1.1 Fractions 1
CHAPTER 1 THE REAL NUMBER
SYSTEM
1.1 Fractions
1.1 Classroom Examples, Now Try Exercises
1. *!
á à (b)
# %& Change both
á à # # "( #! mixed numbers
$ "& $ •' œ •
& $ & $ to improper
á à fractions.
$ &
"( • #! Multiply numerators.
œ
Writing *! as the product of primes gives us &•$ Multiply denominators.
"( • & • %
*! œ # • $ • $ • &. œ Factor.
&•$
N1Þ '! ') #
á à œ , or ## Write as a mixed number.
$ $
# $!
á à Multiply by the
* $ * &
$ "! 4. (a) ƒ œ • reciprocal of the
"! & "! $
á à second fraction.
# & $•$•&
œ
Writing '! as the product of primes gives us #•&•$
$ "
œ , or "
'! œ # • # • $ • &. # #
"# $•% $•" $ Change both
2. œ œ œ
#! &•% &•" & $ " "" "! mixed numbers
(b) # ƒ$ œ ƒ
$! &•' &•" & % $ % $ to improper
N2Þ œ œ œ fractions.
%# (•' (•" (
( "# ( • "# Multiply numerators. Multiply by the
3. (a) • œ "" $
* "% * • "% Multiply denominators. œ • reciprocal of the
% "!
(•#•#•$ second fraction.
œ Factor. $$
$•$•#•( œ
# %!
œ Write in lowest terms.
$ Multiply by the
# ) # *
Change both N4. (a) ƒ œ • reciprocal of the
( * ( )
" $ "! ( mixed numbers second fraction.
(b) $ • " œ • #•$•$
$ % $ % to improper œ
fractions. (•#•%
*
"! • ( Multiply numerators. œ
œ #)
$•% Multiply denominators.
#•&•( Change both
œ Factor. $ # "& $! mixed numbers
$•#•# (b) $ ƒ% œ ƒ
$& & % ( % ( to improper
œ , or & Write as a mixed number.
' ' fractions.
% & %•& Multiply numerators. Multiply by the
N3. (a) • œ "& (
( ) (•) Multiply denominators. œ • reciprocal of the
% $!
%•& second fraction.
œ Factor. "& • (
(•#•% œ
& % • # • "&
œ Write in lowest terms. (
"% œ
)
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.
,2 Chapter 1 The Real Number System
Add numerators; (from )) and one factor of $ (from "#), so it is
" & "&
5. œ denominator # • # • # • $ œ #%.
* * *
does not change. Write each fraction with a denominator of #%.
'
œ & &•# "! $ $•$ *
* œ œ and œ œ
#•$ "# "# • # #% ) )•$ #%
œ Factor.
$•$ Now add.
#
œ & $ "! * "! * "*
$ œ œ œ
"# ) #% #% #% #%
Add numerators;
" $ "$ (b)
N5. œ denominator
) ) ) Change both mixed
does not change. " & "$ %&
$ & œ numbers to improper
% % ) % )
œ fractions.
)
"•% The least common denominator is ), so write each
œ Factor.
#•% fraction with a denominator of ).
"
œ %& "$ "$ • # #'
# and œ œ
) % %•# )
( #
6. (a) Now add.
$! %&
"$ %& #' %& #' %&
Since $! œ # • $ • & and %& œ $ • $ • &, the least œ œ
common denominator must have one factor of # % ) ) ) )
(" (
(from $!), two factors of $ (from %&), and one œ , or )
factor of & (from either $! or %&), so it is ) )
# • $ • $ • & œ *!. $ "
7. (a)
"! %
Write each fraction with a denominator of *!.
Since "! œ # • & and % œ # • #, the least common
( (•$ #" # #•# % denominator is # • # • & œ #!. Write each fraction
œ œ and œ œ
$! $! • $ *! %& %& • # *! with a denominator of #!.
Now add. $ $•# ' " "•& &
œ œ and œ œ
( # #" % #" % #& "! "! • # #! % %•& #!
œ œ œ
$! %& *! *! *! *! Now subtract.
#&
Write *! in lowest terms. $ " ' & "
œ œ
#& &•& & "! % #! #! #!
œ œ
*! & • ") ") Change each mixed
$ " #( $
Change both mixed (b) $ " œ number into an
& " #* ( ) # ) #
(b) % # œ numbers to improper improper fraction.
' $ ' $
fractions. The least common denominator is ). Write each
The least common denominator is ', so write each fraction with a denominator of ). #(
) remains
fraction with a denominator of '. unchanged, and
#* ( (•# "% $ $•% "#
and œ œ œ œ .
' $ $•# ' # #•% )
Now add. Now subtract.
#* ( #* "% #* "% #( $ #( "# #( "# "& (
œ œ œ œ œ , or "
' $ ' ' ' ) # ) ) ) ) )
%$ " & #
œ , or ( N7. (a)
' ' "" *
& $ Since "" œ "" and * œ $ • $, the least common
N6. (a)
"# ) denominator is $ • $ • "" œ **. Write each fraction
Since "# œ # • # • $ and ) œ # • # • #, the least with a denominator of **.
common denominator must have three factors of #
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.
, 1.1 Fractions 3
& &•* %& # # • "" ## ( ( *(! '(*
œ œ and œ œ Ð*(!Ñ œ • œ
"" "" • * ** * * • "" ** #! #! " #
"
Now subtract. œ $$* million, or $$*,&!!,!!!.
#
& # %& ## #$
œ
"" *
** **
œ
**
1.1 Section Exercises
Change each mixed 1. True; the number above the fraction bar is called
" & "$ "( the numerator and the number below the fraction
(b) % # œ number into an
$ ' $ '
improper fraction. bar is called the denominator.
The least common denominator is '. Write each 2. True; & divides the $" six times with a remainder
of one, so $"
& œ '&Þ
"
fraction with a denominator of '. "(
' remains
unchanged, and 3Þ False; this is an improper fraction. Its value is ".
"$ "$ • # #' 4Þ False; the number " is neither prime nor
œ œ .
$ $•# ' composite.
Now subtract. 5Þ False; the fraction "$
$* can be written in lowest
"$ "( #' "( #' "( * " "$ "$ • " "
œ œ œ terms as since œ œ Þ
$ ' ' ' ' ' $ $* "$ • $ $
' #
Now reduce. 6Þ False; the reciprocal of # œ $ is ' œ "$ Þ
* $•$ $ " 7. False; product refers to multiplication, so the
œ œ , or "
' #•$ # # product of "! and # is #!. The sum of "! and # is
8. To find out how many gallons of paint Tran "#.
should buy, divide the total area to be painted by 8. False; difference refers to subtraction, so the
the area that one gallon of paint covers. difference between "! and # is ). The quotient of
%#!! %# # "! and # is &.
œ , or )
&!! & & 9. Since "* has only itself and " as factors, it is a
) #& gal are needed, so he must buy * gal. prime number.
N8. To find out how long each piece must be, divide 10. Since $" has only itself and " as factors, it is a
the total length by the number of pieces. prime number.
" #" % #" " #" & 11. $! œ # • "&
"! ƒ % œ ƒ œ • œ , or #
# # " # % ) ) œ #•$•&
Each piece should be # &) feet long. Since $! has factors other than itself and ", it is a
composite number.
9. (a) In the circle graph, the sector for Europe is the
second largest, so Europe had the second largest 12. &! œ # • #&
$
share of Internet users, "! . œ # • & • &,
(b) As in Example 9(b), so &! is a composite number.
$
"! "!!! œ $!! million. 13. '% œ # • $#
œ # • # • "'
(c) As in Example 9(c),
œ #•#•#•)
$ $ *(! #*"! œ #•#•#•#•%
Ð*(!Ñ œ • œ œ #*" million.
"! "! " "! œ #•#•#•#•#•#
N9. (a) In the circle graph, the sector for Other is the
Since '% has factors other than itself and ", it is a
smallest, so Other had the least number of Internet
users. composite number.
" ( 14. )" œ $ • #(
(b) As in Example 9(b) (using $ for #! ),
œ $•$•*
"
$ "!!! ¸ $$$ million. œ $•$•$•$
(c) As in Example 9(c), Since )" has factors other than itself and ", it is a
composite number.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.
, 4 Chapter 1 The Real Number System
15. As stated in the text, the number " is neither prime && & • "" ""
32. œ œ
nor composite, by agreement. #!! & • %! %!
16. The number ! is not a natural number, so it is ") " • ") "
33. œ œ
neither prime nor composite. *! & • ") &
"' " • "' "
17. &( œ $ • "*, so &( is a composite number. 34. œ œ
'% % • "' %
18. &1 œ $ • "7, so &1 is a composite number. "%% ' • #% '
35. œ œ
19. Since (* has only itself and " as factors, it is a "#! & • #% &
prime number. "$# "# • "" "#
36. œ œ
20. Since )$ has only itself and " as factors, it is a (( ( • "" (
prime number. "' #•) #
37. œ œ
#% $•) $
21. "#% œ # • '#
œ # • # • $", Therefore, C is correct.
so "#% is a composite number. "& $•& &
38. A. œ œ
#( $•* *
22Þ "$) œ # • '*
œ # • $ • #$ , $! '•& &
B. œ œ
&% '•* *
so "$) is a composite number. %! # • #! #!
C. œ œ
23. &!! œ # • #&! (% # • $( $(
œ # • # • "#& && "" • & &
œ # • # • & • #& D. œ œ
** "" • * *
œ # • # • & • & • &,
Therefore, C is correct.
so &!! is a composite number.
% ' %•' #%
24. (!! œ # • $&! 39. • œ œ
& ( &•( $&
œ # • # • "(& & # &•# "!
œ # • # • & • $& 40. • œ œ
* ( *•( '$
œ # • # • & • & • (, # "& # • "& #•$•& &
41. • œ œ œ
so (!! is a composite number. $ "' $ • "' $•#•) )
25. $%&) œ # • "(#* $ #! $ • #! $•&•% %
42. • œ œ œ
œ # • ( • #%( & #" & • #" &•$•( (
œ # • ( • "$ • "* " "# " • "# "•#•' '
43. • œ œ œ
"! & "! • & #•&•& #&
Since $%&) has factors other than itself and ", it is
" "! " • "! "•#•& &
a composite number. 44. • œ œ œ
) ( )•( #•%•( #)
26. "!#& œ & • #!& "& ) "& • )
œ & • & • %" 45. • œ
% #& % • #&
Since "!#& has factors other than itself and ", it is $•&•%•#
a composite number. œ
%•&•&
) "•) " $•#
27. œ œ œ
"' #•) # &
% "•% " ' "
28. œ œ œ , or "
"# $•% $ & &
"& $•& & #" % #" • %
29. œ œ 46. • œ
") $•' ' ) ( )•(
"' %•% % $•(•%
30. œ œ œ
#! &•% & %•#•(
'% % • "' "' $ "
31. œ œ œ , or "
"!! % • #& #& # #
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.
CHAPTER 1 THE REAL NUMBER
SYSTEM
1.1 Fractions
1.1 Classroom Examples, Now Try Exercises
1. *!
á à (b)
# %& Change both
á à # # "( #! mixed numbers
$ "& $ •' œ •
& $ & $ to improper
á à fractions.
$ &
"( • #! Multiply numerators.
œ
Writing *! as the product of primes gives us &•$ Multiply denominators.
"( • & • %
*! œ # • $ • $ • &. œ Factor.
&•$
N1Þ '! ') #
á à œ , or ## Write as a mixed number.
$ $
# $!
á à Multiply by the
* $ * &
$ "! 4. (a) ƒ œ • reciprocal of the
"! & "! $
á à second fraction.
# & $•$•&
œ
Writing '! as the product of primes gives us #•&•$
$ "
œ , or "
'! œ # • # • $ • &. # #
"# $•% $•" $ Change both
2. œ œ œ
#! &•% &•" & $ " "" "! mixed numbers
(b) # ƒ$ œ ƒ
$! &•' &•" & % $ % $ to improper
N2Þ œ œ œ fractions.
%# (•' (•" (
( "# ( • "# Multiply numerators. Multiply by the
3. (a) • œ "" $
* "% * • "% Multiply denominators. œ • reciprocal of the
% "!
(•#•#•$ second fraction.
œ Factor. $$
$•$•#•( œ
# %!
œ Write in lowest terms.
$ Multiply by the
# ) # *
Change both N4. (a) ƒ œ • reciprocal of the
( * ( )
" $ "! ( mixed numbers second fraction.
(b) $ • " œ • #•$•$
$ % $ % to improper œ
fractions. (•#•%
*
"! • ( Multiply numerators. œ
œ #)
$•% Multiply denominators.
#•&•( Change both
œ Factor. $ # "& $! mixed numbers
$•#•# (b) $ ƒ% œ ƒ
$& & % ( % ( to improper
œ , or & Write as a mixed number.
' ' fractions.
% & %•& Multiply numerators. Multiply by the
N3. (a) • œ "& (
( ) (•) Multiply denominators. œ • reciprocal of the
% $!
%•& second fraction.
œ Factor. "& • (
(•#•% œ
& % • # • "&
œ Write in lowest terms. (
"% œ
)
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.
,2 Chapter 1 The Real Number System
Add numerators; (from )) and one factor of $ (from "#), so it is
" & "&
5. œ denominator # • # • # • $ œ #%.
* * *
does not change. Write each fraction with a denominator of #%.
'
œ & &•# "! $ $•$ *
* œ œ and œ œ
#•$ "# "# • # #% ) )•$ #%
œ Factor.
$•$ Now add.
#
œ & $ "! * "! * "*
$ œ œ œ
"# ) #% #% #% #%
Add numerators;
" $ "$ (b)
N5. œ denominator
) ) ) Change both mixed
does not change. " & "$ %&
$ & œ numbers to improper
% % ) % )
œ fractions.
)
"•% The least common denominator is ), so write each
œ Factor.
#•% fraction with a denominator of ).
"
œ %& "$ "$ • # #'
# and œ œ
) % %•# )
( #
6. (a) Now add.
$! %&
"$ %& #' %& #' %&
Since $! œ # • $ • & and %& œ $ • $ • &, the least œ œ
common denominator must have one factor of # % ) ) ) )
(" (
(from $!), two factors of $ (from %&), and one œ , or )
factor of & (from either $! or %&), so it is ) )
# • $ • $ • & œ *!. $ "
7. (a)
"! %
Write each fraction with a denominator of *!.
Since "! œ # • & and % œ # • #, the least common
( (•$ #" # #•# % denominator is # • # • & œ #!. Write each fraction
œ œ and œ œ
$! $! • $ *! %& %& • # *! with a denominator of #!.
Now add. $ $•# ' " "•& &
œ œ and œ œ
( # #" % #" % #& "! "! • # #! % %•& #!
œ œ œ
$! %& *! *! *! *! Now subtract.
#&
Write *! in lowest terms. $ " ' & "
œ œ
#& &•& & "! % #! #! #!
œ œ
*! & • ") ") Change each mixed
$ " #( $
Change both mixed (b) $ " œ number into an
& " #* ( ) # ) #
(b) % # œ numbers to improper improper fraction.
' $ ' $
fractions. The least common denominator is ). Write each
The least common denominator is ', so write each fraction with a denominator of ). #(
) remains
fraction with a denominator of '. unchanged, and
#* ( (•# "% $ $•% "#
and œ œ œ œ .
' $ $•# ' # #•% )
Now add. Now subtract.
#* ( #* "% #* "% #( $ #( "# #( "# "& (
œ œ œ œ œ , or "
' $ ' ' ' ) # ) ) ) ) )
%$ " & #
œ , or ( N7. (a)
' ' "" *
& $ Since "" œ "" and * œ $ • $, the least common
N6. (a)
"# ) denominator is $ • $ • "" œ **. Write each fraction
Since "# œ # • # • $ and ) œ # • # • #, the least with a denominator of **.
common denominator must have three factors of #
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.
, 1.1 Fractions 3
& &•* %& # # • "" ## ( ( *(! '(*
œ œ and œ œ Ð*(!Ñ œ • œ
"" "" • * ** * * • "" ** #! #! " #
"
Now subtract. œ $$* million, or $$*,&!!,!!!.
#
& # %& ## #$
œ
"" *
** **
œ
**
1.1 Section Exercises
Change each mixed 1. True; the number above the fraction bar is called
" & "$ "( the numerator and the number below the fraction
(b) % # œ number into an
$ ' $ '
improper fraction. bar is called the denominator.
The least common denominator is '. Write each 2. True; & divides the $" six times with a remainder
of one, so $"
& œ '&Þ
"
fraction with a denominator of '. "(
' remains
unchanged, and 3Þ False; this is an improper fraction. Its value is ".
"$ "$ • # #' 4Þ False; the number " is neither prime nor
œ œ .
$ $•# ' composite.
Now subtract. 5Þ False; the fraction "$
$* can be written in lowest
"$ "( #' "( #' "( * " "$ "$ • " "
œ œ œ terms as since œ œ Þ
$ ' ' ' ' ' $ $* "$ • $ $
' #
Now reduce. 6Þ False; the reciprocal of # œ $ is ' œ "$ Þ
* $•$ $ " 7. False; product refers to multiplication, so the
œ œ , or "
' #•$ # # product of "! and # is #!. The sum of "! and # is
8. To find out how many gallons of paint Tran "#.
should buy, divide the total area to be painted by 8. False; difference refers to subtraction, so the
the area that one gallon of paint covers. difference between "! and # is ). The quotient of
%#!! %# # "! and # is &.
œ , or )
&!! & & 9. Since "* has only itself and " as factors, it is a
) #& gal are needed, so he must buy * gal. prime number.
N8. To find out how long each piece must be, divide 10. Since $" has only itself and " as factors, it is a
the total length by the number of pieces. prime number.
" #" % #" " #" & 11. $! œ # • "&
"! ƒ % œ ƒ œ • œ , or #
# # " # % ) ) œ #•$•&
Each piece should be # &) feet long. Since $! has factors other than itself and ", it is a
composite number.
9. (a) In the circle graph, the sector for Europe is the
second largest, so Europe had the second largest 12. &! œ # • #&
$
share of Internet users, "! . œ # • & • &,
(b) As in Example 9(b), so &! is a composite number.
$
"! "!!! œ $!! million. 13. '% œ # • $#
œ # • # • "'
(c) As in Example 9(c),
œ #•#•#•)
$ $ *(! #*"! œ #•#•#•#•%
Ð*(!Ñ œ • œ œ #*" million.
"! "! " "! œ #•#•#•#•#•#
N9. (a) In the circle graph, the sector for Other is the
Since '% has factors other than itself and ", it is a
smallest, so Other had the least number of Internet
users. composite number.
" ( 14. )" œ $ • #(
(b) As in Example 9(b) (using $ for #! ),
œ $•$•*
"
$ "!!! ¸ $$$ million. œ $•$•$•$
(c) As in Example 9(c), Since )" has factors other than itself and ", it is a
composite number.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.
, 4 Chapter 1 The Real Number System
15. As stated in the text, the number " is neither prime && & • "" ""
32. œ œ
nor composite, by agreement. #!! & • %! %!
16. The number ! is not a natural number, so it is ") " • ") "
33. œ œ
neither prime nor composite. *! & • ") &
"' " • "' "
17. &( œ $ • "*, so &( is a composite number. 34. œ œ
'% % • "' %
18. &1 œ $ • "7, so &1 is a composite number. "%% ' • #% '
35. œ œ
19. Since (* has only itself and " as factors, it is a "#! & • #% &
prime number. "$# "# • "" "#
36. œ œ
20. Since )$ has only itself and " as factors, it is a (( ( • "" (
prime number. "' #•) #
37. œ œ
#% $•) $
21. "#% œ # • '#
œ # • # • $", Therefore, C is correct.
so "#% is a composite number. "& $•& &
38. A. œ œ
#( $•* *
22Þ "$) œ # • '*
œ # • $ • #$ , $! '•& &
B. œ œ
&% '•* *
so "$) is a composite number. %! # • #! #!
C. œ œ
23. &!! œ # • #&! (% # • $( $(
œ # • # • "#& && "" • & &
œ # • # • & • #& D. œ œ
** "" • * *
œ # • # • & • & • &,
Therefore, C is correct.
so &!! is a composite number.
% ' %•' #%
24. (!! œ # • $&! 39. • œ œ
& ( &•( $&
œ # • # • "(& & # &•# "!
œ # • # • & • $& 40. • œ œ
* ( *•( '$
œ # • # • & • & • (, # "& # • "& #•$•& &
41. • œ œ œ
so (!! is a composite number. $ "' $ • "' $•#•) )
25. $%&) œ # • "(#* $ #! $ • #! $•&•% %
42. • œ œ œ
œ # • ( • #%( & #" & • #" &•$•( (
œ # • ( • "$ • "* " "# " • "# "•#•' '
43. • œ œ œ
"! & "! • & #•&•& #&
Since $%&) has factors other than itself and ", it is
" "! " • "! "•#•& &
a composite number. 44. • œ œ œ
) ( )•( #•%•( #)
26. "!#& œ & • #!& "& ) "& • )
œ & • & • %" 45. • œ
% #& % • #&
Since "!#& has factors other than itself and ", it is $•&•%•#
a composite number. œ
%•&•&
) "•) " $•#
27. œ œ œ
"' #•) # &
% "•% " ' "
28. œ œ œ , or "
"# $•% $ & &
"& $•& & #" % #" • %
29. œ œ 46. • œ
") $•' ' ) ( )•(
"' %•% % $•(•%
30. œ œ œ
#! &•% & %•#•(
'% % • "' "' $ "
31. œ œ œ , or "
"!! % • #& #& # #
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.
