,*APT] #E3 ,FRAC;NS       #E4A
            & ,MIX$ ,NUMB]S

    ,NEW ,SYMBOLS ,9TRODUC$
.=? .<DOTS #A-#D-#E-#F.> ,OP5+ SIMPLE
  FRAC;N 9DICATOR "<NO PR9T EQUIVAL5T">
.=# .<DOTS #C-#D-#E-#F.> ,CLOS+ SIMPLE
  FRAC;N 9DICATOR "<NO PR9T EQUIVAL5T">
.=/ .<DOTS #C-#D.> ,HORIZONTAL FRAC;N
  L9E
.=_/ .<DOTS #D-#E-#F1 DOTS #C-#D.>
  ,DIAGONAL FRAC;N L9E
.=_? .<DOTS #D-#E-#F1 DOTS #A-#D-#E-#F.>
  ,OP5+ MIX$ NUMB] FRAC;N 9DICATOR "<NO
  PR9T EQUIVAL5T">
.=_# .<DOTS #D-#E-#F1 DOTS #C-#D-#E-#F.>
  ,CLOS+ MIX$ NUMB] FRAC;N 9DICATOR "<NO
  PR9T EQUIVAL5T">

    ,KEY ,PO9TS
_4 ^1,SIMPLE ^1FRAC;NS 2G9 ) ! OP5+
  SIMPLE FRAC;N 9DICATOR4 ,! CLOS+
  SIMPLE FRAC;N 9DICATOR 5DS A FRAC;N4
  ,"! >E NO PR9T EQUIVAL5TS = ^!
  SYMBOLS4                            #A
               ,*APT] #E           A#E4A
_4 ,9 A SIMPLE FRAC;N1 FOLL[ ! PR9T &
  USE EI ! ^1HORIZONTAL OR ^7DIAGONAL
  FRAC;N L9E^' TO SEP>ATE ! NUM]ATOR &
  D5OM9ATOR4 ,! DIAGONAL FRAC;N L9E IS
  "S"TS CALL$ A SLA%4
_4 ,:5 A SIMPLE FRAC;N 9CLUDES A
  ^7DIAGONAL FRAC;N L9E^'1 PAY ATT5;N TO
  POSI;N ( ! NUM]ATOR & D5OM9ATOR TO "K
  H[ TO BRL ! SIMPLE FRAC;N4
_4 ^1,MIX$ ^1NUMB]S COMB9E A :OLE NUMB]
  & SIMPLE FRAC;N4
_4 ,AF WRIT+ ! :OLE NUMB]1 USE ! ^7OP5+
  MIX$ NUMB] FRAC;N 9DICATOR^' TO 2G9 !
  FRAC;NAL "P ( ! MIX$ NUMB]4
_4 ,5D ! FRAC;NAL "P ( ! MIX$ NUMB] ) !
  ^7CLOS+ MIX$ NUMB] FRAC;N 9DICATOR^'4
_4 ^1,V>IABLES >E UN"KN NUMB]S WRITT5 Z
  LRS 9 BO? PR9T & ,NEME? ,CODE4
_4 ,SIMPLE FRAC;NS & MIX$ NUMB]S >E 9
  ,NEME? ,CODE & M/ 2 PLAC$ )9 OP5+ &
  CLOS+ ,NEME? ,CODE 9DICATORS4
_4 ,TRY TO AVOID DIVID+ A FRAC;N OR A
  MIX$ NUMB] 2T L9ES 9 ^W PROBLEMS4 ,IF
  Y M/ DIVIDE AN EQUA;N1 D S AT !     #B
               ,*APT] #E           B#E4A
  SIGN ( COMP>ISON4
------------------------------------#E4B
_4 ,IF "! IS AN A2REVIAT$ M1SURE;T UNIT
  ) ! SIMPLE FRAC;N OR MIX$ NUMB]
  "<E4G41 IN1 KM">1 X M/ 2 9CLUD$ )9 !
  ,NEME? ,CODE OP5+ & CLOS+ 9DICATORS )
  NO 3TRAC;NS4
_4 ,EXAMPLE PROBLEMS M/ 2 9CLUD$ :5
  TRANSCRIB+ MAT]IALS4 ,L1VE A BLANK L9E
  ABV & 2L ! EXAMPLE PROBLEM4 ,9 ADDI;N1
  =MAT ! EXAMPLE PROBLEM L ! SURR.D+
  PROBLEMS4

    ,9TRODUC;N
  ,/UD5TS >E USUALLY 9TRODUC$ TO SIMPLE
FRAC;NS 9 ?IRD GRADE & MIX$ NUMB]S 9
F\R? GRADE "<,COMMON ,CORE ,/ATE ,/&>DS
,9ITIATIVE1 #BJAJ2 ,M>YL& ,DE"P;T (
,$UCA;N1 #BJAE">4 ,DUR+ ?IRD GRADE1
/UD5TS LE>N TO R1D & WRITE SIMPLE
FRAC;NS1 & BY ! "T !Y COMPLETE F\R?
GRADE1 !Y >E EXPECT$ TO COMP>E TWO
FRAC;NS ) DI6]5T D5OM9ATORS2 ADD &
SUBTRACT MIX$ NUMB]S ) L              #C
               ,*APT] #E           A#E4B
D5OM9ATORS2 MULTIPLY A FRAC;N BY A :OLE
NUMB]2 & USE DECIMAL NOTA;N = FRAC;NS )
D5OM9ATORS #AJ & #AJJ4 ,9 FIF? GRADE1
/UD5TS LE>N TO MULTIPLY & DIVIDE FRAC;NS
"<,COMMON ,CORE ,/ATE ,/&>DS ,9ITIATIVE1
#BJAJ2 ,M>YL& ,DE"P;T ( ,$UCA;N1
#BJAE">4

    ,SIMPLE ,FRAC;NS
  _% ?3/4# _: IS AN EXAMPLE ( A ^1SIMPLE
^1FRAC;N ) A HORIZONTAL FRAC;N L9E4
  ,! TOP "P ( A SIMPLE FRAC;N IS !
NUM]ATOR1 & ! BOTTOM "P ( ! SIMPLE
FRAC;N IS ! D5OM9ATOR4 ,! ^1NUM]ATOR IS
! NUMB] ( "PS 2+ C.T$1 & ! ^1D5OM9ATOR
IS ! TOTAL NUMB] ( "PS 9 ! :OLE "<,NEW
,OX=D ,AM]ICAN ,DIC;N>Y1 N4D4">4
  ,9 _% ?3/4# _:1 #C IS ! NUM]ATOR & #D
IS ! D5OM9ATOR4
  ,9 ! ,NEME? ,CODE1 SIMPLE FRAC;NS >E
DEF9$ Z HAV+ NUM]ATORS & D5OM9ATORS )\T
A FRAC;N 9 ! NUM]ATOR OR D5OM9ATOR
"<,RULE ,,XII1 ^S#FA">4 ,H["E1 SIMPLE
FRAC;NS C 9CLUDE MA!MATICAL           #D
               ,*APT] #E           B#E4B
EXPRES.NS S* Z _% #4+5 _: & V>IABLES S*
Z ;X4
  ,"! >E NO 8TOPS0 & 8BOTTOMS0 = FRAC;NS
9 ,NEME? ,CODE4 ,WE H 8LEFTS0 & 8"RS0 2C
! NUM]ATOR IS WRITT5 TO ! LEFT ( !
FRAC;N L9E & ! D5OM9ATOR IS WRITT5 TO !
"R ( ! FRAC;N L9E4
  ,FRAC;NS ) A HORIZONTAL FRAC;N L9E >E
5CLOS$ BY OP5+ & CLOS+ SIMPLE FRAC;N
9DICATORS "<,RULE ,,XII1 ^S#FB;A">4 ,"!
>E NO PR9T EQUIVAL5TS =
------------------------------------#E4C
^! 9DICATORS4 ,AN ^7OP5+ SIMPLE FRAC;N
9DICATOR^' IS WRITT5 ) DOTS #A-#D-#E-#F
9 ,NEME? ,CODE4
  .=?

  ^1,T1*+ ^1,TIP3 ,X MAY 2 HELP;L TO
  PO9T \ T ! OP5+ SIMPLE FRAC;N 9DICATOR
  3TA9S ! SAME DOTS Z ! 8TH0 GR\PSIGN4

  ,A ^7CLOS+ SIMPLE FRAC;N 9DICATOR^' IS
WRITT5 ) DOTS #C-#D-#E-#F 9 ,NEME?
,CODE4                                #E
               ,*APT] #E           A#E4C
  .=#

  ^1,T1*+ ^1,TIP3 ,X MAY 2 HELP;L TO
  PO9T \ T ! CLOS+ SIMPLE FRAC;N
  9DICATOR 3TA9S ! SAME DOTS Z ! NUM]IC
  9DICATOR4

  ,SIMIL> TO ,,UEB1 ! SAME DOT 3FIGURA;N
"S"TS REPRES5TS MULTIPLE ITEMS 9 !
,NEME? ,CODE1 DEP5D+ ON ! PLACE;T &
3TEXT ( ! BRL DOT 3FIGURA;N4

  ^1,T1*+ ^1,TIP3 ,REM9D /UD5TS TO ALW
  8CLOSE0 :AT !Y 8OP504 ,?US1 IF !Y 2G9
  ) AN OP5+ SIMPLE FRAC;N 9DICATOR1 !Y W
  NE$ A CLOS+ SIMPLE FRAC;N 9DICATOR AT
  ! 5D ( ! FRAC;N4

  ,A ^7HORIZONTAL FRAC;N L9E^' IS WRITT5
) DOTS #C-#D 9 ! ,NEME? ,CODE4
  .=/

  ^1,T1*+ ^1,TIP3 ,X MAY 2 HELP;L TO
  PO9T \ T ! HORIZONTAL FRAC;N L9E    #F
               ,*APT] #E           B#E4C
  3TA9S ! SAME DOTS Z ! 8ST0 GR\PSIGN4

  ,EXAMPLE #E4A 3TA9S #F EXAMPLES (
SIMPLE FRAC;NS ) HORIZONTAL FRAC;N L9ES4

      ,EXAMPLE #E4A _%
?1/3#  ?5/8#  ?11/12#
?6/7#  ?39/100#  ?30+50/100# _:

------------------------------------#E4D
  ,/UD5TS LE>N TO G5]ATE SIMPLE
EQUIVAL5T FRAC;NS Z WELL Z ADD &
SUBTRACT SIMPLE FRAC;NS ) L D5OM9ATORS 9
?IRD GRADE & ) UNLIKE D5OM9ATORS 9 F\R?
GRADE "<,COMMON ,CORE ,/ATE ,/&>DS
,9ITIATIVE1 #BJAJ">4
  ,"S"TS ! PROBLEMS >E 9 HORIZONTAL
"<L9E>"> =MAT1 & O!R "TS ! PROBLEMS >E
SPATIALLY ALIGN$ "<V]TICAL">4
  ,EXAMPLE #E4B 3TA9S EI<T L9E> PROBLEMS
T 3TA9 "O OR M SIMPLE FRAC;NS ) A
HORIZONTAL FRAC;N L9E4

                                      #G
               ,*APT] #E           A#E4D

      ,EXAMPLE #E4B _%
?2/3#./?1/3# .K ----
?5/6#+1 .K ----
?4/9#+= .K ?7/9#
?23/100# "K ?71/100#
---- -?1/2# .K ?4/13#
?2/6# .K ?1/3#
------------------------------------#E4E
?1/2#@*?4/5# .K ?1@*4/=#
?3/10# .K #.3 _:

  ,"S SIMPLE FRAC;NS 3TA9 A ^7DIAGONAL
FRAC;N L9E^'1 "S"TS CALL$ A SLA%4
  _% ?1_/3# _: IS AN EXAMPLE ( A SIMPLE
FRAC;N ) A DIAGONAL FRAC;N L9E4
  ,X TAKES TWO CELLS TO WRITE A
^7DIAGONAL FRAC;N L9E^' 9 ,NEME? ,CODE4
,X IS WRITT5 ) DOTS #D-#E-#F 9 ! F/
CELL1 FOLL[$ BY DOTS #C-#D 9 ! SECOND
CELL4
  .=_/
  ,EXAMPLE #E4C 3TA9S ?REE EXAMPLES (
SIMPLE FRAC;NS ) A DIAGONAL FRAC;N    #H
               ,*APT] #E           A#E4E
L9E ": ! EXPRES.N OR NUMB] 9 ! NUM]ATOR
APPE>S TO 2 RAIS$ HI<] ?AN ! EXPRES.N OR
NUMB] 9 ! D5OM9ATOR4

      ,EXAMPLE #E4C
_% ?1_/3#  ?5_/4#  ?8+3_/17# _:

  ,DI6]5T RULES APPLY IF ! NUM]ATOR &
D5OM9ATOR >E PR9T$ AT ! SAME LEVEL & !
SIZE ( PR9T IS ! SAME Z ! SURR.D+ MA?
EXPRES.NS4 ,9 O!R ^WS1 A SIMPLE FRAC;N
IS BRLD DI6]5TLY :5 ! NUM]ATOR IS N
RAIS$ HI<] ?AN ! D5OM9ATOR4
  ,9 ? SITUA;N1 SIMPLE FRAC;N 9DICATORS
>E N US$ "<,RULE ,,XII1 ^S#FC;B">4 ,9/1D
2G9 ) A NUM]IC 9DICATOR4
  ,EXAMPLE #E4D 9CLUDES TWO EXAMPLES ":
! NUM]ATOR & D5OM9ATOR >E PR9T$ AT !
SAME LEVEL & ! SIZE ( PR9T IS ! SAME Z !
SURR.D+ MA? EXPRES.NS4
------------------------------------#E4F

      ,EXAMPLE #E4D
_% #1_/4  #3_/2 _:                    #I
               ,*APT] #E           A#E4F

      ,PRACTICE #E4A
    ,9T]L9E ! FOLL[+ SIMPLE FRAC;NS &
  PROBLEMS 3TA9+ SIMPLE FRAC;NS4 ,PAY
  SPECIAL ATT5;N TO ! FRAC;N L9E TO
  DET]M9E IF X IS A HORIZONTAL FRAC;N
  L9E OR A DIAGONAL FRAC;N L9E4 _%
#1_4 ?5_/8#
#2_4 ?3/16#
#3_4 #1_/15#
#4_4 ?21_/100#
#5_4 ?35/100# .K ----
#6_4 ?3/4# .K ?9/12#
#7_4 ?1/3#+?1/3# .K =
#8_4 ?8/20# "K ?3/7#
#9_4 ?5_/6#-?1_/6# .K ----
#10_4 ?1/3#@*?9/16# .K ?1@*9/3@*16#
#11_4 =+?1_/2# .K ?5_/7#
#12_4 ?5_/11#+?3_/13# .K ----
#13_4 ?1/100# .K #.01 _:

    ,WRITE ! FRAC;NS 9 BRL4
#A4 "O OV] N9E @.<DIAGONAL FRAC;N L9E@.>
#B4 ELEV5 OV] SEV5TY                 #AJ
               ,*APT] #E           B#E4F
#C4 TWELVE OV] FIVE @.<DIAGONAL FRAC;N
  L9E@.>
#D4 "O OV] ?REE
------------------------------------#E4G
#E4 F\R OV] N9ETE5
#F4 EI<T OV] "O HUNDR$
#G4 TWO OV] TWO @.<DIAGONAL FRAC;N
  L9E@.>

    ,WRITE ! EXPRES.NS ": ! NUM]ATOR &
  D5OM9ATOR >E AT ! SAME LEVEL4
#H4 "O SLA% ?REE
#I4 ?REE SLA% SEV5
#AJ4 ?IRTE5 SLA% F\RTE5

    ,MIX$ ,NUMB]S
  ,A ^1MIX$ ^1NUMB] 2G9S ) A :OLE NUMB]4
,X IS FOLL[$ BY A SIMPLE FRAC;N1
COMPRIS$ ( NUMB]S1 USUALLY WRITT5 9
SMALL] PR9T "<,RULE ,,XII1 ^S#FD">4
  _% #2_?1_/9_# _: & _% #5_?3/4_# _: >E
EXAMPLES ( MIX$ NUMB]S4
  ,9 ! ,NEME? ,CODE1 MIX$ NUMB]S 2G9 ) !
NUM]IC 9DICATOR & ! :OLE NUMB]4 ,!   #AA
               ,*APT] #E           A#E4G
FRAC;NAL "P ( ! MIX$ NUMB] IS !N 5CLOS$
)9 ! ^7MIX$ NUMB] FRAC;N 9DICATORS^'
"<,RULE ,,XII1 ^S#FD">4 ,EI ! HORIZONTAL
FRAC;N L9E OR ! DIAGONAL FRAC;N L9E IS
US$ )9 ! FRAC;NAL "P "<,RULE ,,XII1
^S#FD">4
  ,X TAKES TWO CELLS TO WRITE AN ^7OP5+
MIX$ NUMB] FRAC;N 9DICATOR^' 9 ,NEME?
,CODE4 ,X IS WRITT5 ) DOTS #D-#E-#F 9 !
F/ CELL1 FOLL[$ BY DOTS #A-#D-#E-#F 9 !
SECOND CELL4 ,"! IS NO PR9T EQUIVAL5T =
! OP5+ MIX$ NUMB] FRAC;N 9DICATOR4
  .=_?
  ,X AL TAKES TWO CELLS TO WRITE A
^7CLOS+ MIX$ NUMB] FRAC;N 9DICATOR^' 9
,NEME? ,CODE4 ,X IS WRITT5 ) DOTS
#D-#E-#F 9 ! F/ CELL1 FOLL[$ BY
------------------------------------#E4H
DOTS #C-#D-#E-#F 9 ! SECOND CELL4 ,"! IS
NO PR9T EQUIVAL5T = ! CLOS+ MIX$ NUMB]
FRAC;N 9DICATOR4
  .=_#

    ^1,T1*+ ^1,TIP3 ,REM9D /UD5TS    #AB
               ,*APT] #E           A#E4H
  TO ALW CLOSE :AT !Y OP54 ,?US1 IF !Y
  2G9 ) AN OP5+ MIX$ NUMB] FRAC;N
  9DICATOR1 !Y W NE$ A CLOS+ MIX$ NUMB]
  FRAC;N 9DICATOR AT ! 5D ( ! FRAC;NAL
  "P4
    ^1,T1*+ ^1,TIP3 ,X MAY 2 HELP;L TO
  PO9T \ T ! SAME DOT 3FIGURA;N IS US$ =
  FRAC;N L9ES 9 ! FRAC;NAL "P ( MIX$
  NUMB]S Z 9 SIMPLE FRAC;NS4

  ,EXAMPLE #E4E 3TA9S FIVE EXAMPLES (
MIX$ NUMB]S4 ,PAY SPECIAL ATT5;N TO !
MIX$ NUMB] FRAC;N 9DICATORS & FRAC;N
L9E4
  ,IF ! FRAC;N L9E IS HORIZONTAL 9 PR9T1
!N A HORIZONTAL FRAC;N L9E IS US$ 9 BRL
"<,RULE ,,XII1 ^S#FD">4 ,SIMIL>LY1 IF !
FRAC;N L9E IS DIAGONAL 9 PR9T1 !N A
DIAGONAL FRAC;N L9E IS US$ 9 BRL "<,RULE
,,XII1 ^S#FD">4

      ,EXAMPLE #E4E _%
#2_?1/6_#  #3_?4/5_#
#8_?17_/20_#  #25_?2/3_#             #AC
               ,*APT] #E           B#E4H
#100_?3_/10_# _:

  ,EXAMPLE #E4F 3TA9S ?REE EXAMPLES (
L9E> PROBLEMS T 3TA9 "O OR M MIX$
NUMB]S4 ,NOTICE T ! PREVI\S RULES
EXPLA9+ AB SPAC+ ( L9E> PROBLEMS APPLY
:5 MIX$ NUMB]S >E 9CLUD$ 9 ! PROBLEM4
------------------------------------#E4I

      ,EXAMPLE #E4F _%
#3_?1/7_# "K #3_?2/7_#
#5_?1/2_#+2 .K ----
#7_?3_/8_# .K #7_?9_/24_# _:

  ,EXAMPLE #E4G 3TA9S ?REE EXAMPLES (
L9E> PROBLEMS T 3TA9 "O OR M MIX$ NUMB]S
& A SIGN ( OMIS.N4 ,NOTICE T ANY NUM]AL
9 ! MIX$ NUMB] OR ! 5TIRE MIX$ NUMB] MAY
2 REPRES5T$ BY A SIGN ( OMIS.N4 ,AL
NOTICE 9 ! F/ EXAMPLE T A NUM]IC
9DICATOR IS N NE$$ 2F ! :OLE NUMB] 9 !
SUBTRAH5D4

                                     #AD
               ,*APT] #E           A#E4I

      ,EXAMPLE #E4G _%
#9_?1/5_#-2_?1/4_# .K =
#1_?=/4_#+?1/3# .K #1_?7/12_#
#7_?2/9_#@*= .K #7_?2/9_# _:
-----------------------------------#E4AJ

    ,PRACTICE #E4B
    ,9T]L9E ! MIX$ NUMB]S & PROBLEMS T
  3TA9 "O OR M MIX$ NUMB]S4 _%
#1_4 #3_?1/2_#
#2_4 #12_?5_/8_#
#3_4 #5_?6/10_#
#4_4 #384_?2_/11_#
#5_4 #1_?3/5_#
#6_4 #1_?1/3_#-?1/2#
#7_4 #15_?1/4_#+7_?3/8_#
#8_4 #3+2_?9_/10_#
#9_4 #1,563_?3/7_#-582_?1/3_#
#10_4 ?1/2#+1_?2/3_# _:



                                     #AE
               ,*APT] #E          A#E4AJ

    ,N[ WRITE ! FOLL[+ MIX$ NUMB]S &
  PROBLEMS 3TA9+ "O OR M MIX$ NUMB]S 9
  BRL4
#A4 TWO & "O OV] SEV5 @.<DIAGONAL FRAC;N
  L9E@.>
#B4 F\R & "O OV] ?REE
#C4 EI<T & "O OV] TWO @.<DIAGONAL FRAC;N
  L9E@.>
#D4 "O & ?REE OV] SIXTE5
#E4 ?REE & F\R OV] SEV5 PLUS "O & "O OV]
  SEV5 EQUALS L;G-DA%
#F4 ;8 M9US "O OV] F\R EQUALS TWO & "O
  OV] TWO
#G4 F\R & TWO OV] ?REE M9US ?REE & "O
  OV] ?REE EQUALS ;8 @.<BO? FRAC;NS USE
  DIAGONAL FRAC;N L9ES@.>
#H4 TWELVE & FIVE OV] N9E M9US TWO OV]
  ;8 EQUALS TWELVE & ?REE OV] N9E EQUALS
  TWELVE & "O OV] ?REE



                                     #AF
               ,*APT] #E           #E4AA

    ,SPATIALLY ,ALIGN$ ,PROBLEMS )
    ,SIMPLE ,FRAC;NS & ,MIX$ ,NUMB]S
  ,:5 A SPATIALLY ALIGN$ PROBLEM 3TA9S
FRAC;NS1 ! FRAC;N 9DICATORS & FRAC;N
L9ES M/ 2 V]TICALLY ALIGN$ "<,RULE
,,XXIV1 ^S#AGH;E">4
  ,EXAMPLE #E4H 3TA9S TWO PROBLEMS )
SIMPLE FRAC;NS4 ,! SIMPLE FRAC;N
9DICATORS & HORIZONTAL FRAC;N L9ES >E
ALIGN$ V]TICALLY 9 EA* PROBLEM4 ,NOTICE
T BO? PROBLEMS >E NUMB]$1 & "! IS "O
BLANK COLUMN 2T ! P]IOD ( ! PROBLEM
NUMB] & ! 2G9N+ ( ! SEP>A;N L9E4
  ,ALL O!R RULES = SPATIALLY ALIGN$
PROBLEMS APPLY4 ,?US1 ! M9US_/PLUS SIGN
IS PLAC$ "O CELL TO ! LEFT ( ! OP5+
SIMPLE FRAC;N 9DICATOR1 & ! SEP>A;N L9E
EXT5DS "O CELL TO ! LEFT & "O CELL TO !
"R ( ! WIDE/ L9E 9 EA* >RANGE;T4



                                     #AG
               ,*APT] #E          A#E4AA

      ,EXAMPLE #E4H _%

#15_4   ?1/5#   #16_4   ?8/9#
       -?1/4#          +?2/3#
      33333333        33333333

_:
  ,EXAMPLE #E4I 3TA9S SPATIAL PROBLEM )
SIMPLE FRAC;NS4 ,! F/ FRAC;N HAS A
TWO-DIGIT NUM]ATOR & D5OM9ATOR1 & !
SECOND FRAC;N 3TA9S A "O-DIGIT NUM]ATOR
& D5OM9ATOR4 ,S T "EY?+ IS L9$ UP 9 !
PROBLEM1 ! NUM]ATORS >E "R JU/IFI$1 & !
D5OM9ATORS >E LEFT JU/IFI$ "<,RULE
,,XXIV1 ^S#AGH;E">4







                                     #AH
               ,*APT] #E           #E4AB

      ,EXAMPLE #E4I _%

  ? 7/8 #
 -?11/16#
3333333333

_:

  ^1,T1*+ ^1,TIP3 ,9/1D ( REF]R+ TO "R
  JU/IFI$ & LEFT JU/IFI$1 X MAY 2 HELP;L
  TO PO9T \ T ! NUM]ATOR & D5OM9ATOR W
  SNU7LE ! FRAC;N L9E4

  ,EXAMPLE #E4AJ 3TA9S TWO ADDI;NAL
NUMB]$ PROBLEMS ) SIMPLE FRAC;NS4 ,PAY
SPECIAL ATT5;N TO ! ALIGN;T ( ! FRAC;N
L9ES & SIMPLE FRAC;N 9DICATORS Z WELL Z
! PLACE;T ( ! NUM]ATORS & D5OM9ATORS4




                                     #AI
               ,*APT] #E          A#E4AB

      ,EXAMPLE #E4AJ _%

#7_4   ?3/10 #   #8_4   ?16/17#
      +?1/100#         -? 9/34#
     3333333333       3333333333

_:
  ,:5 A SPATIALLY ALIGN$ PROBLEM 3TA9S
MIX$ NUMB]S1 ! :OLE NUMB]S M/ AL 2
V]TICALLY ALIGN$ "<,RULE ,,XXIV1
^S#AGH;F">4 ,ALL O!R RULES = SPATIALLY
ALIGN$ PROBLEMS ) FRAC;NS APPLY4 ,=
EXAMPLE1 ! NUM]ATORS ( ! FRAC;NAL "PS W
2 "R JU/IFI$ & ! D5OM9ATORS ( ! FRAC;NAL
"PS W 2 LEFT JU/IFI$4
-----------------------------------#E4AC
  ,EXAMPLE #E4AA 3TA9S F\R PROBLEMS )
MIX$ NUMB]S4 ,NOTICE H[ ! :OLE NUMB]S1
FRAC;N 9DICATORS1 & HORIZONTAL FRAC;N
L9ES >E ALIGN$ V]TICALLY 9 EA* PROBLEM4
,AL NOTICE T ! NUM]IC 9DICATOR IS N TO 2
US$ 9 FRONT ( ! :OLE NUMB] ( ! MIX$
NUMB] 9 SPATIALLY >RANG$ PROBLEMS4   #BJ
               ,*APT] #E          A#E4AC

      ,EXAMPLE #E4AA _%

  3_?3/4 _#    24_?11/13_#
 +1_?5/12_#   - 1_? 2/3 _#
333333333333 33333333333333

  8_?1/4_#    6_?2/3_#
 +2_?1/5_#   -  ?1/9 #
33333333333 33333333333

_:

    ,PRACTICE #E4C
    ,9T]L9E ! FOLL[+ PROBLEMS T 3TA9
  FRAC;NS & MIX$ NUMB]S4 _%

#1_4   ?7/12#
      +?3/12#
     333333333

#2_4   ?11/15#
      -? 6/15#
     3333333333                      #BA
               ,*APT] #E           #E4AD

#3_4   8_? 4/24_#
      -3_?11/6 _#
     3333333333333

#4_4    8_?17_/20_#
      +14_? 3_/5 _#
     333333333333333

_:
  ,USE YR BRLWRIT] TO WRITE ! PROBLEMS T
3TA9 FRAC;NS & MIX$ NUMB]S 9 SPATIAL
=MAT4
      @.<,! FOLL[+ #F PROBLEMS >E WRITT5
    ) TWO PROBLEMS ON EA* L9E4@.>

?REE OV] FIVE M9US "O OV] TWO
EI<T OV] N9E M9US TWO OV] N9E
?REE OV] F\R PLUS "O OV] SIXTE5
SIXTE5 OV] "O HUNDR$ M9US "O OV] T5
?REE & FIVE OV] SEV5 PLUS "O & "O OV]
  SEV5
SIXTE5 & "O OV] ?REE M9US TWO & "O OV]
  F\R                                #BB
               ,*APT] #E          A#E4AD

    ,^W ,PROBLEMS ) ,SIMPLE ,FRAC;NS &
    ,MIX$ ,NUMB]S
  ,Y LE>N$ 9 ,*APT] #B T FREE/&+ NUMB]S
S* Z #DH & #AJJ 9 ^W PROBLEMS >E N
3SID]$ MA? NOTA;N4 ,9 3TRA/1 SIMPLE
FRAC;NS & MIX$ NUMB]S 9 ^W PROBLEMS >E
3SID]$ MA? NOTA;N4 ,?US1 !Y M/ 2 WRITT5
9 ,NEME? ,CODE1 & ,NEME? ,CODE SWIT*
9DICATORS M/ 2 US$ "<,,BANA ,GUID.E1
#BJAH1 ;P4 #A">4
  ,REVIEW ! ^W PROBLEMS 9 ,EXAMPLE #E4AB
& #E4AC4 ,NOTICE T ! SIMPLE FRAC;NS &
MIX$ NUMB]S >E 9 ,NEME? ,CODE4 ,!Y >E
PLAC$ )9 OP5+ & CLOS+ ,NEME? ,CODE
9DICATORS4 ,AL NOTICE T ! OP5+
-----------------------------------#E4AE
,NEME? ,CODE 9DICATOR1 ! MA? MAT]IAL1 &
! ,NEME? ,CODE T]M9ATOR >E KEPT ON !
SAME BRL L9E :5 POSSIBLE4



                                     #BC
               ,*APT] #E          A#E4AE

      ,EXAMPLE #E4AB
#A4 ,ME<AN HAS _% ?3/4# _: ( A BOX ( C&Y
  & EATS _% ?1/3# _: ( :AT %E HAS4 ,:AT
  FRAC;N ( ! BOX DID %E EAT8

      ,EXAMPLE #E4AC
#B4 ,JACKSON & 8 BRO!R WALK$
  _% #1_?1/5_# _: MILES F _! HOME TO
  S*OOL4 ,JOSEPH WALKS _% ?7/8# _: MILES
  F 8 HOME TO S*OOL4 ,H[ M* F>!R DOES
  ,JOSEPR WALK TO S*OOL ?AN ,JACKSON & 8
  BRO!R8

  ,IF "! IS AN A2REVIAT$ M1SURE;T UNIT )
! SIMPLE FRAC;N OR MIX$ NUMB] "<E4G41
IN1 KM">1 X M/ 2 9CLUD$ )9 ! ,NEME?
,CODE OP5+ & CLOS+ 9DICATORS ) NO
3TRAC;NS "<,,BANA ,GUID.E1 #BJAH1 ;P4
#H">4
  ,REVIEW ! ^W PROBLEMS 9 ,EXAMPLES
#E4AD & #E4AE4 ,PAY SPECIAL ATT5;N TO !
A2REVIAT$ M1SURE;T UNIT & H[ !Y >E
9CLUD$ Z "P ( ! MAT]IAL 9 ,NEME?     #BD
               ,*APT] #E          B#E4AE
,CODE4 ,9 ,EXAMPLE #E4AE1 AL NOTICE H[ !
3TRAC;N IS N US$ 9 ! A2REVIAT$ M1SURE;T
UNIT4
-----------------------------------#E4AF

      EXAMPLE #E4AD
#AE4 ,ANA/ASIA RODE H] SCOOT] =
  _% #2_?1/4_# MI _: ON ,SATUR"D &
  _% #1_?5/8_# MI _: ON ,TUES"D4 ,H[ _M
  MILES DID %E RIDE ALT8

      ,EXAMPLE #E4AE
#AF4 ,JE;G HAS A PIECE ( /R+ T IS
  _% #11_?1_/2_# IN4 _: L;G4 ,HE CUTS (F
  _% #8_?1_/6_# IN4 _: TO USE Z "P ( A
  SCI;E EXP]I;T4 ,H[ _M 9*ES ( /R+ DOES
  HE H LEFT8

  ,9 ,EXAMPLE #E4AF1 ! M1SURE;T UNIT IS
N A2REVIAT$4 ,"!=E1 X IS N 9CLUD$ )9 !
,NEME? ,CODE OP5+ & CLOS+ 9DICATORS4 ,AL
NOTICE T ! S+LE PROBLEM IS N NUMB]$4
,?US1 X TR1T$ Z A P>AGRAPH & 2G9S 9 CELL
#C ) RUN-OV]S 9 CELL #A4             #BE
               ,*APT] #E          A#E4AF

      ,EXAMPLE #E4AF
  ,LAUR5 CUTS AN _% #18_?1/2_# _: FOOT
WOOD5 BO>D 9TO TWO PIECES4 ,"O BO>D IS
#B FEET LESS ?AN TWICE ! L5G? ( ! O!R4
,:AT IS ! L5G? ( ! L;G] WOOD5 BO>D8
-----------------------------------#E4AG

    ,PRACTICE #E4D
    ,N[ WRITE ! FOLL[+ ^W PROBLEMS 9
  BRL4
#D4 ,MR4 ,/EFANIK IS ORD]+ PIZZA = !
  EI<TE5 /UD5TS 9 8 HOMEROOM4 ,IF HE
  E/IMATES T EA* /UD5T W EAT "O OV] F\R
  @.<DIAGONAL FRAC;N L9E@.> ( A PIZZA1
  H[ _M PIZZAS %D HE ORD]8
#E4 ,MAYA 0 FIVE FT4 TWO & "O OV] TWO
  IN4 TALL LA/ YE>4 ,%E GREW "O & "O OV]
  F\R IN4 9 ! PA/ YE>4 ,H[ TALL IS %E
  N[8
#F4 ,TWO *N WEI< SEV5TY-TWO & "O OV]
  ?REE LBS4 & EI<TY-SIX & TWO OV] FIVE
  LBS4 ,:AT IS ! DI6];E 2T _! WEI<TS8
                                     #BF
               ,*APT] #E          A#E4AG

    ,"W%EETS ) ,EXAMPLE ,PROBLEMS
  ,"S"TS "W%EETS 9CLUDE AN EXAMPLE
PROBLEM4 ,L1VE A BLANK L9E ABV & 2L !
EXAMPLE "<.1,BRL .1,=MATS1 #AJ4H4C">4
,!N TRANSCRIBE ! EXAMPLE L ! O!R
PROBLEMS "<.1,BRL .1,=MATS1 #AJ4H4D">4
  ,:5 A COLON FOLL[S ! ^W 8EXAMPLE0 D N
BOLD1 "UL9E1 OR ITALICIZE X "<.1,BRL
.1,=MATS1 #AJ4H4B">4 ,H["E1 :5 NO
PUNCTUA;N FOLL[S ! ^W 8EXAMPLE01 FOLL[
PR9T =MATT+ "<.1,BRL .1,=MATS1
#AJ4H4A">4
  ,IF Y NE$ TO DIVIDE A PROBLEM1 D S AT
! SIGN ( COMP>ISON "<,RULE ,,XXV1
^S#AIE;E">4
  ,! "W%EET 9 ,EXAMPLE #E4AG 9CLUDES AN
EXAMPLE PROBLEM AF ! DIREC;NS4 ,NOTICE
H[ ! EXAMPLE 2G9S 9 CELL #A L ! O!R
PROBLEMS4 ,AL NOTICE T "! IS A BLANK L9E
BO? 2F & AF ! EXAMPLE PROBLEM4
  ,S9CE ! ^W 8EXAMPLE0 IS FOLL[$ BY A
COLON1 X IS N BOLD$ 9 BRL4
  ,! EXAMPLE MA? PROBLEM W N FIT     #BG
               ,*APT] #E          B#E4AG
ON "O BRL L9E4 ,?US X HAS BE5 DIVID$ ACR
TWO L9ES4 ,PAY SPECIAL ATT5;N TO H[ !
PROBLEM 0 DIVID$ S T ! SECOND L9E 2G9S )
A SIGN ( COMP>ISON4
  ,ADDI;NALLY1 S9CE ! DIREC;NS >E FOLL[$
BY NUMB]$ PROBLEMS1 !Y 2G9 9 CELL #E )
RUN-OV]S 9 CELL #C4
-----------------------------------#E4AH

      ,EXAMPLE #E4AG

7777777777777777777777777777777777777777
            ,FUN ) ,FRAC;NS

    ,%[ YR "W & WAT* ! SIGNS C>E;LLY6

,EXAMPLE3 _% ?2/4#+?1/4#
  .K ?2+1/4# .K ?3/4#

#1_4 ?5/8#+?1/8# .K =
#2_4 ?33/100#-?11/100# .K =
#3_4 ?6/7#-?2/7# .K = _:
GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG
                                     #BH
               ,*APT] #E           #E4AI

    ,*APT] ,SUMM>Y
      ,NEME? ,CODE ,SWIT* ,9DICATORS
_4 ,! MA? PROBLEM M/ 2 ON ! SAME L9E ) !
  OP5+ ,NEME? ,CODE 9DICATOR & ,NEME?
  ,CODE T]M9ATOR :5"E POSSIBLE4

      ,SIMPLE ,FRAC;NS
_4 ,FRAC;NS ) A HORIZONTAL FRAC;N L9E >E
  5CLOS$ BY OP5+ & CLOS+ SIMPLE FRAC;N
  9DICATORS "<,RULE ,,XII1 ^S#FB;A">4
_4 ,9 ,NEME? ,CODE1 ! NUM]ATOR IS WRITT5
  TO ! LEFT ( ! FRAC;N L9E & ! D5OM9ATOR
  IS WRITT5 TO ! "R ( ! FRAC;N L9E4
_4 ,IF ! FRAC;N L9E IS HORIZONTAL 9
  PR9T1 !N A HORIZONTAL FRAC;N L9E IS
  US$ 9 BRL "<,RULE ,,XII1 ^S#FD">4
_4 ,SIMIL>LY1 IF ! FRAC;N L9E IS
  DIAGONAL 9 PR9T1 !N A DIAGONAL FRAC;N
  L9E IS US$ 9 BRL "<,RULE ,,XII1
  ^S#FD">4


                                     #BI
               ,*APT] #E          A#E4AI

      ,MIX$ ,NUMB]S
_4 ,MIX$ NUMB]S 2G9 ) ! NUM]IC 9DICATOR
  & ! :OLE NUMB]4
_4 ,! FRAC;NAL "P ( ! MIX$ NUMB] IS
  5CLOS$ )9 ! MIX$ NUMB] FRAC;N
  9DICATORS "<,RULE ,,XII1 ^S#FD">4

      ,L9E> ,PROBLEMS ,T ,3TA9 ,SIMPLE
      ,FRAC;NS & ,MIX$ ,NUMB]S
_4 ,PROBLEMS 3TA9+ SIMPLE FRAC;NS &_/OR
  MIX$ NUMB]S WRITT5 9 HORIZONTAL =MAT 9
  PR9T >E WRITT5 HORIZONTALLY 9 BRL4
_4 ,ALL O!R RULES = L9E> PROBLEMS APPLY4

      ,SPATIALLY ,ALIGN$ ,PROBLEMS ,T
      ,3TA9 ,SIMPLE ,FRAC;NS & ,MIX$
      ,NUMB]S
_4 ,! FRAC;N 9DICATORS & FRAC;N L9ES M/
  2 V]TICALLY ALIGN$ "<,RULE ,,XXIV1
  ^S#AGH;E">4
_4 ,! NUM]ATORS >E "R JU/IFI$1 & !
  D5OM9ATORS >E LEFT JU/IFI$ "<,RULE
  ,,XXIV1 ^S#AGH;E">4                #CJ
               ,*APT] #E          B#E4AI
_4 ,ALL O!R RULES = SPATIALLY ALIGN$
  PROBLEMS APPLY4

      ,V>IABLES
_4 ,V>IABLES REPRES5T AN UN"KN NUMB]4
_4 ,WRITE A V>IABLE 9 AN EQUA;N OR
  EXPRES.N :]"E X O3URS4
-----------------------------------#E4BJ

      ,^W ,PROBLEMS ,T ,3TA9 ,SIMPLE
      ,FRAC;NS & ,MIX$ ,NUMB]S
_4 ,SIMPLE FRAC;NS & MIX$ NUMB]S 9 ^W
  PROBLEMS >E WRITT5 9 ,NEME? ,CODE
  "<,,BANA ,GUID.E1 #BJAH1 ;P4 #A">4 ,9
  ADDI;N1 ,NEME? ,CODE SWIT* 9DICATORS
  >E US$4
_4 ,! ,NEME? ,CODE SWIT* 9DICATORS & MA?
  MAT]IAL >E KEPT ON ! SAME BRL L9E :5
  POSSIBLE4

      ,"W%EETS ,T ,3TA9 ,AN ,EXAMPLE
      ,PROBLEM
_4 ,L1VE A BLANK L9E ABV & 2L ! EXAMPLE
  "<.1,BRL .1,=MATS1 #AJ4H4C">4      #CA
               ,*APT] #E          A#E4BJ
_4 ,TRANSCRIBE ! EXAMPLE PROBLEM L ! O!R
  PROBLEMS "<.1,BRL .1,=MATS1 #AJ4H4D">4
_4 ,:5 A COLON FOLL[S ! ^W 8EXAMPLE0 D N
  BOLD1 "UL9E1 OR ITALICIZE X "<.1,BRL
  .1,=MATS1 #AJ4H4B">4
_4 ,:5 NO PUNCTUA;N FOLL[S ! ^W
  8EXAMPLE01 FOLL[ PR9T =MATT+ "<.1,BRL
  .1,=MATS1 #AJ4H4A">4
-----------------------------------#E4BA

    ,*APT] #E3 ,ANSW] ,KEY
      ,ANSW] #E4A
#A4 FIVE OV] EI<T @.<DIAGONAL FRAC;N
  L9E@.>
#B4 ?REE OV] SIXTE5
#C4 "O OV] FIFTE5 @.<DIAGONAL FRAC;N
  L9E@.>
#D4 TW5TY-"O OV] "O HUNDR$ @.<DIAGONAL
  FRAC;N L9E@.>
#E4 ?IRTY-FIVE OV] "O HUNDR$ EQUALS
  L;G-DA%
#F4 ?REE OV] F\R EQUALS N9E OV] TWELVE
#G4 "O OV] ?REE PLUS "O OV] ?REE EQUALS
  ;8                                 #CB
               ,*APT] #E          A#E4BA
#H4 EI<T OV] TW5TY IS LESS ?AN ?REE OV]
  SEV5
#I4 FIVE OV] SIX M9US "O OV] SIX EQUALS
  L;G-DA% @.<,BO? FRAC;NS USE DIAGONAL
  FRAC;N L9ES@.>
      @.<,9 PROBLEM #AJ1 HYPH5S >E US$
    TO KEEP ! NUM]ATOR & D5OM9ATOR TGR
    ": AN OP]ATOR IS 9CLUD$ )9 !
    FRAC;N4@.>
#AJ4 "O OV] ?REE "TS N9E OV] SIXTE5
  EQUALS "O-"TS-N9E OV] ?REE-"TS-SIXTE5
#AA4 ;8 PLUS "O OV] TWO EQUALS FIVE OV]
  SEV5 @.<,BO? FRAC;NS USE DIAGONAL
  FRAC;N L9ES@.>
#AB4 FIVE OV] ELEV5 PLUS ?REE OV] ?IRTE5
  EQUALS L;G-DA%
#AC4 "O OV] "O HUNDR$ EQUALS PO9T Z]O "O
-----------------------------------#E4BB

      ,ANSW] #E4A "<3T9U$"> _%
#1_4 ?1_/9#
#2_4 ?11/70#
#3_4 ?12_/5#
#4_4 ?1/3#                           #CC
               ,*APT] #E          A#E4BB
#5_4 ?4/19#
#6_4 ?8/100#
#7_4 ?2_/2#
#8_4 #1_/3
#9_4 #3_/7
#10_4 #13_/14 _:

      ,ANSW] #E4B
#A4 ?REE & "O OV] TWO
#B4 TWELVE & FIVE OV] EI<T @.<DIAGONAL
  FRAC;N L9E@.>
#C4 FIVE & SIX OV] T5 @.<DIAGONAL FRAC;N
  L9E@.>
#D4 ?REE HUNDR$ EI<TY-F\R & TWO OV]
  ELEV5 @.<DIAGONAL FRAC;N L9E@.>
#E4 "O & ?REE OV] FIVE
#F4 "O & "O OV] ?REE M9US "O OV] TWO
#G4 FIFTE5 & "O OV] F\R PLUS SEV5 & ?REE
  OV] EI<T
#H4 ?REE PLUS TWO & N9E OV] T5
  @.<DIAGONAL FRAC;N L9E@.>
#I4 "O ?\S& @.<COMMA@.> FIVE HUNDR$
  SIXTY-?REE & ?REE OV] SEV5 M9US FIVE
  HUNDR$ EI<TY-TWO & "O OV] ?REE     #CD
               ,*APT] #E          B#E4BB
#AJ4 "O OV] TWO PLUS "O & TWO OV] ?REE
-----------------------------------#E4BC

      ,ANSW] #E4B "<3T9U$"> _%
#1_4 #2_?1_/7_#
#2_4 #4_?1/3_#
#3_4 #8_?1_/2_#
#4_4 #1_?3/16_#
#5_4 #3_?4/7_#+1_?1/7_# .K ----
#6_4 =-?1/4# .K #2_?1/2_#
#7_4 #4_?2_/3_#-3_?1_/3_# .K =
#8_4 #12_?5/9_#-?2/=# .K #12_?3/9_#
  .K #12_?1/3_# _:

      ,ANSW] #E4C
      @.<,! FOLL[+ #D PROBLEMS >E ALL
    SPATIAL EXPRES.NS4@.>
#A4 SEV5 OV] TWELVE PLUS ?REE OV] TWELVE
#B4 ELEV5 OV] FIFTE5 M9US SIX OV] FIFTE5
#C4 EI<T & F\R OV] TW5TY-F\R M9US ?REE &
  ELEV5 OV] SIX
#D4 EI<T & SEV5TE5 OV] TW5TY PLUS F\RTE5
  & ?REE OV] FIVE @.<,BO? FRAC;NS USE
  DIAGONAL FRAC;N L9ES@.>            #CE
               ,*APT] #E           #E4BD

      ,ANSW] #E4C "<3T9U$"> _%

  ?3/5#     ?8/9#
 -?1/2#    -?2/9#
33333333  33333333

  ?3/4 #    ?16/100#
 +?1/16#   -? 1/10 #
333333333 33333333333

  3_?5/7_#    16_?1/3_#
 +1_?1/7_#   - 2_?1/4_#
33333333333 333333333333

_:

      ,ANSW] #E4D
#D4 ,MR4 ,/EFANIK IS ORD]+ PIZZA = ! #AH
  /UD5TS 9 8 HOMEROOM4 ,IF HE E/IMATES T
  EA* /UD5T W EAT _% ?1/4# _: ( A PIZZA1
  H[ _M PIZZAS %D HE ORD]8

                                     #CF
               ,*APT] #E          A#E4BD
#E4 ,MAYA 0 _% #5 FT4 #2_?1/2_# IN4 _:
  TALL LA/ YE>4 ,%E GREW
  _% #1_?1/4_# IN4 _: 9 ! PA/ YE>4 ,H[
  TALL IS %E N[8
#F4 ,TWO *N WEI< _% #72_?1/3_# LBS4 ,'&
  #86_?2/5_# LBS4 _: ,:AT IS ! DI6];E 2T
  _! WEI<TS8
















                                     #CG