*,# '
Aritmética
/= 1H00 L
U, )..& #(#%& # ,# /.%# % 10 $; % "&,9*(% 8% /&$. "# *.)# '(# # ' #"/#,/ , 2 %#'; L# *.)# (*, ,# 8#.# ,*&.$ %&.#.# (&%& "& '(*' # ' #"/#,/ ,K '* "# "&,9*(% % "# /.%# .# % 25 $(.&' "# #"(.# % "# *.)# % 06 $; #= 14 %#' %= 17 %#'
#= 30 $ )= 3H $ /= 60 $ %= 120 $ = N;A;
)= 1500 L = 1H4H L
)= 15 %#' /= 16 %#' = 1H %#'
7= S (*, 200 $&,%#' % "#' /#"' 60 '&, % , ,& '&" "&' .'(#,(' % 5 '&"'; C:,(#' $&,#' % 5 '&"' ' 8%, #[#%*. 8#.# 8&. /#%# 20 $&,%#' % 5 '&"' ## ,# % 1 '&" <1 % , '&"= #= 260 %= 200 H=
)= 250 = 220
/= 240
U,# ).*9#%# % 30 &)..&' ' /&$8.&$(, #/. 30 $; % ,# #,# , 30 %#'> # "&' 5 %#' % $8#%& " (.#)#& ' #$,(# 5 &)..&' 10 %#' %'8+' ' #$,(# 5 &)..&' $#'; C:" ' " (*$8& $8"#%& , #/. "# &).#
#= 10 )= 15 /= 20 %= 25 = 30 != S '#) 10 &)..&' 8.&%/, H0 /&$8#' , 5 %#' ' *. 8.&%/*. /&, %*/&' &)..&' H00
82
Quinto Año
/&$8#' 8.& ' "' %# , 8"#& % 20 %#' 8#.# "# ,(.9# 8&. "& ' %/*% /&,(.#(#. ,# /#,(*%#% #%*/*&,#" % &)..&' (.#)##.#, %'% " ,&,& %# #'(# 2 %#' #,(' %" %# % ,(.9#; C#"/"#. " ,$.& % &)..&' ' /&,(.#(#.&,; #= 10 %= 40
)= 20 = 50
/= 30
10= D&' #")#[*"' % "&' /#"' " '9,%& /&$*,# # (.#)##. , %# %'8+' " 8.*$.& (.$*,#, % /&,'(.*. ,# 8#.% (.#)##,%& ,(&' 7 %#' %'8+' " 8.*$.& *,*/*& " (.#)#&; P#.# #/. '( (.#)#& /#%# ,& '&"& " 8.*$. #")#[*" .*. 2 %#' $:' " '9,%&; + (*$8& %$&.# /#%# #")#[*" , /&,'(.*. ,# 8#.% #= H 10 )= H 11 /= ! 12 %= 10 12 = 12 14 11= P#.# #/. $*" (*#' ' ,/'*(#, 25 9 % $#(.*# 8.*$# 8.%*+,%&' Hi , "# #).*/#/*J,> , ,# (*# ' %'8.%*/*# " 20i #" (*"*#.'; S* .,*$&' "&' %'8.%*/*&' % 1000 (*#' "#' $8"#$&' /&$& $#(.*# 8.*$#; C:,(#' (*#' 8&%.$&' #/. #= 1H4 %= 150
)= 115 = 160
/= 125
12= S* " (.#)#& /& 8&. < 1= &$).' , < 1= %#' ' " (.#)#& /& 8&. < 2= &$).'
, < 1= %#' /&$& -# ' # <# 1=; W#""#. , ,/*J, % -A; #= 2# 1 %= 2# 1
)= #2 1 /= # 1 = # 1
13= P&. 6!1500 '&"' ' #, /&$8.#%& /#,(*%#%' *9#"' % 3 /"#'' %*'(*,(#' % "#%.*""&'; L# 8.*$.# # .#J, % 655 '&"' " /*,(& "# '9,%# # 7500 '&"' " $*""#. "# (./.# # !0 $*" '&"' "&' 10 $*""#.'; C:,(&' "#%.*""&' ' /&$8.#.&, #= 120 $*""&,' /= 100 %= !0
)= H0 = 150
14= 25 &)..&' #/, S; H % ,# &).# , 10 %#'; A 8#.(*. % ' $&$,(& ' /&,(.#(#, -, &)..&' $#' /#%# %# (.$*,:,%&' 2 %#' #,(' % "# /# , (.$*,#.#, "&' 25 &)..&' '* )*.#, /&,(*,#%& ""&' '&"&'; W#""#. -, #= 3 %= 6
)= 4 = H
/= 5
15= 300 8#,(#"&,' % %&)" /&'(.# 8% '. /&/*%&' 8&. 24 &$).' & 32 $.' , 20 %#' (.#)##,%& ! &.#' %*#.*#'; C:,(#' $.' %), .&.#. # 21 &$).' #, # /&'. 200 8#,(#"&,' % (.*8" /&'(.# , 1H %#' (.#)##,%& H &.#' %*#.*#' #= 10 %= 13
)= 11 = 14
/= 12
TEMA: TANTO !OR CIENTO
Aritmética
83
Quinto Año
5efinición#7 E' "# ."#/*J, % ,# %(.$*,#%# /#,(*%#% .'8/(& # 100 ,*%#%'; E$? E*'(, !5i % /#(J"*/&' , " P.> ' %/*. % /#%# 100 8.'&,#' !5 8.&'#, " ."*9*J, /#(J"*/#; Notación#7
a -" -# 8&. /*,(& % -)? # i % ) 100 ( b )
!ropie(a(es#7 1= T&%# /#,(*%#% ' " 100i % '* $*'$#? N B 100i N 2= T&%& 8&./,(# ' 8% '$#. & .'(#. # ,# $*'$# /#,(*%#%; E$? 43 i M 27i M 15i M B 55i M Relaciones entre el tanto por ciento 8 las fracciones#7
S* # ,# /#,(*%#% "# %**%*$&' , 5 8#.(' *9#"' ' (,%.:? /&$& /#%# 8#.( B 15 %" (&(#"> %+/*$& ' .8.',(# #" 20i %" (&(#";
Tanto por ciento (el tanto por ciento#7 E' " /:"/"& %" 8&./,(# % &(.& 8&./,(#; E$?
b c M E" #i %" )i %" /i % M 100 100 100 a
A$,(& ,*/& B
2 2 D'/,(& Ú,*/& B 100 100
E$? D&' #$,(&' '/'*&' %" 30 40i A$,(& Ú,*/&? 30 40 30<40=100 B H2i D'/,(& Ú,*/&? 30 40 30<40=100 B 5Hi c c O@S? P#.# (.#,'#//*&,' /&$./*#"'? @ D&,%? <= P B P./*& % ,(# <= G B G#,#,/*# <= P/ B P./*& % /&'(& & /&$8.# <= P B P.%*%# <= P" B P./*& % "*'(# & *# <= D B D'/,(&
Aritmética
84
Quinto Año
!RO%*EMA& !ARA *A C*A&E 1= + 8&./,(# %" (.*8" %" 50i % , $[/& ' " 30i %" 20i %" 20 8&. 50 %" $*'$& ,$.& 2= + 8&./,(# % , ,$.& (*, 8&. 20i #" 50i % 60 ' " 72i % &(.& ,$.& (*, 8&. 40i # 6 8&. 10 % 20 3= U,# )&(""# % *,& /'(# S; H40 8.& "# )&(""# '&"# /'(# S; 600 $,&' " *,&; E, 8&./,(# ' $#&. " /&'(& %" *,& /&, .'8/(& #" /&'(& % "# )&(""# 4= X#, .#&,# % "# '*9*,( $#,.#? -P#.# /#,/"#. $* %%# /&, P8 $ 8.'(#. %*,.& % R&'# /&, "# /#" $* %%# /&, ""# #$,(# , 40 i> 8.& '* " 8#9& S; 420 000 $* %%# (&(#" %*'$*,*.: , 30i C:,(& " %) X#, # P8 5= E, , /,(.& % '(%*&' " %8#.(#$,(& % '.*/*& '&/*#" .)## "#' 8,'*&,' # "&' '(%*#,(' % )#&' ./.'&' /&,J$*/&' , , 30i #$,(# , , 40i # "&' %$:'; S* " $&,(& (&(#" % 8,'*&,' %# %*'$*,*%& , , 10i /&, '(# 8&"(*/#; + 8&./,(# % 8,'*&,' <(&(#"= .8.',(# "& 8#9#%& 8&. "&' '(%*#,(' % )#&' ./.'&' 6= U,# :).*/# 8.&%/ "#8*/.&' /& /&'(& ' %*'(.*) % "# '*9*,( $#,(#? 60i , $#(.*# 8.*$# 30i , $#,& % &).# " .'(& , 9#'(&' 9,.#"' "&' Aritmética
,% 9#,#,%& " 20i %" /&'(&; D)*%& # ,# #.*#/*J, % 8./*&' '' /&'(&' #$,(#.&, % "# '*9*,( $#,.#? 50i , $#(.*# 8.*$#> 40i , $#,& % &).# '' 9#'(&' 9,.#"' , , 20i '* #&.# ' 9#,#,/*# '.: " 30i %" /&'(&; E, (#,(& 8&. /*,(& #$,(#.# " 8./*& % ,(# % "&' "#8*/.&' 7= D , ./*8*,( "",& % #9# .(*.& " 40i % "& ,& .(*.& % "& .(*.#%& %"& " 40i % "& ,& %"&; E,(&,/' #&.# %#, 3! "*).&'; C:,(&' "*).&' ,& %&" H= A" ,%. , #.(/"& ' #/, 2 %'/,(&' '/'*&' %" 10i 20i 8.& #, ' 9#,# " 20i #""#. " /&'(& % %*/& #.(*/"& '#)*,%& #" *#. *,*/*#"$,( ' 8./*& " /&'(& ' *,/.$,(# , S; 500 != T,9& /*.(# /#,(*%#% % %*,.&; S* " 8.*$. %# 9#'(& " 43i; + 8&./,(# % "# $ %# %)& 9#'(#. #" '9,%& %# 8#.# $ % /&, " 2H5i %" %*,.& &.*9*,#" 10= E, ,# &.(# , /&$./*#,( %*'$*, " 8./*& % , #.(/"& , 25i $&(*& 8&. " /#" "# %$#,%# #$,(# , 60i; E, 8&./,(# #.# "# ./#%#/*J, 11= S&). " 8./*& % ,(# % , #.(/"& ' .)#& " 20i " 30i 85
Quinto Año
#, %#.# , $#.9, % 9#,#,/*# %" 40i %" /&'(&; W#""#. " 8./*& % ,(# ' %*& '* " 8./*& % /&'(& % S; !4000; 12= U, /&$./*#,( (*, 3 T ,% 2 % ""&' , 360 '&"' /#%# ,& 9#,#,%& , ,& % ""&' " 30 8&. 70 % ' /&'(& 8.%*,%& , " &(.& " 20i % ' 8./*& % ,(#; S* " (./. T " /&'(& " 10i %" 8.*$.& $#' " H0i %" '9,%&; + 8&./,(# % 9#,#,/*# & 8+.%*%# %) &)(,. 8#.# , " ,(# (&(#" ,& 9#, ,* 8*.%# 13= X&'+ # # ,# (*,%# /&$8.# /*.(& #.(*/"& 8&. " 8./*& % S; 6!7 "9& % 5 #[&' .9.'# # "# $*'$# (*,%#> /&$8.# " $*'$& #.(*/"& 8#9#,%& #&.# !00i %" 8./*& #,(.*&.; S* " "(..& %" #.(/"& %/# %'/,(& %" 235i $#' " 1Hi *,%*/#,%& 2 %'/,(&' '/'*&'; C:" " 8./*& % "*'(# %" '9,%& #.(/"& /&$8.#%& 14= U, #.(#/(& #" ,%.' ' " %'/,(# " 10i "9& " ./#.9# " 10i> 8.& ' " %" # %'/&,(#. " 10i 8#9:,%&' S; H!10; C:" .# " 8./*& &.*9*,#" 15= U, (*%& #" '. "##%& 8*.% " 2 8&. 13 % ' #,/.# 05 8&. 5 % ' "&,9*(% %#,%& !! $(.&' /#%.#%&'; S %'# '#). (#,(& 8&. /*,(& %" :.# &.*9*,#" .8.',(# " :.# *,#" %" (*%& '* #,(' %" "##%& " (*%&
Aritmética
'* #,(' %" "##%& " (*%& (,# 26 $(.&' % #,/.#; 16= D&' ./*8*,(' A @ /&,(*,, *,& " ./*8*,( -A '(: $%*& "",& " -@ , 13 % ' &"$,; S /&$8"(#, /&, #9# "#' /#8#/*%#%' % A @ .(*+,%&' "# $/"# , , (./. ./*8*,( -C; '#)*,%& "# /#8#/*%#% % @ .' " %&)" % A> %(.$*,#. " (#,(& 8&. /*,(& % *,& /&,(*, "# $/"# , C; 17= k*$,# ,%*J ' /#..& 9#,#,%& " 60i % "# ,(#; S* "& )*.# ,%*%& " 60i %" /&'(& #).# 8.%*%& 11340 '&"'; C:,(& " /&'(& " /#..& # k*$,# 1H= E, , '#"J, % /"#'' "30i % #"$,&' '&, $.'> '* " 25i % $.' #"(#,> '&"& ' /,(# /&, 1H $.'; C:,(&' #"$,&' (*, " '#"J, 1!= U, )&#%&. (*,, 2H0 8"#' 8#/(#%#' % "#' /#"' 9#,& 35; C:,(#' 8"#' $#' /&$& $:*$& %) .#"*#. 8#.# '' 8"#' 9#,#%#' '#, " 30i %" (&(#" 20= U, /&$./*#,( %'# 8.&$&/*&,#. '' ,(#' &./*,%& , %'/,(& " 20i 8.& /&$& , .#"*%#% ,& *. .)##. " 8./*& %).: 8.*#$,( ')*."&'; E, 8&./,(# "& #.:
86
Quinto Año
!RO%*EMA& !ARA *A CA&A 1= S* # ,# /#,(*%#% ' " #$,(# ' 20i # "# ,# /#,(*%#% ' " %*'$*, ' 20i ' 8% #*.$#. /&, .'8/(& # "# /#,(*%#% *,*/*#" ? #= )= /= %= =
2= S* , ,# .,*J, '&/*#" " 75i % "&' &$).' ' *9#" #" 45i % "#' $.'; + 8&./,(# %" (&(#" % 8.'&,#' '&, $.' )= 625i %= 43i
3= E, , /&..#" " 40i '&, 8#(&'> " 35i '&, /&,&' " .'(& 8#&'; S* " ,$.& % 8#(&' ' (.*8"*/# ' %8"*/# " % "&' &(.&' 2; 8&./,(# %" ,& (&(#" '&, 8#(&' #= 20H3i /= 2!16i = N;A;
)= 406i %= 50i
4= S* , "#%& % , /#%.#%& #$,(# , 20i; E, 8&./,(# #$,(# ' :.# #= 20i /= 36i = 4Hi
)= 30i %= 44i
5= S* "# )#' % , (.*:,9"& ' (.*8"*/# ' #"(.# ' %8"*/#; lE, 8&./,(# #$,(# ' :.# Aritmética
)= 300i %= 500i
6= S* " "#.9& " #,/& % , ./(:,9"& #$,(#, , 20i 25i .'8/(*#$,( ' :.# #$,(# , 2400 $2; #""#. " :.# *,*/*#" %" ./(:,9"&;
A$,(# 10i D*'$*, 10i N& #.*# D*'$*, 4i D*'$*, Hi
#= 375i /= 565i = 36i
#= 200i /= 400i = !00i
#= 3600 $ 2 /= 3200 = 7200
)= 4H00 %= 4500
7= W#""#. " 25i %" 120i %" 60i %" 15 8&. 45 % 1500; #= 150 /= H0 = !0
)= 120 %= 60
H= A" '"%& % , $8"#%& ' " #/ , #$,(& %" 20i #" /&$,#. " #[& , " $' % "*& , #$,(& %" 10i '&). " (&(#"; + 8&./,(# % ' '"%& %" #[& #,(.*&. '(#.: ./*)*,%& , A9&'(& #= 12Hi /= 103i = 132i
)= 130i %= 125i
!= S* 9#'(#.# " 30i %" %*,.& (,9& 9#,#.# " 2Hi % "& $ %#.#; P.%.# S; 156; + /#,(*%#% % %*,.& (,9& #= 3500
)= 2000 87
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