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3.1 Adding and Subtracting Like Fractions 83
CHAPTER 3 ADDING ANDSUBTRACTINGFRACTIONS
3.1 Adding and Subtracting LikeFractions
3.1 Margin Exercises
1. (a) # $
& &
The denominators are the same, so and are # $& & like
fractions.
(b)# $
$ %
The denominators are different, so and are# $$ %
unlike fractions.
(c)( ""
"# "#
The denominators are the same, so and are( """# "#
like fractions.
(d)$ $
) "'
The denominators are different, so and are$ $) "'
unlike fractions.
2. (a) $ " $ " %
) ) ) ) œ œ
œ œ% ƒ % "
) ƒ % #
Lowestterms
(b) #
*
&
*
Add denominator stays the same.numerators. The # & (
* *œ
(c) $ " $ " %
"' "' "' "' œ œ
œ œ% ƒ % "
"' ƒ % %
Lowestterms
(d) $ " % $ " %
"! "! "! "! œ
œ œ œ) ) ƒ # %
"! "! ƒ # &
Lowestterms
3. (a) & " & " % % ƒ # #
' ' ' ' ' ƒ # $ œ œ œ œ
(b) "'
"!
(
"!
"' ( *
"! "!œ
(c) "& & "& & "! "
$ $ $ $ $ œ œ œ $
(d) #&
$#
'
$#
#& ' "*
$# $#œ
3.1 Section Exercises
1. $ # $ # &
) ) ) ) œ œ
3. # $ # $ &
' ' ' ' œ œ
5. " " " " #
% % % % œ œ
œ œ# ƒ # "
% ƒ # #
Lowestterms
7. *
"!
$
"!
* $ * $ "# "# ƒ # ' "
"! "! "! "! "! ƒ # & & œ œ œ œ œ "
9. #
*
"
*
# " # " $ $ ƒ $ "
* * * * * ƒ $ $ œ œ œ œ
11. ' % $ ' % $ "$
#! #! #! #! #! œ œ
13. % # & % # & ""
"& "& "& "& "& œ œ
15. $ ( # $ ( # "#
) ) ) ) ) œ œ
œ œ œ ""# ƒ % $ "
) ƒ % # #
17. # ) "# # ) "# ##
&% &% &% &% &% œ œ
œ œ## ƒ # ""
&% ƒ # #(
84 Chapter 3 Adding and Subtracting Fractions
19. ( % ( % $
) ) ) ) œ œ
21. "! % "! % '
"" "" "" "" œ œ
23. * $ * $ ' ' ƒ # $
"! "! "! "! "! ƒ # & œ œ œ œ
25. $"
#"
(
#"
$" ( $" ( #% #% ƒ $ ) "
#" #" #" #" #" ƒ $ ( ( œ œ œ œ œ "
27. #(
%!
"*
%!
#( "* #( "* ) ) ƒ ) "
%! %! %! %! %! ƒ ) & œ œ œ œ
29. %( & %( & %# %# ƒ ' ( "
$' $' $' $' $' ƒ ' ' ' œ œ œ œ œ "
31. ($ ( ($ ( '' '' ƒ ' "" "
'! '! '! '! '! ƒ ' "! "! œ œ œ œ œ "
33. Three steps to add like fractions are:
1. Add the numerators of the fractions to findthe numerator of the sum (the answer).
2. Use the denominator of the fractions asthe denominator of the sum.
3. Write the answer in lowest terms.
35. Add the two amounts to find the total fractionraised.
# & # & (
* * * * œ œ
They raised of their target goal.(*
37. Add the two amounts to find the total fractionsaved.
$ # $ # &
( ( ( ( œ œ
He saved of the amount needed.&(
39. First, add the two fractions of land that werepurchased.
* $ * $ "#
"! "! "! "! œ œ
Next, subtract the fraction of land that was plantedin carrots.
"# ( "# ( & "
"! "! "! "! # œ œ œ
acre is planted in squash."#
3.2 Least Common Multiples3.2 Margin Exercises
1. (a) The multiples of are&
&ß "!ß "&ß #!ß #&ß $!ß $&ß %!ßá .
The multiples of are(b) )
)ß "'ß #%ß $#ß %!ß %)ß &'ßá .
Look at the answers for (a) and (b). is the(c) %!only number found in both lists, so it is the leastcommon multiple.
2. (a) and # &
The multiples of are&
&ß "!ß "&ß #!ßá .
The first multiple of that is divisible by is ,& # "!so the least common multiple of the numbers #and is .& "!
and (b) $ *
The multiples of are*
*ß ")ß #(ß $'ß %&ßá .
The first multiple of that is divisible by is , so* $ *the least common multiple of the numbers and $ *is .*
and (c) ' )
The multiples of are)
)ß "'ß #%ß $#ß %!ß %)ßá .
The first multiple of that is divisible by is ,) ' #%so the least common multiple of the numbers 'and is .) #%
and (d) % (
The multiples of are(
(ß "%ß #"ß #)ß $&ß %#ßá .
The first multiple of that is divisible by is ,( % #)so the least common multiple of the numbers %and is .( #)
3. (a) and "& ")
"& œ $
") œ #
•• •&
$ $
To find the LCM of and , we'll start with the"& ")factors of , ") # • •$ $ "& $ &. Now only has and asfactors and we already have a in the LCM, so we$just need to include a .&
LCM œ # $ $ & œ *!• • •
The least common multiple of and is ."& ") *!
3.2 Least Common Multiples 85
(b) and "# #!
"# œ #
#! œ #
• •• •# $
# &
LCM œ # • • •# $ & œ '!
The least common multiple of and is ."# #! '!
4. and (a) $ '
) &
) œ #
& œ "
• ••# #
&
LCM œ # • • •# # & œ %!
The least common multiple of and is .) & %!
and (b) & "
' "%
' œ #
"% œ #
••$
(
LCM œ # • •$ ( œ %#
The least common multiple of and is .' "% %#
and (c) % & (
* ") #%ß ß
* œ $
") œ #
#% œ #
•• •• • •
$
$ $
# # $
LCM œ # • • • •# # $ $ œ (#
The least common multiple of , , and is .* ") #% (#
5. (a) %ß )ß *
% œ #
) œ # #
* œ $
•• ••
#
#
$
LCM œ # #• • • •# $ $ œ (#
The least common multiple of , , and is .% ) * (#
(b) $ß 'ß )
$ œ "
' œ #
) œ #
••• •
$
$
# #
LCM œ # • • •# # $ œ #%
The least common multiple of , , and is .$ ' ) #%
(c) *ß $'ß %)
* œ $
$' œ #
%) œ #
•• • •• • • •
$
# $ $
# # # $
LCM œ # • • • • •# # # $ $ œ "%%
The least common multiple of , , and is .* $' %) "%%
(d) "&ß #!ß $!ß %!
"& œ $
#! œ #
$! œ #
%! œ #
•• •• •• • •
&
# &
$ &
# # &
LCM œ # • • • •# # $ & œ "#!
The least common multiple of , , , and is"& #! $! %!"#!.
6. least common multiple of and is the(a) The ' "&product of the numbers on the left side.
LCM œ # • •$ & œ $!
(b) The least common multiple of and is the#! $'product of the numbers on the left side.
LCM œ # • • • •# $ $ & œ ")!
7. and (a) $ß 'ß "!
# $ ' "!y
$ $ $ &y
& " " &y y
" " "
The LCM of , , and is$ ' "!
# • •$ & œ $!.
(b) "& and %!
# "& %!y
# "& #!y
# "& "!y
$ "& &y
& & &" "
The LCM of and is"& %!
# • • • •# # $ & œ "#!.
(c) * and #%
# * #%y
# * "#y
# * 'y
$ * $$ $ "y
" "
The LCM of and is* #%
# • • • •# # $ $ œ (#.
86 Chapter 3 Adding and Subtracting Fractions
(d) and )ß #"ß #%
# ) #" #%y
# % #" "#y
# # #" 'y
$ " #" $y
( " ( "y y
" " "
The LCM of , , and is) #" #%
# • • • •# # $ ( œ "').
8. ?
(a) "
% "'œ
Divide by , getting . Now multiply both the"' % %numerator and the denominator by of by ."
% %
" " %
% % "'œ œ
•
•
%
%
?
(b) #
$ "&œ
Divide by , getting . Now multiply both the"& $ &numerator and the denominator by of by .#
$ &
# # "!
$ $ "&œ œ
•
•
&
&
?
(c) (
"' $#œ
Divide by , getting . Now multiply both the$# "' #numerator and the denominator by of by .(
"' #
( ( "%
"' "' $#œ œ
•
•
#
#
?
(d) '
"" $$œ
Divide by , getting . Now multiply both the$$ "" $numerator and the denominator by of by .'
"" $
' ' ")
"" "" $$œ œ
•
•
$
$
3.2 Section Exercises1. and $ '
Multiples of :'
'ß "#ß ")ß #%ß $!ßá
is the first number divisible by . ' $ Ð' ƒ $ œ #Ñ
The least common multiple of and is .$ ' '
3. and $ &
Multiples of :&
&ß "!ß "&ß #!ß #&ßá
is the first number divisible by . "& $ Ð"& ƒ $ œ &Ñ
The least common multiple of and is .$ & "&
5. and % *
Multiples of :*
*ß ")ß #(ß $'ß %&ßá
is the first number divisible by . $' % Ð$' ƒ % œ *Ñ
The least common multiple of and is .% * $'
7. and # (
Multiples of :(
(ß "%ß #"ß #)ß $&ßá
is the first number divisible by . "% # Ð"% ƒ # œ (Ñ
The least common multiple of and is .# ( "%
9. and ' "!
Multiples of :"!
"!ß #!ß $!ß %!ß &!ßá
is the first number divisible by . $! ' Ð$! ƒ ' œ &Ñ
The least common multiple of and is .' "! $!
11. and #! &!
Multiples of :&!
&!ß "!!ß "&!ß #!!ß #&!ßá
is the first number divisible by ."!! #!Ð"!! ƒ #! œ &Ñ
The least common multiple of and is .#! &! "!!
13. %ß "!
# % "!# # &y
& " &y
" "
The LCM of and is% "!
# • •# & œ #!.
15. "#ß #!
# "# #!# ' "!$ $ &y
& " &y
" "
The LCM of and is"# #!
# • • •# $ & œ '!.
3.2 Least Common Multiples 87
17. 'ß *ß "#
# ' * "#y
# $ * 'y y
$ $ * $$ " $ "y y
" " "
The LCM of , , and is' * "#
# • • •# $ $ œ $'.
19. %ß 'ß )ß "!
% œ #
' œ #
) œ #
"! œ #
••• ••
#
$
# #
&
LCM œ # • • • •# # $ & œ "#!
The LCM of , , , and is .% ' ) "! "#!
21. "#ß "&ß ")ß #!
"# œ #
"& œ $
") œ #
#! œ #
• ••• •• •
# $
&
$ $
# &
LCM œ # • • • •# $ $ & œ ")!
The LCM of , , , and is ."# "& ") #! ")!
23. )ß "!ß "#ß "'ß $'
) œ #
"! œ #
"# œ #
"' œ #
$' œ #
• ••• •• • •• • •
# #
&
# $
# # #
# $ $
LCM œ # • • • • • •# # # $ $ & œ (#!
The LCM of , , , , and is .) "! "# "' $' (#!
25. ?#
$ #%œ #% ƒ $ œ )
# # "'
$ $ #%œ œ
•
•
)
)
27. ?$
% #%œ #% ƒ % œ '
$ $ ")
% % #%œ œ
•
•
'
'
29. ?&
' #%œ #% ƒ ' œ %
& & #!
' ' #%œ œ
•
•
%
%
31. ?"
# 'œ ' ƒ # œ $
" " $
# # 'œ œ
•
•
$
$
33. ?$
% "'œ "' ƒ % œ %
$ $ "#
% % "'œ œ
•
•
%
%
35. ?(
) $#œ $# ƒ ) œ %
( ( #)
) ) $#œ œ
•
•
%
%
37. ?$
"' '%œ '% ƒ "' œ %
$ $ "#
"' "' '%œ œ
•
•
%
%
39. ?)
& #!œ #! ƒ & œ %
) ) $#
& & #!œ œ
•
•
%
%
41. ?*
( &'œ &' ƒ ( œ )
* * (#
( ( &'œ œ
•
•
)
)
43. ?(
% %)œ %) ƒ % œ "#
( ( )%
% % %)œ œ
•
•
"#
"#
45. ?)
"" "$#œ "$# ƒ "" œ "#
) ) *'
"" "" "$#œ œ
•
•
"#
"#
47. ?$
"' "%%œ "%% ƒ "' œ *
$ $ #(
"' "' "%%œ œ
•
•
*
*
49. Answers will vary. A sample answer follows:It probably depends on how large the numbers are.If the numbers are small, the method usingmultiples of the largest number seems best. If thenumbers are larger, or if there are more than twonumbers, then the factorization method will bebetter.
51. #& $)
%!! ")!!ß
Multiples of :")!!
")!!ß $'!!ß &%!!ßá
is the first number divisible by .$'!! %!!Ð$'!! ƒ %!! œ *Ñ
The LCM of the denominators is .$'!!
88 Chapter 3 Adding and Subtracting Fractions
53. "!* #$
"&"# $*#ß
# "&"# $*## (&' "*'# $() *)$ ")* %*y
$ '$ %*y
$ #" %*y
( ( %*( " (y
" "
The LCM of the denominators is
# • • • • • • •# # $ $ $ ( ( œ "! &)%Þ,
55. Fractions with the same denominators are likefractions and fractions with different denominatorsare unlike fractions.
56. To subtract like fractions, first subtract thenumerators to find the numerator of thedifference. Write the denominator of the likefractions as the of the difference.denominatorFinally, write in terms.lowest
57. The least common multiple (LCM) of twonumbers is the whole number divisible bysmallestboth of those numbers.
58. The smallest number in both lists is , so is%! %!the least common multiple of and .) "!
59. &ß (ß "%ß "!
& œ "
( œ "
"% œ #
"! œ #
••••
&
(
(
&
LCM œ # • •& ( œ (!
The LCM of , , , and is .& ( "% "! (!
60. #&ß ")ß $!ß &
#& œ &
") œ #
$! œ #
& œ "
•• •• ••
&
$ $
$ &
&
LCM œ # • • • •$ $ & & œ %&!
The LCM of , , , and is .#& ") $! "& %&!
61. ) œ #
$ œ "
& œ "
% œ #
"! œ #
• •••••
# #
$
&
#
&
LCM œ # • • • •# # $ & œ "#!
The LCM of , , , , and is . is a) $ & % "! "#! #%!common multiple, but it is not the commonleastmultiple.
62. The least common multiple can be no smaller thanthe largest number in a group and the number"('! && Ð"('! ƒ && œ $#ÑÞ is a multiple of
3.3 Adding and Subtracting UnlikeFractions
3.3 Margin Exercises
1. LCD(a)
Step 1
Step 2
" $
# ) Ð œ )Ñ
" " %
# # )œ œ
" $ % $ % $ (
# ) ) ) ) ) œ œ œ
•
•
%
%
LCD(b)
Step 1
Step 2
$ "
% ) Ð œ )Ñ
$ $ '
% % )œ œ
$ " ' " ' " (
% ) ) ) ) ) œ œ œ
•
•
#
#
LCD(c)
Step 1
Step 2
$ $
& "! Ð œ "!Ñ
$ $ '
& & # "!œ œ
$ $ ' $ ' $ *
& "! "! "! "! "! œ œ œ
•
•
#
(d) Step 1
Step 2
" &
"# ' Ð œ "#Ñ
& & "!
' ' "#œ œ
" & " "! " "! ""
"# ' "# "# "# "# œ œ œ
LCD
•
•
#
#
2. (a)
Step 1
Step 2
Step 3
$ " $ #
"! & "! "! œ
$ # $ # &
"! "! "! "! œ œ
& "
"! #œ
(b)
Step 1
Step 2
& " "& )
) $ #% #% œ
"& ) "& ) #$
#% #% #% #% œ œ
(c) Step 1
Step 2
Step 3
" " " $ "! &
"! $ ' $! $! $! œ
$ "! & $ "! & ")
$! $! $! $! $! œ œ
") $
$! &œ
3.3 Adding and Subtracting Unlike Fractions 89
3. (a) & & "&
) ) #%œ œ
œ œ" " #
"# "# #%"(
#%
•
••
•
$
$#
#
(b) ( (
"' "'œ
œ œ" " %
% % "'""
"'
•
•
%
%
4. (a)
Step 1
Step 2
& " & #
) % ) ) œ
& # & # $
) ) ) ) œ œ
(b)
Step 1
Step 2
% $ "' "&
& % #! #! œ
"' "& "' "& "
#! #! #! #! œ œ
5. (a) ( ( #"
) ) #%œ œ
œ œ# # "'
$ $ #%&
#%
•
••
•
$
$)
)
(b) & & "!
' ' "#œ œ
œ" "
"# "#* $
"# %œ
•
•
#
#
3.3 Section Exercises
1. $ " ' "
% ) ) ) œ
œ' "
)
œ(
)
LCD is 8
3. # # ' #
$ * * * œ
œ' #
*
œ)
*
LCD is 9
5. * $ * '
#! "! #! #! œ
œ* '
#!
œ œ"& $
#! %
LCD is 20
In lowest terms
7. $ $ #% "&
& ) %! %! œ
œ#% "&
%!
œ$*
%!
LCD is 40
9. # & ) "&
* "# $' $' œ
œ) "&
$'
œ#$
$'
LCD is 36
11. " $ & *
$ & "& "& œ
œ& *
"&
œ"%
"&
LCD is 15
13. " # " * ) "#
% * $ $' $' $' œ
œ* ) "#
$'
œ#*
$'
LCD is 36
15. $ # $ ' ) $
"! & #! #! #! #! œ
œ' ) $
#!
œ"(
#!
LCD is 20
17. % " " ) & "!
"& ' $ $! $! $! œ
œ) & "!
$!
œ#$
$!
LCD is 30
19. " " #
% % )œ œ
œ" "
) )$
)
•
•
#
#LCD is 8
21. & & #!
"# "# %)œ œ
œ œ" " $
"' "' %)#$
%)
•
••
•
%
%$
$
LCD is 48
23. & " & #
' $ ' ' œ
œ& #
'
œ œ$ "
' #
LCD is 6
In lowest terms
90 Chapter 3 Adding and Subtracting Fractions
25. # " % "
$ ' ' ' œ
œ% "
'
œ œ$ "
' #
LCD is 6
In lowest terms
27. # " "! $
$ & "& "& œ
œ"! $
"&
œ(
"&
LCD is 15
29. & " & $
"# % "# "# œ
œ& $
"#
œ œ# "
"# '
LCD is 12
In lowest terms
31. ) ( %! #"
* "& %& %& œ
œ%! #"
%&
œ"*
%&
LCD is 45
33. ( ( $&
) ) %!œ œ
œ œ% % $#
& & %!$
%!
•
••
•
&
&)
)
LCD is 40
35. & & #!
"# "# %)œ œ
œ œ" " $
"' "' %)"(
%)
•
••
•
%
%$
$
LCD is 48
37. The widest blades have widths " and ". The" $%
difference is
" œ œ œ$ % $ % $ "
% % % % % inch.
39. Subtract the area used for general admission fromthe total area.
% % $#
& & %!œ œ
œ œ$ $ "&
) ) %!"(
%!
•
••
•
)
)&
&
LCD is 40
The fraction of the total area used for reservedseating is ."(
%!
41. Add the fractions to find the total length." " # & "! "'
) % & %! %! %! œ
œ& "! "'
%!
œ$"
%!
LCD is 40
The total length of the screw is inch.$"%!
43. First add to find the amount used." $ ) * ) * "(
$ ) #% #% #% #% œ œ œ
Then subtract to find the amount remaining.$ "( ") "( ") "( "
% #% #% #% #% #% œ œ œ
The fraction of the tank of fuel that remains is ."#%
45. Answers will vary. A sample answer follows:You cannot add or subtract until all the fractionalpieces are the same size. For example, halves arelarger than fourths, so you cannot add until" "
# %
you rewrite as ." ## %
47. Add the time spent in class and study." " % $ % $ (
' ) #% #% #% #% œ œ œ
of the student's day was spent in class and(#%
study.
49. One way to compare fractions accurately is torewrite each fraction with a common denominator.
" ) " % " $ " # (
$ #% ' #% ) #% "# #% #%œ ß œ ß œ ß œ ß
The fraction with the largest numerator is thelargest fraction. This is or , which is "Work) "
#% $
and Travel."
To find the number of hours, multiply.
"
$ y• •#% œ œ œ )
" )
$ " ""
)#%y
hours were spent on work and travel. Note that)this is just the numerator in the equivalent fraction)#% .
Now add the fractions for "Work and Travel" and"Class."
" " # " # " $ "
$ ' ' ' ' ' # œ œ œ œ
The fraction of the day spent on these activities is"# Þ
3.4 Adding and Subtracting Mixed Numbers 91
51. First add the lengths on either side of the hole.$ $ $ $ ' $
) ) ) ) % œ œ œ
Then subtract this length from the total length."& $ "& "# "& "# $
"' % "' "' "' "' œ œ œ
The diameter is inch.$"'
3.4 Adding and Subtracting MixedNumbers
3.4 Margin Exercises
1. Whole number part
Remainder
(a) *
#% Ã
# *)
" Ã
* "
# #œ %
Whole number part
Remainder
(b) )
$# Ã
$ )'
# Ã
) #
$ $œ #
4 (c) $
%
% • % œ "'"' $ œ "*
4$ "*
% %œ
(d) $(
)
$ ) œ #%#% ( œ $"
•
$ œ( $"
) )
2. (a) Estimate: Exact:
( ' œ '
#
*
( (
) )
# œ #" #
% )
)*
)
) œ ) " œ ** " "
) ) )
(b) #& "#$ $
& "!
Estimate: #' "# œ $)
Exact:
#& "# œ #& "# œ $($ $ ' $ *
& "! "! "! "!
(c) Estimate: Exact:
& % œ %
$
#
( (
* *
# œ ## '
$ *
#"
*
3. (a) Estimate: Exact:
"! * œ *
)
")
$ $
% %
( œ (" #
# %
"'&
%
"' œ "' œ "' " œ "(& & " "
% % % %
(b) Estimate: Exact:
"' "& œ "&
"$
#*
% "#
& "&
"# œ "## "!
$ "&
#(##
"&
#( œ #( œ #( " œ #)## ## ( (
"& "& "& "&
4. (a) Estimate: Exact:
( ( œ (
&
#
" #
$ '
% œ %& &
' '
Regroup:
( œ ( œ ' " œ ' œ '# # # ' # )
' ' ' ' ' '
')
'
%&
'
# œ #$ "
' #
(b) Estimate: Exact:
& % œ %
$
#
& "!
) "'
# œ #"& "&
"' "'
Regroup:
% œ % œ $ " œ $ œ $"! "! "! "' "! #'
"' "' "' "' "' "'
continued
92 Chapter 3 Adding and Subtracting Fractions
$#'
"'
#"&
"'
"""
"'
(c) Estimate: Exact:
"& "&
'
* '
%
*
Regroup:
"& œ "% " œ "% œ "%* *
* *
"%*
*
'%
*
)&
*
5. (a) $ œ œ$ #( #(
) ) )
# œ œ" & #!
# # )%( (
) )œ &
(b) ' œ œ$ #( )"
% % "#
% œ œ# "% &'
$ $ "##& "
"# "#œ #
3.4 Section Exercises1. Estimate: Exact:
' & œ &
$
*
" $
# '
$ œ $" #
$ '
)&
'
3. Estimate: Exact:
( ( œ (
%
""
" #
$ '
% œ %" "
' '
"" œ ""$ "
' #
5. Estimate: Exact:
" œ
%
&
& "&
) #%
$ œ $( "%
"# #%
$#*
#%
$ œ $ œ $ " œ %#* #* & &
#% #% #% #%
7. Estimate: Exact:
#& #%
"*
%%
&
'
")&
'
%#"!
'
%# œ %# " œ %$"! % #
' ' $
9. Estimate: Exact:
$% $$ œ $$
"*
&$
$ '
& "!
") œ ")" &
# "!
&"""
"!
&" œ &" " œ &#"" " "
"! "! "!
11. Estimate: Exact:
#$ ## œ ##
"&
$)
$ #"
% #)
"& œ "&$ "#
( #)
$($$
#)
$( œ $( " œ $)$$ & &
#) #) #)
13. Estimate: Exact:
"$ "# œ "#
"*
"&
%(
) "'
"& $!
") œ ")$ ")
& $!
"% œ "%( #"
"! $!
%%&&
$!
%% œ %% " œ %&&& #& &
$! $! '
3.4 Adding and Subtracting Mixed Numbers 93
15. Estimate: Exact:
"& "% œ "%
"#
$
( (
) )
"# œ "#" #
% )
#&
)
17. Estimate: Exact:
"$ "# œ "#
"
"#
# "!
$ "&
" œ "" $
& "&
""(
"&
19. Estimate: Exact:
#) #) œ #)
'
##
$ *
"! $!
' œ '" #
"& $!
##(
$!
21. Estimate: Exact:
"( "(
(
"! '
&
)
Regroup:
"( œ "' " œ "' )
)
"')
)
'&
)
"!$
)
23. Estimate: Exact:
"* ") œ ")
'
"$
$ "&
% #!
& œ &% "'
& #!
Regroup:
") œ "( " œ "( œ "("& "& #! "& $&
#! #! #! #! #!
"($&
#!
&"'
#!
"#"*
#!
25. Estimate: Exact:
#! "* œ "*
"#
)
# )
$ "#
"" œ ""$ *
% "#
Regroup:
"* œ ") " œ ") œ ")) ) "# ) #!
"# "# "# "# "#
")#!
"#
""*
"#
(""
"#
27. ( œ œ& '" '"
) ) )
" œ œ" $ "#
# # )($ "
) )œ *
29. % œ œ# "% #)
$ $ '
' œ œ& %" %"
' ' ''* $ "
' ' #œ "" œ ""
31. # œ œ# ) "'
$ $ '
" œ œ" ( (
' ' '#$ &
' 'œ $
33. $ œ œ" "$ $*
% % "#
$ œ œ# "" %%
$ $ "#)$ ""
"# "#œ '
35. " œ œ$ "" ""
) ) )
' œ œ$ #( &%
% % )'& "
) )œ )
37. $ œ œ" ( #"
# # '
# œ œ# ) "'
$ $ '&
'
94 Chapter 3 Adding and Subtracting Fractions
39. ) œ œ$ $& (!
% % )
& œ œ( %( %(
) ) )#$ (
) )œ #
41. ( œ œ" #* )(
% % "#
% œ œ# "% &'
$ $ "#$" (
"# "#œ #
43. * œ œ" %' ")%
& & #!
$ œ œ$ "& (&
% % #!"!* *
#! #!œ &
45. ' œ œ$ %& "$&
( ( #"
# œ œ# ) &'
$ $ #"(* "'
#" #"œ $
47. Find the least common denominator. Change thefraction parts so that they have the samedenominator. Add the fraction parts. Add thewhole number parts. Write the answer as a mixednumber.
49. Subtract from to determine the"% #&" $# %
difference.
feetEstimate: #' "& œ ""
Exact: #& œ œ$ "!$ "!$
% % %
"% œ œ" #* &)
# # %%& "
% %œ ""
The current world record is feet taller than the"""%
first world record.
51. The longest wrench is and the second to##$%
" shortest wrench is . Subtract to find the")&
)"
difference.
inchesEstimate: #$ "* œ %
Exact: ## œ ##$ '
% )
") œ ")& &
) )
%"
)
The longest wrench is inches longer than the%")
second to shortest wrench.
53. The three longest wrenches measure , ,## ##$ "% %
" "
and ". Add to find the total length.#"
inchesEstimate: #$ ## #" œ ''
Exact: ##$
%
##"
% #"
'& œ '& " œ ''%
%
The total length of the three longest wrenches is'' inches.
55. The largest hose clamp is and the smallest#$%
" hose clamp is . Subtract to find the difference.*
"'"
inchesEstimate: $ " œ #
Exact: # œ #$ "#
% "'
œ* *
"' "'
#$
"'
The largest hose clamp is inches larger than# $"'
the smallest hose clamp.
57. Add the four measurements.
ftEstimate: "' "* #% $" œ *!
Exact: "& œ "&" #
# %
") œ ")$ $
% %
#% œ #%" "
% %
$! œ $!" #
# %
)( œ )( # œ )*)
%
Andre needs feet of fencing to go around the)*garden.
3.4 Adding and Subtracting Mixed Numbers 95
59. Add the lengths of the four sides.
in.Estimate: #% $& #% $& œ "")
Exact: #$ œ #$$ $
% %
$% œ $%" #
# %
#$ œ #$$ $
% %
$% œ $%" #
# %
""% œ ""% # œ ""'"! # "
% % #
The craftsperson needs inches of lead""'"#
stripping.
61. Subtract the amounts used from the total amount.
Estimate: "!! "! "% * "* "# "! "% œ "# gallons
Exact: Add up the amounts used.
"! œ "!" #
% )
"$ œ "$" %
# )
) œ )( (
) )
") œ ")$ '
% )
"# œ "#$ $
) )
* œ *" %
# )
"% œ "%" "
) )
)% œ )% $ œ )(#( $ $
) ) )
Now subtract from 100.
"!! œ **)
)
)( œ )($ $
) )
"#&
)
There are gallons of water remaining."#&)
63. Subtract the known lengths from the total length.
ftEstimate: &#( "!) "&" "$* œ "#*
Exact: Add up the lengths of the known sides.
"!( œ "!(# "'
$ #%
"&! œ "&!$ ")
% #%
"$) œ "$)& "&
) #%
$*& œ $*& # œ $*(%* " "
#% #% #%
Now subtract from .&#( "#%
&#("
#%
$*("
#%"$!
The length of the fourth side is feet."$!
65. Add the weights.
tonsEstimate: &* #% "( #* &) œ ")(
Exact:
&) œ &)" "#
# #%
#$ œ #$& "&
) #%
"' œ "'& #!
' #%
#* œ #*" '
% #%
&) œ &)" )
$ #%
")% œ ")% # œ ")''" "$ "$
#% #% #%
The total weight is tons.")'"$#%
67. First add the two given portions of the line.
#$
)
#$
)
%'
)
Then subtract to find the unknown length.
* œ * œ )( ( #$
"' "' "'
% œ % œ %' "# "#
) "' "'
%""
"'
The unknown length is inches.%"""'
96 Chapter 3 Adding and Subtracting Fractions
69. Add the length of the sections at each end.
' œ '" #
% )
" œ "( (
) )
( œ ( " œ )* " "
) ) )
Then subtract this total from the total length of thearrow.
#* œ #*" %
# )
) œ )" "
) )
#"$
)
The unknown length is inches.#"$)
71. ?
(a) &
* &%œ &% ƒ * œ '
& & $!
* * &%œ œ
•
•
'
'
(b)(
"# %)œ %) ƒ "# œ %
( ( #)
"# "# %)œ œ
?
•
•
%
%
(c)&
) %!œ %! ƒ ) œ &
& & #&
) ) %!œ œ
?
•
•
&
&
(d)""
& "#!œ "#! ƒ & œ #%
"" "" #'%
& & "#!œ œ
?
•
•
#%
#%
72. When rewriting unlike fractions as like fractionswith the least common multiple as a denominator,the new denominator is called the least commondenominator, or LCD.
73. (a) & " "& ) "& ) #$
) $ #% #% #% #% œ œ œ
(b) "* & &( #& &( #& $# )
#! "# '! '! '! '! "& œ œ œ œ
(c) ( #)
"# %)œ
$ *
"' %)œ
œ$ '
#% %)%$
%)
(d) ' ")
( #"œ
œ# "%
$ #"%
#"
74. A common method for adding or subtractingmixed numbers is to add or subtract the fractionparts and then add or subtract the whole numberparts.
75. Another method for adding or subtracting mixednumbers is to first change the mixed numbers toimproper fractions. After adding or subtracting,write the answer as a mixed number in lowestterms. This method is difficult to use if the mixednumbers are .large
76. (a) First Method:
% œ %& &
) )
$ œ $$ '
% )
( œ ( " œ )"" $ $
) ) )
Second Method: % œ œ& $( $(
) ) )
$ œ œ$ "& $!
% % )'( $
) )œ )
(b) First Method: "# œ "# œ ""# "' &'
& %! %!
) œ ) œ )( $& $&
) %! %!
$#"
%!
Second Method: "# œ œ# '# %*'
& & %!
) œ œ( (" $&&
) ) %!"%" #"
%! %!œ $
Both methods give the same answer.Preferences will vary. A sample answer follows:When a problem requires regrouping, it is easier tochange all the numbers to improper fractions.Otherwise, adding the whole numbers and thenadding the fractions seems easier.
3.5 Order Relations and the Order of Operations 97
3.5 Order Relations and the Order ofOperations
3.5 Margin Exercises1. (a)–(c)
2. number line,(a) is to the left of on the " œ "& "% %
so is less than ." &%
" &
%
(b) is to the right of on the) # $ "$ $ # #œ # œ "
number line, so is greater than .) $$ #
) $
$ #
number line, so is(c) is to the left of on the ! " !less than ."
! "
(d) is to the right of on the"( " )) ) %œ # œ #
number line, so is greater than ."( )) %
"( )
) %
3.
, so .
(a) ( $
) %$ '
% )œ
( ' ( $
) ) ) %
**
**LCD is 8
(b) "$ "&
) *"$ ""( "& "#!
) (# * (#œ œ
""( "#! "$ "&
(# (# ) *
**
**
and
, so .
LCD is 72
and
, so .
(c) * (
% $* #( ( #)
% "# $ "#œ œ
#( #) * (
"# "# % $
**
**LCD is 12
and
, so .
(d) * "%
"! "&* #( "% #)
"! $! "& $!œ œ
#( #) * "%
$! $! "! "&
**
**LCD is 30
4. (a) Œ" "
# #œ
%
• • •" " " "
# # # "'œ
(b) Œ$ $
% %œ
#
•$ *
% "'œ
(c) Œ"
#
œ" y y
# y y
œ"
")
$
• • • • •
• • • •
• • • •
Œ Œ Œ# " " " # #
$ # # # $ $œ
" " # #
# # $ $
#
" "
" "
(d) Œ"
&
œ" y y
&y y
œ"
*
#
• • • •
• • •
• • •
Œ Œ Œ& " " & &
$ & & $ $œ
" & &
& $ $
#
" "
" "
5. (a) & $ # & $ #
* % $ * $ œ
y
%y
y
y
œ & "
* #
œ "! *
") ")
œ"
")
Œ Œ"
#
"
"
LCD is 18
(b) $ # $ #
% $ % $œ
y
y
œ$
%y
y
œ$
"!
Œ Œ• •
•
$ $
& &
#
&
"
"
#
"
(c) ( # " ( # "
) $ # ) $ # œ
y
y
œ ( "
"# %
œ ( $
"# "#
œ œ% "
"# $
Œ Œ Œ#
%
"
•"
#
LCD is 12
98 Chapter 3 Adding and Subtracting Fractions
(d) ˆ ‰ Œ&'
#
%$
œ ƒ& %
' $
œ&
'
œ& y
' y
œ#&
%)
•
•
••
• •
• •
&
'
& $
' %
& $
' %
"
#
3.5 Section Exercises1.–12.
13.
, so .
" $
# )" %
# )œ
% $ " $
) ) # )
**
**LCD is 8
15.
, so .
& ""
' "#& "!
' "#œ
"! "" & ""
"# "# ' "#
**
**LCD is 12
17.
and
, so .
& $
"# )& "! $ *
"# #% ) #%œ œ
"! * & $
#% #% "# )
**
**LCD is 24
19.
and
, so .
( ""
"# ")( #" "" ##
"# $' ") $'œ œ
#" ## ( ""
$' $' "# ")
**
**LCD is 36
21.
, so .
"" &
") *& "!
* ")œ
"" "! "" &
") ") ") *
**
**LCD is 18
23.
and
, so .
$( "$
&! #!$( (% "$ '&
&! "!! #! "!!œ œ
(% '& $( "$
"!! "!! &! #!
**
**LCD is 100
25. Œ" "
$ $œ
#
••
•
" " " "
$ $ $ *œ œ
27. Œ& & #&
) ) '%œ
#
••
•
& & &
) ) )œ œ
29. Œ$ $ *
% % "'œ
#
•$
%œ
31. Œ% % '%
& & "#&œ
$
• •% %
& &œ
33. Œ$ $ $ $ $ )" "
# # # # # "' "'œ œ œ &
%
• • •
35. Œ$ $ $ $ $ )"
% % % % % #&'œ œ
%
• • •
37. Answers will vary. A sample answer follows:A number line is a horizontal line with a range ofnumbers placed on it. The lowest number is on theleft and the greatest number is on the right. It canbe used to compare the size or value of numbers.
39. # %Ð$Ñ œ "' %Ð$Ñ
œ "' "#
œ %
%
41. $
œ "# '
$œ "# #
œ "!
• •# œ $ % ' '
$ $#
43. Œ"
#
œ"
%
œ"
%
#
• • •
•
•
% œ %" "
# #
%
%
"œ "
45. Œ$
%
œ$ y
% y
œ$
"'
#
• • •
• •
• •
Œ Œ" $ $ "
$ % % $œ
$ "
% $
"
"
3.5 Order Relations and the Order of Operations 99
47. Œ%
&
œ% %y y y y
&y y y y
œ%
*
#
• • • •
• • •
• • •
Œ Œ Œ& % % & &
' & & ' 'œ
& &
& ' '
#
# # " "
" " $ $
49. ' œ '# " # "
$ # $ #
œ'yy
y y
$y y y y
œ"
$
Œ Œ Œ Œ# $
• • •
• • • • •
• • • •
# " "
$ # #
# # " " "
$ # # #
#"
" "
" " " "
51.
$ " # $ $ " # $
& $ & % & $ & œ
y
y
y
%y
œ " $
& "!
œ # $
"! "!
œ œ& "
"! #
Œ Œ Œ Œ"
"
"
#
LCD is 10
53. " " $ " "
# # ) # # œ
œ # " $
% % )
œ $ $
% )
œ œ' $ $
) ) )
Π#
•" $
# )
LCD is 8
55. Œ" "
$ '
œ$
'
œ$y
'y
œ"
%
• •
•
•
•
" # " "
# ' ' #œ
"
#
"
#
Œ
"
#
57. * # " * ) " * *
) $ "# ) "# "# ) "#ƒ œ ƒ œ ƒ
œ*y
)y y
œ œ "$ "
# #
Œ Œ"
# "
$
•"#y
*
59. Œ Œ( $ $ ( ' $
) % # ) ) # ƒ œ ƒ
œ ƒ" $
) #
œ"
)y
y
œ"
"#
%
"
•#
$
61. $ " "
) % #
œ$
)y y
y
y
y
œ $
Œ • •
• •
$# $ " # $#
$ ) % % $œ
$
% $
Œ
" "
"
"
)"
$#y
63. Œ Œ Œ Œ$ " " % $ $ " %
% # ' $ % ' ' $ ƒ œ ƒ
œ$
%
œ * #
"' '
œ * "
"' %
œ * %
"' "'
œ&
"'
# #
• •
•
$ # $
% ' %
$
%
Œy
y
y
y
"
#
"
#
LCD is 16
65. Œ Œ Œ Œ( " # $ ( # # $
) % $ % ) ) $ % œ
œ & # *
) $
y
y
y
œ & $
) )
œ œ# "
) %
# #
"
" )
$
•"'y
100 Chapter 3 Adding and Subtracting Fractions
67. Œ Œ ŒŒ Œ ŒŒ
$ # & " "
% $ * % )
œ $ ' & " "
% * * % )
œ$
%
œ$y y
% yy
œ " "
"' $#
œ œ# " "
$# $# $#
#
#
#
• •
• • •
" " "
* % )
$ " " "
% * % )
" "
$"
69.
and
, so .
% &
#& $!% #% & #&
#& "&! $! "&!œ œ
#% #& % &
"&! "&! #& $!
**
**LCD is 150
in Atlanta is greater. &$!
71. When comparing the size of two numbers, thesymbol means and the symbol is less than means .is greater than
72. (a) To identify the greater of two or morefractions, we must first write the fractions as likefractions and then compare the numerators.The fraction with the greater isnumeratorthe greater fraction.
Answers will vary. A sample answer follows:(b)" " " " $ $ & &% # "! "' ) # ' ( ß ß ß
73. 1. Do all operations inside parentheses or othergrouping symbols.
2. Simplify any expressions with andexponentsfind any roots.square
3. or proceeding from left to right.Multiply divide
4. or proceeding from left to right.Add subtract
74.
Œ ŒŒ ŒŒ
# % $ &
$ & "! % ƒ
œ ƒ# ) $ &
$ "! "! %
œ ƒ# & &
$ "! %
œ % &
* "!
œ % &
*
y y
y
œ % #
* &
œ #! ")
%& %&
œ#
%&
#
#
#
•
•
%
&
%
&
"
& "
#
"!y
LCD is 45
For Exercises 75–80, see the number line followingExercise 80.
75. Œ# #
$ $œ
#
•# %
$ *œ
76. Œ$ $
# #œ
#
•$ * "
# % %œ œ #
77. Œ$ $
& &œ
$
• •$ $ #(
& & "#&œ
Use to help place the answer on the#& ""#& &œ
number line.
78. Œ& &
% %œ
#
•& #& *
% "' "'œ œ "
Use to help place the answer on the" ") ""' #œ
number line.
79. % # # œ % # %
œ ' %
œ #
#
Summary Exercises on Fractions 101
80. Œ Œ Œ ŒŒ
& ( " " & ( # "
) ) % % ) ) ) % ƒ œ ƒ
œ ƒ& & "
) ) %
œ #& &
'% )
œ #& &
'% )y
y
œ #& &
'% #
œ # #& "
'% #
œ # #& $#
'% '%
œ #&(
'%
# #
#
•
•
%
"
%
"#
"
Use to help place the answer on the# œ #&' ('% )
number line.
Summary Exercises on Fractions1. is a proper $
% fraction since the numerator is lessthan the denominator.
3. is an improper "!"! fraction since the numerator isgreater than or equal to the denominator.
5. $! $! ƒ ' &
$' $' ƒ ' 'œ œ
7. "& "& ƒ & $
$& $& ƒ & (œ œ
9. $
%
y y
y y•
• •
• •
# $ # " " "
$ # " #œ œ œ
% $
" "
# "
11. &'y
• ••
•
& & ( & $&
) " ) " " "œ œ œ œ $&
&'y(
"
13. $& "! $&
%& "& %&ƒ œ
œ
œ œ œ "( ( "
$ ' '
•
•
•
•
"&
"!
"
#
$& "&y y
y y
(
$
"
#%& "!
15. ( # ( # #" "'
) $ ) $ #% #% œ œ
œ œ œ "#" "' $( "$
#% #% #%
• •
• •
$ )
$ )
17. ( & # ( "! ) ( "! )
"# ' $ "# "# "# "# œ œ
œ œ ##& "
"# "#
19. ( & #" "! #" "! ""
) "# #% #% #% #% œ œ œ
21. $"# • #"
%
Estimate: % • # œ )
Exact: $"
#• •# œ œ œ (
" ( * '$ (
% # % ) )
23. ) • •& ## $$ )
Estimate: ) • • •' # œ %) # œ *'
=Exact: )y
y• • • •& # œ œ "!(
# $ ) "( "* $#$ #
$ ) " $ ) $ $
"
"
25. ' ƒ #()
Estimate: ( ƒ # œ $"#
Exact: ' ƒ # œ ƒ( && #
) ) "
œ&&
)
œ œ $&& (
"' "'
•"
#
27. Estimate: Exact:
' & œ &
%
"!
# )
$ "#
% œ %" $
% "#
*""
"#
29. Estimate: Exact:
"&
""
#'
"% œ "%$ *
& "&
"! œ "!# "!
$ "&
#% œ #% " œ #&"* % %
"& "& "&
31. Estimate: Exact:
"% "% œ "$
(
(
)
)
( œ ($ $
) )
'&
)
102 Chapter 3 Adding and Subtracting Fractions
33. " # " " ) $
& $ % & "# "# œ
œ"
&
y
y
ΠΥ
•
•
& " & "
"# & "# "#œ œ
"
"
35. # # & # % &
$ $ ' $ * ' œ
œ "# ) "&
") ") ")
œ #! "&
") ")
œ&
")
Π#
37. *ß ")ß #%
* œ $
") œ #
#% œ #
•• •• • •
$
$ $
# # $
LCM œ # • • • •# # $ $ œ (#
The LCM of , , and is .* ") #% (#
39. ?&
' %#œ %# ƒ ' œ (
& & $&
' ' %#œ œ
•
•
(
(
41. ?""
"# '!œ '! ƒ "# œ &
"" "" &&
"# "# '!œ œ
•
•
&
&
43.
and
, so .
"' #$
#! $!"' %) #$ %'
#! '! $! '!œ œ
%) %' "' #$
'! '! #! $!
**
**LCD is 60
Chapter 3 Review Exercises
1. & " & " '
( ( ( ( œ œ
2. % $ % $ (
* * * * œ œ
3. " $ # " $ # ' $
) ) ) ) ) % œ œ œ
4. & $ & $ # "
"' "' "' "' ) œ œ œ
5. & $ & $ ) %
"! "! "! "! & œ œ œ
6. & $ & $ # "
"# "# "# "# ' œ œ œ
7. $' "! $' "! #' "$
'# '# '# '# $" œ œ œ
8. ') %$ ') %$ #& "
(& (& (& (& $ œ œ œ
9. Add to find what fraction of his total incomecomes from the two activities.
( % ( % ""
"# "# "# "# œ œ
of his total income comes from the two"""#
activities.
10. Subtract to find the answer.& $ & $ # "
) ) ) ) % œ œ œ
They completed Web page less in the afternoon."%
11. &ß #
Multiples of :&
&ß "!ß "&ß #!ß #&ß $!ßá
is the first number divisible by . ( )"! # "! ƒ # œ &The least common multiple of and is .& # "!
12. $ß %
Multiples of :%
%ß )ß "#ß "'ß #!ß #%ßá
is the first number divisible by . ( )"# $ "# ƒ $ œ %The least common multiple of and is .$ % "#
13. # "! "# #!# & ' "!y
$ & $ &y y
& & " &y
" " "
LCM œ # • • •# $ & œ '!
14. # $ ) %y
# $ % #y
# $ # "y y
$ $ " "y y
" " "
LCM œ # • • •# # $ œ #%
15. 'ß )ß &ß "&
' œ #
) œ #
& œ "
"& œ $
•• •••
$
# #
&
&
LCM œ # • • • •# # $ & œ "#!
The LCM of , , , and is .' ) & "& "#!
Chapter 3 Review Exercises 103
16. # "& * #!yy
# "& * "!yy
$ "& * &y
$ & $ &y y
& & " &y
" " "
LCM œ # • • • •# $ $ & œ ")!
17. ?#
$ "#œ "# ƒ $ œ %
# # )
$ $ "#œ œ
•
•
%
%
18. ?$
) &'œ &' ƒ ) œ (
$ $ #"
) ) &'œ œ
•
•
(
(
19. ?#
& #&œ #& ƒ & œ &
# # "!
& & #&œ œ
•
•
&
&
20. ?&
* )"œ )" ƒ * œ *
& & %&
* * )"œ œ
•
•
*
*
21. ?%
& %!œ %! ƒ & œ )
% % $#
& & %!œ œ
•
•
)
)
22. ?&
"' '%œ '% ƒ "' œ %
& & #!
"' "' '%œ œ
•
•
%
%
23. " " $ #
# $ ' ' œ
œ$ #
'
œ&
'
24. " $ $ ) "# "&
& "! ) %! %! %! œ
œ) "# "&
%!
œ œ$& (
%! )
25. & "!
"# #%œ
œ& &
#% #%"& &
#% )œ
26. # " ) $ &
$ % "# "# "# œ œ
27. ( #"
) #%œ
œ" )
$ #%"$
#%
28. "" $$
"# $'œ
œ% "'
* $'"(
$'
29. Add the fractions.# "#
& $!œ
" &
' $!œ
œ" "!
$ $!#( *
$! "!œ
of the total students participated in these*"!
activities.
30. Add the fractions." )
$ #%œ
$ *
) #%œ
œ" '
% #%#$
#%
of her business comes from these three#$#%
categories.
31. Estimate: Exact:
"* ") œ ")
"%
$$
& &
) )
"$ œ "$$ '
% )
$" œ $" " œ $#"" $ $
) ) )
32. Estimate: Exact:
#$ ## œ ##
"&
$)
# '
$ *
"& œ "&% %
* *
$( œ $( " œ $)"! " "
* * *
104 Chapter 3 Adding and Subtracting Fractions
33. Estimate: Exact:
"$
*
"!
$#
"# œ "#$ %)
& )!
) œ )& &!
) )!
"! œ "!& #&
"' )!
$! œ $! " œ $""#$ %$ %$
)! )! )!
34. Estimate: Exact:
$# $" œ $"
"&
"(
$ *
% "#
"% œ "%# )
$ "#
"("
"#
35. Estimate: Exact:
$% $% œ $$
"'
")
$
$
"& œ "&# #
$ $
")"
$
36. Estimate: Exact:
#"& #"&
"$'
(*
(
"' "$'
(*(
"'
37. & œ œ# #( &%
& & "!
$ œ œ( $( $(
"! "! "!*" "
"! "!œ *
38. % œ œ$ "* &(
% % "#
& œ œ# "( ')
$ $ "#"#& &
"# "#œ "!
39. & œ œ& #!
" %
" œ œ$ ( (
% % %"$ "
% %œ $
40. ' œ œ" "$ $*
# # '
% œ œ& #* #*
' ' '"! % #
' ' $œ " œ "
41. ) œ œ" #& &!
$ $ '
# œ œ& "( "(
' ' '$$ $ "
' ' #œ & œ &
42. & œ œ& '& "$!
"# "# #%
# œ œ& #" '$
) ) #%'( "*
#% #%œ #
43. Subtract the uphill and downhill distances fromthe total distance.
milesEstimate: "* ' ( œ '
Exact: Add the uphill and downhill distances.
& œ && "&
) #%
( œ (" )
$ #%
"##$
#%
Now subtract from .")$%
") œ ") œ "($ ") %#
% #% #%
"# œ "# œ "##$ #$ #$
#% #% #%
&"*
#%
The level portion of the course was miles.&"*#%
44. Add to find the total weight.
tonsEstimate: #* #& œ &%
Exact: #) œ #)# )
$ "#
#% œ #%$ *
% "#
&# œ &# " œ &$"( & &
"# "# "#
The total weight of the newspapers was tons.&$ &"#
Chapter 3 Review Exercises 105
45. Add to find the total weight.
poundsEstimate: * "! ( œ #'
Exact: ) œ )( #"
) #%
* œ *" "#
# #%
' œ '$ ")
% #%
#$ œ #$ # œ #&&" $ "
#% #% )
The total weight of the three largemouth bass was#&"
) pounds.
46. Subtract the two parcels from the total amount sheneeds.
acresEstimate: * # $ œ %
the two parcels.Exact: Add
" œ """ ""
"' "'
# œ #$ "#
% "'
$ œ $ " œ %#$ ( (
"' "' "'
Now subtract from .) 12
) œ )" )
# "'
% œ %( (
"' "'
%"
"'
She needs to buy an additional acres.% ""'
47.–50.
51.
and
, so .
# $
$ %# ) $ *
$ "# % "#œ œ
) * # $
"# "# $ %
**
**LCD is 12
52.
, so .
$ (
% )$ '
% )œ
' ( $ (
) ) % )
**
**LCD is 8
53.
and
, so .
" (
# "&" "& ( "%
# $! "& $!œ œ
"& "% " (
$! $! # "&
**
**LCD is 30
54.
and
, so .
( )
"! "&( #" ) "'
"! $! "& $!œ œ
#" "' ( )
$! $! "! "&
**
**LCD is 30
55.
, so .
* &
"' )& "!
) "'œ
* "! * &
"' "' "' )
**
**LCD is 16
56.
and
, so .
( )
#! #&( $& ) $#
#! "!! #& "!!œ œ
$& $# ( )
"!! "!! #! #&
**
**LCD is 100
57.
and
, so .
"* #*
$' &%"* &( #* &)
$' "!) &% "!)œ œ
&( &) "* #*
"!) "!) $' &%
**
**LCD is 108
58.
and
, so .
"* (
"$# &&"* *& ( )%
"$# ''! && ''!œ œ
*& )% "* (
''! ''! "$# &&
**
**LCD is 660
59. Œ" "
# #œ
#
•" "
# %œ
60. Œ# #
$ $œ
#
•# %
$ *œ
61. Π$ $ #(
"! "! "!!!œ
$
• •$ $
"! "!œ
62. Œ$ $ $ $ $ )"
) ) ) ) ) %!*'œ œ
%
• • •
106 Chapter 3 Adding and Subtracting Fractions
63. ) œ" )
% "
œ)yy
" y y
œ"
#
Π#
• •
• •
• •
" "
% %
" "
% %
#"
" #
64. "# œ$ "#
% "
œ" y
œ œ '#( $
% %
Π#
• •
• •
• •
$ $
% %
$ $
% %
"#y$
"
65. Œ#
$
œ#y y y y
$y y y y
œ"
"'
#
• • • •
• • •
• • •
Œ$ # # $ $
) $ $ ) )œ
# $ $
$ ) )
#
" " " "
" " % %
66. ( " $ ( " '
) ) % ) ) )ƒ œ ƒ
œ ƒ( (
) )
œ(y
)y
y
y
œ "
Œ Œ
"
"
"
"
•)
(
67. Œ"
#
œ"
#
œ$
"'
#
• • •
• •
Œ Œ" " " " " #
% # # # % % œ
" $
# %
68.
Œ Œ" & $ "
% ) % % œ
œ " ""
'% )
œ " ))
'% '%
œ œ ")* #&
'% '%
$
• •" " & '
% % ) ) Œ
LCD is 64
69. [3.1] ( " ( " ' $
) ) ) ) % œ œ œ
70. [3.1] ( $ ( $ % #
"! "! "! "! & œ œ œ
71. [3.3] #* & #* "! #* "! "*
$# "' $# $# $# $# œ œ œ
72. [3.3]
" " & % # & % # & ""
% ) "' "' "' "' "' "' œ œ œ
73. [3.4] ' œ '# %
$ '
% œ %" $
# '
#"
'
74. [3.4] * œ *" #
# %
"' œ "'$ $
% %
#& œ #& " œ #'& " "
% % %
75. [3.4] ( œ ')
)
" œ "& &
) )
&$
)
76. [3.4] # œ #$ %)
& )!
) œ )& &!
) )!
œ& #&
"' )!
"! œ "! " œ """#$ %$ %$
)! )! )!
77. [3.4] $#&
"# "(
"&&
"#
78. [3.3] ( $ $ ( $ '
## ## "" ## ## ## œ
œ( $ '
##
œ œ"' )
## ""
79. [3.5] Œ"
% y yy
y y y
œ"
#&!
#
• • • • •Œ# " " # # #
& & & &œ
% %
$
#"
#
" " "
Chapter 3 Review Exercises 107
80. [3.5] $ " " $ # "
) # % ) % %ƒ œ ƒ
œ ƒ$ $
) %
œ$y
)y y
y
œ"
#
Œ Œ
"
# "
"
•%
$
81. [3.5] Œ#
$
œ œ#y
$ y
œ#
*
#
• •
• •
Œ Œ Œ" " # # "
$ ' $ ' ' œ
# "
$ #
#
"
"
3 16 2
82. [3.5] Œ Œ Œ Œ# # & # ' &
$ $ * $ * * œ
œ # # # "
$ $ $ *
œ ) "
#( *
œ œ) $ ""
#( #( #(
$ $
• •
83. [3.5]
, so .
# (
$ "## )
$ "#œ
) ( # (
"# "# $ "#
**
**LCD is 12
84. [3.5]
and
, so .
) "&
* )) '% "& "$&
* (# ) (#œ œ
'% "$& ) "&
(# (# * )
**
**LCD is 72
85. [3.5]
, so .
"( $'
$! '!"( $%
$! '!œ
$% $' "( $'
'! '! $! '!
**
**LCD is 60
86. [3.5]
and
, so .
& "(
) $!& (& "( ')
) "#! $! "#!œ œ
(& ') & "(
"#! "#! ) $!
**
**LCD is 120
87. [3.2] "#ß ")
# "# ")# ' *y
$ $ *$ " $y
" "
LCM œ # • • •# $ $ œ $'
88. [3.2] , , , ' ) "! "#
' œ #
) œ #
"! œ #
"# œ #
•• ••• •
$
# #
&
# $
LCM œ # • • • •# # $ & œ "#!
89. [3.2] , , * "% #"
# * "% #"y y
$ * ( #"y
$ $ ( (y y
( " ( (y
" " "
LCM œ # • • •$ $ ( œ "#'
90. [3.2] ?#
$ #(œ #( ƒ $ œ *
# # ")
$ $ #(œ œ
•
•
*
*
91. [3.2] ?*
"# "%%œ "%% ƒ "# œ "#
* * "!)
"# "# "%%œ œ
•
•
"#
"#
92. [3.2] ?%
& (&œ (& ƒ & œ "&
% % '!
& & (&œ œ
•
•
"&
"&
93. [3.4] S ubtract the amounts of wire mesh from thetotal length of the roll.
feetEstimate: *$ "% ## œ &(
Exact: Add the amounts of wire mesh.
"$ œ "$" %
# )
## œ ##$ $
) )
$&(
)
Now subtract from .*#$%
continued
108 Chapter 3 Adding and Subtracting Fractions
*# œ *# œ *"$ ' "%
% ) )
$& œ $& œ $&( ( (
) ) )
&'(
)
After completing the two jobs, feet of wire&'()
mesh remain.
94. [3.4] Multiply to determine the total amount ofsugar available. Then subtract the amounts ofsugar used.
Estimate: % • &! œ #!! pounds of sugar#!! '* (( $$ œ #" pounds of sugar
Exact: Add the amounts of sugar used.
') œ ')" %
# )
(' œ ('& &
) )
$$ œ $$" #
% )
"(( œ "(( " œ "()"" $ $
) ) )
Now subtract from .#!!
#!! œ "**)
)
"() œ "()$ $
) )
#"&
)
remain.#"&) pounds of sugar
Chapter 3 Test
1. & " & " ' $
) ) ) ) % œ œ œ
2. " ( " ( ) "
"' "' "' "' # œ œ œ
3. ( $ ( $ % #
"! "! "! "! & œ œ œ
4. ( & ( & # "
"# "# "# "# ' œ œ œ
5. #ß $ß %
# # $ %y
# " $ #y y
$ " $ "y y
" " "
LCM œ # • •# $ œ "#
The LCM of , , and is .# $ % "#
6. 'ß $ß &ß "&
' œ #
$ œ "
& œ "
"& œ $
••••
$
$
&
&
LCM œ # • •$ & œ $!
The LCM of , , , and is .' $ & "& $!
7. 'ß *ß #(ß $'
' œ #
* œ $
#( œ $
$' œ #
••• •• • •
$
$
$ $
# $ $
LCM œ # • • • •# $ $ $ œ "!)
The LCM of , , , and is .' * #( $' "!)
8. $ " $ #
) % ) ) œ
œ œ$ # &
) )
LCD is 8
9. # & ) "&
* "# $' $' œ
œ œ) "& #$
$' $'
LCD is 36
10. ( # #" "'
) $ #% #% œ
œ œ#" "' &
#% #%
LCD is 24
11. # $ "' "&
& ) %! %! œ
œ œ"' "& "
%! %!
LCD is 40
12. ( %# &
$ '
Estimate: ) & œ "$
Exact: ( % œ ( %# & % &
$ ' ' '
œ "" œ "" " œ "#* $ "
' ' #
13. "' ""# #
& $
Estimate: "' "# œ %
Exact: "' œ "' œ "&# ' #"
& "& "&
"" œ "" œ ""# "! "!
$ "& "&
%""
"&
Chapter 3 Test 109
14. ") * "#$ # "
% & $
Estimate: "* * "# œ %!
Exact: ") œ ")$ %&
% '!
* œ *# #%
& '!
"# œ "#" #!
$ '!
$* œ $* " œ %!)* #* #*
'! '! '!
15. #% ")$
)
Estimate: #% ") œ '
Exact: #% œ #$)
)
") œ ")$ $
) )
&&
)
16. Probably addition and subtraction of fractions ismore difficult because you have to find the leastcommon denominator and then change thefractions to the same denominator.
17. Round mixed numbers to the nearest wholenumber. Then add or subtract to estimate theanswer. The estimate may vary from the exactanswer but it lets you know if your answer isreasonable.
18. Add the number of pounds.
poundsEstimate: "! )& $( ) œ "%!
Exact:
"! œ "!$ *
) #%
)% œ )%" "#
# #%
$' œ $'& #!
' #%
) œ )" )
$ #%
"$) œ "$) # œ "%!%* " "
#% #% #%
The total number of pounds used was ."%! "#%
19. Subtract the number of gallons that were usedfrom the amount the contractor had when hearrived to find the number of gallons remaining.
gallonsEstimate: "%) '* $( ' œ $'
the number of gallons that were used.Exact: Add
') œ ')" %
# )
$( œ $($ $
) )
& œ &$ '
% )
""! œ ""! " œ """"$ & &
) ) )
Now subtract from ."%("#
"%( œ "%( œ "%'" % "#
# ) )
""" œ """ œ """& & &
) ) )
$&(
)
The number of gallons remaining is .$&()
20.
, so .
$ "(
% #%$ ")
% #%œ
") "( $ "(
#% #% % #%
**
**LCD is 24
21.
and
, so .
"* "(
#% $'"* &( "( $%
#% (# $' (#œ œ
&( $% "* "(
(# (# #% $'
**
**LCD is 72
22. Œ"
$
œ"
#(
œ œ ##
"
$
• • • •
•
&% œ &%" " "
$ $ $
&%
"
Œ
y
y
"
#
23. Œ Œ$ (
% )
œ *
"'
œ œ#( "% "$
%) %) %)
#
• •" $ $ (
$ % % #%œ
(
#%LCD is 48
24. % œ % ( ( "% (
) "' "' "'
œ %
œ%
"
œ œ "( $
% %
ΠΥ
•
(
"'
(yy
"'
"
%y
110 Chapter 3 Adding and Subtracting Fractions
25. & % $ & % $
' $ ) ' $ ) œ
œ & "
' #
œ & $
' '
œ& $
'
œ œ " œ ") # "
' ' $
Œ Œyy y
y"
" #
"
LCD is 6
Cumulative Review Exercises(Chapters 1–3)
1. , ,& ' * % )$ #
is in the & millions place.$ is in the ten-thousands place.* is in the thousands place.# is in the hundreds place.
2. To the nearest ten: ,&* )!$
Next digit is or less. Tens place does not change.%All digits to the right of the underlined place arechanged to zero. &* )!!,
,To the nearest hundred: &* )!$
Next digit is or less. Hundreds place does not%change. All digits to the right of the underlinedplace are changed to zero. &* )!!,
,To the nearest thousand: &* )!$
Next digit is or more. All digits to the right of&the underlined place are changed to zero. Add to"* ! ". Write and carry . '! !!!,
3. Estimate: Exact:
, ,,
#! !!!
"! !!!
"! !!!
# % # ( '$ " '
* ) ) (
" % $ ) *
/ // ,/ /
//
/ ,
" "'"$ """'
4. Estimate: Exact:
# & ! ! $ # " "
%! " ! ! ! ! ! $& " " # $ ) &" ! &
( $( !
$ )$ &
$ &$ &
!
, ,
5. ,,,
, ,
*" " # # ""$(& )**
&#" (%#
$&( *')
" #&& '!*
6. , ,/ / //
/, ,, ,
$ )*' & ! #
" !*% ) ! (
# )!" ' * &
& %"%
*"#
7. &Ð)ÑÐ%Ñ œ %!Ð%Ñ œ "'!
8. *'#
‚ $)%
$ )%)
(' *'
#)) '
$'* %!),
à %‚ *'#
à ) ‚ *'#
à $ ‚ *'#
9. " $ &
) " ! ) !)
# )# %
% !% !
!
10. ,"$ %'( ƒ &
,
# ' * $ #
& "$ % ' (
R$ % "
11. To find the perimeter, add the six measurements ofthe four sides.
ftEstimate: #! * & #! * & œ ')
Exact: ") * & ") * & œ '% ft
The perimeter of this parking space is feet.'%
12. To find the area of the parking space, multiply thelength ( ) and the width ( ).") "%
Estimate: #! • "! œ #!! ft#
Exact: ")
‚ "%
(#
")
#&#
The area of the parking space is ft .#&# #
Cumulative Review Exercises (Chapters 1–3) 111
13. Multiply the length and the width to find the area.
Estimate: # • $ œ ' yd#
Exact: "$
% y
y• •# œ œ œ %
# ( ) "% #
$ $ $ $%"
#
The area of the pool table is yd .%#$
#
14. Multiply the height of the person by ."$!
Estimate: (! • "$! œ *"!! in.
Exact:
(!"
%• •"$! œ œ œ *"$#
#)" "$! $' &$! "
% " % #
,
A human could jump *"$#"# inches high.
15. # "%%% • • • • • • •$ œ # # # # $ $ œ "' * œ#
16. & #Ð)Ñ œ #& #Ð)Ñ
œ #& "'
œ *
#
17. È#& & • •* ' œ & & * '
œ & %& '
œ &! ' œ %%
18. # % # # "# "!
$ & $ $ "& "& œ
œ#
$
œ%
%&
ΠΥ#
"&
19. $ " " $ # $
% $ # % ' 'ƒ œ ƒ
œ ƒ$ &
% '
œ$
%y
y
Œ Œ
#
$
•' *
& "!œ
20. ( $ $ ( $ $
) % ) ) % ) œ
œ ( * $
) "' )
œ "% * '
"' "' "'
œ "% * '
"' "'
œ #$ '
"' "'
œ#$ '
"'
œ œ ""( "
"' "'
Œ Œ#
•$
%
21. $ "
% #
y
y
y
y• •
•
•
# $ # " "
$ $ # "œ œ œ
%
"
#
"
"
22. %# $'y
$
%• •
•
•
( ( #" ( "%(
) " ) " % %œ œ œ œ
%#y#"
%
23. #& "! $
%! $& "'ƒ œ #
y#&y
y y
&"
) #%! "!
•$& $&
œ œ"'
24. * ƒ œ "$# * "
$ " #•$ #(
# #œ œ
25. Estimate: Exact:
$ $ œ $
&
)
$ $
) )
% œ %" %
# )
((
)
26. Estimate: Exact:
##
%
#'
#" œ #"( #"
) #%
% œ %& "!
"# #%
#& œ #& " œ #'$" ( (
#% #% #%
27. Estimate: Exact:
& & œ %
#
$
)
)
# œ #$ $
) )
#&
)
28–31.
32.
and
, so .
$ &
& )$ #% & #&
& %! ) %!œ œ
#% #& $ &
%! %! & )
**
**LCD is 40
33.
, so .
"( $
#! %$ "&
% #!œ
"( "& "( $
#! #! #! %
**
**LCD is 20
112 Chapter 3 Adding and Subtracting Fractions
34.
and
, so .
( ""
"# ")( #" "" ##
"# $' ") $'œ œ
#" ## ( ""
$' $' "# ")
**
**LCD is 36
